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The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this paper is to prove a well-known conjecture on the regularity index of \(n+3\) almost equimultiple fat points in \(\mathbb{P}^n\).
Let \( R = k[x_0 ,\dots, x_n]\) be the standard graded polynomial ring over an algebraically closed field \(k\).
Let \( X = \{P_1, \dots , P_s\} \subset \mathbb{P}^n\) be a set of \(s\) distinct points, and let \(m_1,\ldots,m_s\) be positive integers. Denote \(Z:=m_1P_1+\ldots+m_sP_s\) the zero-scheme corresponding to the ideal of all forms of \(R\) vanishing at \(P_i\) with multiplicity at least \(m_i\), for \(i=1,\ldots,s.\) This zero-scheme \(Z\) is called a set of fat points and its defining ideal is \(I_Z =\wp_1^{m_1} \cap \dots \cap \wp_s^{m_s}\).
It is well known that the Hilbert function \(H_{R/I}(t) := \dim_{k}(R/I)_t=\dim_{k}R_t-\dim_{k}I_t\) is strictly increasing until it reaches the multiplicity \({e(R/I)}: =\sum_{i=1}^{s}{{m_i+n-1}\choose{n}}\) at which it stabilizes. The least integer \(t\) for which \(H_{R/I}(t)= {e(R/I)}\) is called the regularity index of \(Z\) and it is denoted by \(\mathrm{reg}(Z)\). It is well-known that \(\mathrm{reg}(Z)\) is equal to the Castelnuovo regularity index \(\mathrm{reg}(R/I)\).
In 1996, N. V. Trung formulated the following conjecture (independently given also by G. Fatabbi and A. Lorenzini):
Conjecture. Let \(Z=m_1P_1+\ldots+m_sP_s\) be a set of fat points in \(\mathbb{P}^n\). For \(j=1,\ldots,n\), let
\[
T_j:=\max\{[{\frac {1}{n}}(m_{i_1}+m_{i_2}+\ldots+m_{i_q}+j-2)]| P_{i_1},P_{i_2},\dots,P_{i_q} \text{ lie~ on~ a \(j\)-plane}\}
\]
then \(\mathrm{reg}(Z)\leq T_{Z}:=\max\{T_j|j=1,\dots,n\}\).
A set of fat points \(Z\) is called almost equimultiple if the multiplicities of the points are equal to \(m\) or \(m-1\) for a given integer \(m\geq 2\). In this paper the authors prove the conjecture for any set of \(n+3\) almost equimultiple, non degenerate fat points in \(\mathbb{P}^n\). The authors also show several results which will be used to prove the conjecture, and explain why the case of \(n+3\) fat points is more complicated than the case of \(n+2\) fat points. regularity index; zero-scheme; Hilbert function; fat points Tu, N. C.; Hung, T. M.: On the regularity index of n+3 almost equimultiple fat points in pn. Kyushu J. Math. 67, 203-213 (2013) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Polynomials, factorization in commutative rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Computational aspects in algebraic geometry, Multiplicity theory and related topics On the regularity index of \(n+3\) almost equimultiple fat points in \(\mathbb P^n\) | 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 Let F be a totally real number field and let \({\mathbb{A}}_ F\) be its adele ring. Let \(\pi\) be a cuspidal automorphic representation of \(GL_ 2({\mathbb{A}}_ F)\) such that for each infinite place v, \(\pi_ v\) is a discrete series representation of weight \(k_ v\) and central character \(t\to t^{-w}\), where w is an integer independent of v. Then \(\pi\) corresponds to a holomorphic Hilbert modular new form of weight \((k_ v).\)
In this paper the authors prove the following: Suppose all \(k_ v\equiv w mod 2.\) Then there exists a number field \(T\subset {\mathbb{C}}\) and a collection \(\rho (\pi)=\{\rho_{\lambda}\}\), where for each \(\ell\)-adic completion \(T_{\lambda}\) of T, \(\rho_{\lambda}\) is a continuous representation of Gal\((\bar F/F)\) in \(GL_ 2(T_{\lambda})\) such that the L-functions \(L_ v(s,\rho_{\lambda})=L_ v(s,\pi)\) for all finite places v prime to \(\ell\) of F at which \(\pi_ v\) is unramified. Furthermore, the system \(\rho\) (\(\pi)\) is motivic.
The main part of the proof is to construct, for each quadratic CM extension E/F, the \(\ell\)-adic representations associated to the L- function of the base change to \(GL_ 2({\mathbb{A}}_ E)\) of certain cuspidal representations of the quasi-split unitary group in two variables relative to E/F in the étale cohomology of local systems on a Shimura surface associated to a certain unitary group in three variables.
Recently, \textit{R. Taylor} [in Automorphic forms, Shimura varieties, and L-functions II, Academic Press, Boston, 323-336 (1990)] obtained a similar result, but the construction is quite different. totally real number field; holomorphic Hilbert modular new form; \(\ell \)- adic representations associated to the L-function; local systems on a Shimura surface Blasius D. and Rogawski J., Galois representations for Hilbert modular forms, Bull. Amer. Math. Soc. 21 (1989), 65-69. Adèle rings and groups, Zeta functions and \(L\)-functions of number fields, Langlands-Weil conjectures, nonabelian class field theory, Global ground fields in algebraic geometry, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Representation-theoretic methods; automorphic representations over local and global fields Galois representations for Hilbert forms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0722.00006.]
A scheme consisting of \(e\) points in a projective space is called a Cayley-Bacharach scheme if the subsets of \(e-1\) points have all the same postulation (i.e. Hilbert function), this condition being a weaker one than that arising in Harris' uniform position lemma [cf. \textit{J. Harris}, Math. Ann. 249, 191-204 (1980; Zbl 0449.14006)]. The main technical result of the paper is the determination of the relation between the Cayley-Bacharach property of \(X\) and the structure of the canonical module of the coordinate ring of \(X\). A characterization of CB-schemes in terms of liason is given, the so called ``weak inequalities'' are proved for Hilbert functions of CB-schemes, some ``strong inequalities'' are conjectured and relations with some new results of R. Stanley about the Hilbert function of graded Cohen-Macaulay domains are explained. Cayley-Bacharach scheme; postulation; Hilbert function; liason Geramita, A., Kreuzer, M., Robbiano, L.: Cayley--Bacharach schemes and Hilbert Functions, Preprint (1990) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Linkage, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Algebraic moduli problems, moduli of vector bundles Cayley-Bacharach schemes and their canonical modules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the book under review, the author provides an interesting introduction to the theory of hyperplane arrangements. This theory is a very active and attractive area of research, combining ideas from different branches of modern mathematics including algebraic geometry, algebraic topology, combinatorics, commutative algebra, theory of singularities, etc. The first question that one might ask is about interesting line arrangements in the real projective plane having some extreme properties, for instance the maximal number of triple intersection points (i.e., points where exactly \(3\) lines from the arrangement meet), or what is the number of bounded regions cut off by a given arrangement, and how to compute this number using only the combinatorial data of the arrangement. On the other hand, one can also ask about the topology of the complements of complex line arrangements and whether some topological properties are described by the combinatorics. This is only a fractional part of possible questions that one can ask and usually these problems are very easy to formulate. On the other side, it is really tough to find the answers on them, and really often one needs to use a bunch of different techniques to attack these problems successfully. It seems that the multidisciplinarity of the theory of hyperplane arrangements is a real magnet which attracts mathematicians from different branches. The main idea behind this book is to deliver enjoyable introduction to the theory presenting both classical aspects of the theory and very recent developments. It is worth pointing out that the text is accessible even for advanced undergraduate students (especially the first three chapters) and motivating since the book focuses on current research problems and provides some open problems. Now I would like to present (only shortly) the content of the book.
In the first chapter, the author provides a short and general introduction to the subject focusing on the interplay between combinatorics, geometry, and topology. This includes Robert's formulae on the number of regions given by a line arrangements in \(\mathbb{R}^{2}\), and Melchior's Inequality on the number of the intersection points of an essential line arrangement in the real projective plane.
In the second chapter, the author provides basics on hyperplane arrangements and their combinatorics, introducing intersection lattices, the Möbius function, and the key objects of studies, namely the characteristic polynomial and the Poincaré polynomial of a given arrangement. In the next sections, the author delivers the so-called Deletion-Restriction Theorem and he introduces several classes of hyperplane arrangements, namely supersolvable, graphical (given by graphs), and reflection arrangements. One of the main results of this chapter states the factorization property of the Poincaré polynomial for supersolvable arrangements.
In the third chapter, the author introduces the notion of Orlik-Solomon algebras associated to line arrangements, and he provides a fundamental result which tells us that these algebras are isomorphic to the cohomology algebras of the complements of complex hyperplane arrangements.
The fourth chapter is devoted to the topology of the complements of complex line arrangements. This includes considerations about the minimality of complements, the fundamental groups of complements, and arrangements whose complements are the so-called \(K(\pi, 1)\)-spaces. In the spirit of \(K(\pi,1)\)-spaces, the author recalls Deligne's result which says that real simplicial hyperplane arrangements give rise to the mentioned spaces. Another interesting result tells us that a central arrangement is supersolvable iff it is of a fiber type (please consult Definition 4.4). However, the main role in this chapter plays a beautiful result due to \textit{J. Huh} [J. Am. Math. Soc. 25, No. 3, 907--927 (2012; Zbl 1243.14005)] which tells us that the coefficients of the Poincaré polynomial of a given arrangements have the log-concavity property, which means that the Betti numbers of the complement of the arrangement form a log-concave sequence.
In Chapter 5, the author introduces the notion of Milnor fibers associated to central arrangements and he discusses about the monodromy action on the cohomology of Milnor fibers. The main result of this chapter is Theorem 5.3 which is a very general vanishing result on the twiseted cohomology of the complement of a given hyperplane arrangement with coefficients in a rank one local system -- this result comes from [\textit{D. Cohen} et al., Ann. Inst. Fourier 53, No. 6, 1883--1896 (2003; Zbl 1054.32016)].
In Chapter 6, the author introduces the notion of characteristic varieties and resonance varieties. The main result of this chapter is the Tangent Cone Theorem for complements -- see Theorem 6.1. Next, the author discusses the relation between the characteristic varieties and the homology of finite abelian covers, which leads to the polynomial periodicity properties of the first Betti numbers of such covers -- this is provided by Theorem 6.3, and this also applies to smooth surfaces obtained as the minimal desingularizations of Hirzebruch-Kummer covers of \(\mathbb{P}^{2}\) branched along line arrangements -- see Theorem 6.4. In Section 6.4, the author focuses on the so-called deleted \(B_{3}\)-line arrangement providing in great detail a complete discussion about the characteristic varieties of this arrangement and the so-called translated components -- see Definition 6.5.
In Chapter 7, the author focuses on logarithmic connections and mixed Hodge structures related to hyperplane arrangements. The first result is a new version of the Tangent Cone Theorem in the setting of logarithmic connections. After that, the author discusses the mixed Hodge structures on the cohomology of the hyperplane complements and of the Milnor fibers. After defining the spectrum \(\mathrm{Sp}(Q)\) of a given configuration (see Formulae (7.7), page 135 therein), the author presents a fundamental result due to \textit{N. Budur} and \textit{M. Saito} [Math. Ann. 347, No. 3, 545--579 (2010; Zbl 1195.14070)] which tells us that the spectrum of a central hyperplane arrangement is determined by the intersection lattice. Next, the author studies the so-called polynomial count varieties, which roughly means that for a variety \(Y\) defined over \(\mathbb{Q}\) there exists a polynomial \(P_{Y} \in \mathbb{Z}[t]\) such that for all except finitely many primes \(p\) and for any field \(\mathbb{F}_{q}\) with \(q = p^{s}\), the number of points of \(Y\) over \(\mathbb{F}_{q}\) is given by \(P_{Y}(q)\). The first result into this direction tells us that for a hyperplane arrangement in \(\mathbb{C}^{n}\) defined over \(\mathbb{Q}\), which means that the equations of these hyperplanes can be written as linear equations with \textit{integral} coefficients, the complement has polynomial count given by the characteristic polynomial of the arrangement. It is extremely interesting that the property of polynomial count is related to Hodge theoretical properties -- the property of being cohomologically Tate (see page 141 therein). Relations between being cohomologically Tate, having polynomial count, and the triviality of the monodromy action on all the cohomology groups, when our variety \(Y\) is the Milnor fiber, are discussed in Theorem 7.10, and Propositions 7.6 and 7.8. In the last part of this chapter, the author studies the relation between the polynomial count property and compactly supported Deligne-Hodge polynomials, especially focusing on the case of line arrangements having only double and triple points.
In the last chapter of the book, the author focuses on the freeness, de Rham cohomology of Milnor fibers, and Alexander polynomials. Firstly, the author discusses the notion of freeness for reduced projective hypersurfaces emphasizing the relation with the Jacobian syzygy of the defining equations. The main result of the first part of this chapter is the famous Saito's freeness criterion (in a general setting, not only for hyperplane arrangements). After that, the author studies different classes of hyperplane arrangements which are free, for instance reflection arrangements and supersolvable arrangements. Next, the author recalls a beautiful result due to \textit{H. Terao} [Invent. Math. 63, 159--179 (1981; Zbl 0437.51002)] which tells us that for free hyperplane arrangements the Poincaré polynomial splits into linear factors over the integers and the splitting type of this polynomial corresponds to the sequence of exponents. This also means that the exponents of free arrangements are combinatorially determined by the intersection lattices. In the second part, the author studies the notion of freeness for reduced curves and line arrangements in \(\mathbb{P}^{2}\), especially in the context of Terao's conjecture [\textit{P. Orlik} and \textit{H. Terao}, Arrangements of hyperplanes. Berlin: Springer-Verlag (1992; Zbl 0757.55001)]. Let us recall that Terao's conjecture predicts that the freeness is combinatorial in nature, which means that if we have two line arrangements \(\mathcal{L}\) and \(\mathcal{L}'\) in the complex projective plane such that their Levi graphs are isomorphic and \(\mathcal{L}\) is free, then \(\mathcal{L}'\) is also free. For instance, Proposition 8.3 gives us some constraints on possible exponents and the maximal multiplicity of the intersection points of free line arrangements which satisfy Terao's conjecture. In the third section, the author discusses a spectral sequence approach to the computation of the Alexander polynomial for a plane curve. The main point of this section is the algorithm which allows to compute the Alexander polynomial for a free plane curve having weighted homogeneous singularities relatively fast -- a general case is also considered. The last section is devoted to explicitly computed examples of bases for some eigenspaces of the monodromy actions on \(H^{1}(F)\) for the Milnor Fibers of monomial line arrangements (or CEVA arrangements) in the complex projective plane.
It is also worth pointing out that the author provides, after each chapter, a series of nice exercises where the role of classical line arrangements is emphasized. I think that this is an engaging reading which is written in a nice way. Of course, one can find some grammar flaws, there are some small inaccuracies (like in the first chapter where the author is actually discussing the dual Sylvester-Gallai property), and there are some places written slightly chaotic (like Theorem 4.6). However, it does not change the fact that this is a very nice introduction to the subject. hyperplane arrangements; Milnor fibers; Orlik-Solomon algebras; de Rham cohomology; matroids; local systems; characteristic polynomials; Betti numbers; characteristic varieties; resonance varieties; monodromy; intersection lattices; mixed Hodge structures; spectral sequences; polynomial count varieties; Alexander polynomials; freeness; Terao' conjecture; free divisors; logarithmic connections; logarithmic differential forms Dimca, A., Hyperplane arrangements: an introduction, Universitext, (2017), Springer-Verlag Research exposition (monographs, survey articles) pertaining to algebraic geometry, Configurations and arrangements of linear subspaces, Transcendental methods, Hodge theory (algebro-geometric aspects), de Rham cohomology and algebraic geometry, Planar arrangements of lines and pseudolines (aspects of discrete geometry), Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Relations with arrangements of hyperplanes Hyperplane arrangements. An introduction | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Nonsingular maximal intersections of three real six-dimensional quadrics are considered. Such algebraic varieties are referred to for brevity as real four-dimensional \(M\)-triquadrics. The dimensions of their cohomology spaces with coefficients in the field of two elements are calculated. six-dimensional quadric; triquadric; spectral curve; spectral bundle; index function; index orientation; complete involution; cohomology group; Stiefel-Whitney class Semialgebraic sets and related spaces Real four-dimensional \(M\)-triquadrics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In [\textit{K. Thas}, Proc. Japan Acad., Ser. A 90, No. 1, 21--26 (2014; Zbl 1329.14009)] it was explained how one can naturally associate a Deitmar scheme (which is a scheme defined over the field with one element, \(\mathbb{F}_1\)) to a so-called ``loose graph'' (which is a generalization of a graph). Several properties of the Deitmar scheme can be proven easily from the combinatorics of the (loose) graph, and known realizations of objects over \(\mathbb{F}_1\) such as combinatorial \(\mathbb{F}_1\)-projective and \(\mathbb{F}_1\)-affine spaces exactly depict the loose graph which corresponds to the associated Deitmar scheme. In this paper, we first modify the construction of [loc. cit.], and show that Deitmar schemes which are defined by finite trees (with possible end points) are ``defined over \(\mathbb{F}_1\)'' in Kurokawa's sense; we then derive a precise formula for the Kurokawa zeta function for such schemes (and so also for the counting polynomial of all associated \(\mathbb{F}_q\)-schemes). As a corollary, we find a zeta function for all such trees which contains information such as the number of inner points and the spectrum of degrees, and which is thus very different than Ihara's zeta function (which is trivial in this case). Using a process called ``surgery,'' we show that one can determine the zeta function of a general loose graph and its associated {Deitmar, Grothendieck}-schemes in a number of steps, eventually reducing the calculation essentially to trees. We study a number of classes of examples of loose graphs, and introduce the \textit{Grothendieck ring of}\(\mathbb{F}_1\)\textit{-schemes} along the way in order to perform the calculations. Finally, we include a computer program for performing more tedious calculations, and compare the new zeta function to Ihara's zeta function for graphs in a number of examples. field with one element; Deitmar scheme; loose graph; zeta function; Ihara zeta function Mérida-Angulo, M.; Thas, K., Deitmar schemes, graphs and zeta functions, J. Geom. Phys., 117, 234-266, (2017) Schemes and morphisms, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Deitmar schemes, graphs and zeta 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 fat point subscheme \(\mathbb W\) of a projective space \(\mathbb P^n\) over a field \(K\) of characteristic zero is a scheme defined by a saturated ideal the form \(I_{\mathbb W} = \wp_1^{m_1} \cap \dots \cap \wp_s^{m_s}\), where \(\wp_1 ,\dots, \wp_s\) are the homogeneous ideals of distinct points in \(\mathbb P^n\) and \(m_i \geq 1\) for all \(i\). Let \(R_{\mathbb W} = K[x_0\dots,x_n]/I_{\mathbb W}\) be the homogeneous coordinate ring of \(\mathbb W\). The Hilbert functions of such schemes, namely the function \(h_{\mathbb W}(t) = \dim_K [R_{\mathbb W}]_t\), have been of great interest for several decades. The Kähler differential module \(\Omega_{R_\mathbb W / K}^1\) was introduced by the first author and others for the study of finite sets of points.
In this paper, the authors extend this to the theory of fat point schemes. They also define and study the Kähler differents of \(\mathbb W\), which are the Fitting ideals of the Kähler differential module. They give a connection to the Cayley-Bacharach property, and at the end they prove the Segre bound for the regularity of a set of points in \(\mathbb P^4\), and they prove that under an additional hypothesis this bound holds for equimultiple fat point schemes in \(\mathbb P^4\). fat point scheme; Kähler differential; Kähler different; Hilbert function; Hilbert polynomial; regularity Kreuzer, M.; Linh, T.N.K.; Long, L.N., Kähler differentials and Kähler differents for fat point schemes, J. pure appl. algebra, 219, 4479-4509, (2015) Modules of differentials, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Projective techniques in algebraic geometry Kähler differentials and Kähler differents for fat point schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In der Abhandlung: De binis quibuslibet functionibus etc. (Crelle's J. XII.) entwickelt Jacobi über die Determinanten von Functionssystemen zwei Sätze (teor. 4, 5 p. 40), indem er diese Functionen unter der Form von Brüchen mit gleichem Nenner voraussetzt. Der Verfasser vorliegender Note zeigt, dass sich ausser diesen Theoremen noch andere sehr wichtige Sätze herleiten lassen, wenn man andere Typen von Determinanten von Functionen betrachtet und immer voraussetzt, dass diese Functionen Brüche mit gleichm Nenner oder auch Producte mit gemeinsamem Factor seien. Determinants; theorems; dissertation of Jacobi; function systems; fractions; same denominator; products; common factor Determinants, permanents, traces, other special matrix functions, Classical propositional logic, Implicit function theorems, Jacobians, transformations with several variables, Jacobian problem, Rings of fractions and localization for commutative rings, Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial) Some theorems on the determinants. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 sparse resultant for a system of \(n+1\) Laurent polynomials in \(n\) variables and \(m\) parameters defines a generically non-zero projection operator from a space of dimension \(m+n\) to one of dimension \(m\) [cf. \textit{I. M. Gelfand, M. M. Kapranov} and \textit{A. V. Zelevinsky} ``Discriminants, resultants and multidimensional determinants'' (1994; Zbl 0827.14036); \textit{B. Sturmfels}, J. Algebr. Comb. 3, No. 2, 207-236 (1994; Zbl 0798.05074)].
The paper under review provides a general method to construct a family of projection operators allowing to compute all isolated roots of algebraic systems even in the presence of infinite solutions in the toric variety associated to the given data. The main tool used by the authors is a simple perturbation scheme based on a lifting of Newton polytopes of the polynomials. Their approach represents a generalization of the method introduced by \textit{J. Canny} [J. Symb. Comput. 9, No. 3, 241-250 (1990; Zbl 0704.12004)] for homogeneous dense polynomials, and a simplification of \textit{M. Roja}'s work on sparse polynomials [J. Symb. Comput. 28, No. 1-2, 155-186 (1999; Zbl 0943.65060)]. The novelty consists in using linear perturbations of very few coefficients of the given polynomials. These perturbations are constructed as determinants of resultant matrices. Consequently, the complexity is simply exponential in the dimension \(n\) and polynomial in the sparse resultant degree. The authors detail efficient (deterministic as well as randomized) algorithms for computing the coefficients of the perturbation polynomials, derive tight bounds on the number of isolated roots for specialized coefficients, and estimate the asymptotic complexity of these algorithms. The paper ends with several examples and discussion of promising directions for future work. polynomial system solving; sparse resultant; degeneracy; mixed subdivision; asymptotic complexity; toric variety; perturbation scheme; lifting of Newton polytopes D'andrea, C.; Emiris, I. Z.: Computing sparse projection operators. Contemporary mathematics 286, 121-139 (2001) Computational aspects in algebraic geometry, Analysis of algorithms and problem complexity, Computational aspects and applications of commutative rings, Symbolic computation and algebraic computation Computing sparse projection 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 an algebraic scheme over an algebraically closed field and \(\ell\) a prime number invertible on \(X\). According to Theorem 1.1 of [\textit{P. Deligne}, Cohomologie étale. Seminaire de géométrie algébrique du Bois-Marie SGA 4 1/2 par P. Deligne, avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier. York: Springer-Verlag (1977; Zbl 0345.00010)], the étale cohomology groups \(\operatorname{H}^i(X,\mathbb Z /\ell\mathbb Z)\) are finite-dimensional. Using an \(\ell\)-adic variant of Artin's good neighborhoods and elementary results on the cohomology of pro-\(\ell\) groups, we express the cohomology of \(X\) as a well controlled colimit of that of toposes constructed on \(\mathsf{B}G\) where the \(G\) are computable finite \(\ell\)-groups. From this, we deduce that the Betti numbers modulo \(\ell\) of \(X\) are algorithmically computable (in the sense of Church and Turing). The proof of this fact, along with certain related results, occupies the first part of this paper. This relies on the tools collected in the second part, which deals with computational algebraic geometry. étale cohomology; Galois cohomology; cohomological descent; spectral sequence; simplicial scheme; profinite group; Eilenberg-MacLane space; Artin's neighborhood; stack; effective algebraic geometry; computability Madore, DA; Orgogozo, F, Calculabilité de la cohomologie étale modulo \(\mathcall \), Algebra Number Theory, 9, 1647-1739, (2015) Étale and other Grothendieck topologies and (co)homologies, Computational aspects of higher-dimensional varieties, Galois cohomology, Computational aspects of field theory and polynomials, Computability and recursion theory, Computational aspects in algebraic geometry, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Generalizations (algebraic spaces, stacks), Homotopy theory and fundamental groups in algebraic geometry, Simplicial sets, simplicial objects (in a category) [See also 55U10], Limits, profinite groups, Eilenberg-Mac Lane spaces, General theory of spectral sequences in algebraic topology Calculability of étale cohomology modulo \(\ell\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The present paper develops new algorithms for computing zeta functions of algebraic varieties over finite fields. It builds on previous work of the author for hyperelliptic curves and improves substantially on previously known algorithms if it comes to complexity, both on the time- and space-scale. Quite surprisingly, the algorithms do not require very sophisticated machinery, such as \(p\)-adic or \(\ell\)-adic cohomology theories. There are also applications of zeta functions of schemes over \(\mathbb Z\). zeta function; algorithm; arithmetic scheme; hypersurface Harvey, David, Computing zeta functions of arithmetic schemes, Proc. Lond. Math. Soc. (3), 111, 6, 1379-1401, (2015) \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Zeta and \(L\)-functions in characteristic \(p\), Number-theoretic algorithms; complexity, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Computing zeta functions of arithmetic 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 We describe a class of spectral curves and find explicit formulas for the Darboux coordinates of Hitchin systems corresponding to classical simple groups on hyperelliptic curves. We consider in detail the systems with rank 2 groups on genus 2 curves. spectral curves; Darboux coordinates; Hitchin systems; genus-2 curves Relations of finite-dimensional Hamiltonian and Lagrangian systems with algebraic geometry, complex analysis, special functions, Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.), Relationships between algebraic curves and integrable systems Hitchin systems on hyperelliptic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A (contravariant) pseudofunctor over a category \(\mathcal C\) is assigning an object \(X\) of \(\mathcal C\) to a category \(\mathcal D(X)\) and a morphism \(f \: X \to Y\) to a functor \(\mathcal D(Y) \to \mathcal D(X)\). For example, the twisted inverse image \((-)^!\) is a pseudofunctor on the category of schemes. In the present paper, the authors construct a new pseudofunctor \((-)^\sharp\) on the category of formal schemes with codimension function. They assign such a scheme to a category of Cousin complexes. However, their definition of Cousin complexes is different from Hartshorne's one. Roughly speaking, they call \(M^\bullet\) a Cousin complex on a formal scheme~\(X\) if it has a similar structure as a Cousin complex of a sheaf on~\(X\) with Hartshorne's definition. Cousin complex; twisted inverse image; codimension function; formal scheme Lipman, J., Nayak, J., Sastry, S.: Pseudofunctorial behavior of Cousin complexes on formal schemes. Variance and duality for Cousin complexes on formal schemes. Contemporary Mathematics, vol. 375, pp. 3--133. American Mathematical Society, Providence (2005) Local cohomology and algebraic geometry, Foundations of algebraic geometry, (Co)homology theory in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Categorical algebra, Radical theory Pseudofunctorial behavior of Cousin complexes on 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 Determining when a 0-dimensional scheme in projective space \(\mathbb P^n_K\) (where \(K\) is a field) which is the complete intersection of \(n\) hypersurfaces has played an important role in both Algebraic Geometry and Commutative Algebra. For example, in projective 2-space, work of \textit{E. Davis} and \textit{P. Maroscia} [Lect. Notes Math. 1092, 253--269 (1984; Zbl 0556.14025)] provided a characterization of such schemes using the Hilbert function symmetry and the Cayley-Bacharach property. In [\textit{E. D. Davis} et al., Proc. Am. Math. Soc. 93, 593--597 (1985; Zbl 0575.14040)] and [\textit{M. Kreuzer}, Math. Ann. 292, No. 1, 43--58 (1992; Zbl 0741.14030)], it was then shown that this also provides a characterization for arithmetically Gorenstein schemes. There have also been characterizations of 0-dimensional local complete intersections using the Kähler different (see [\textit{G. Scheja} and \textit{U. Storch}, J. Reine Angew. Math. 278/279, 174--190 (1975; Zbl 0316.13003)]).
In this paper the authors uses the Kähler and Dedekind differents of \(R/K[x_0]\) and \(R/K\) to further the characterizations of 0-dimensional complete intersections. A sample characterization is the following theorem:
Theorem. Let \(\mathbb X\) be a smooth 0-dimensional subscheme of \(\mathbb P^n_K\). Then \(\mathbb X\) is a complete intersection if and only if \(\mathbb X\) is a Cayley-Bacharach scheme and the Hilbert function of the Kähler different of \(\mathbb X\) in degree \(r_{\mathbb X}\) is non-zero, where \(r_{\mathbb X}\) is the regularity index of the Hilbert function of \(\mathbb X\).
Most of the characterizations can be found in Section 5 of the paper. Indeed, in this section the authors present three characterizations: one generalizing the work of Scheja and Storch; one using a value of the Hilbert function of the Kähler different of the scheme to separate the classes of complete intersections from arithmetically Gorenstein schemes; and one is the above stated theorem. Moreover, using the Dedekind different rather than the Kähler different, the authors characterize 0-dimensional arithmetically Gorenstein schemes.
One particular gem of this paper is its useful exposition with many included necessary definitions and references to the literature. zero-dimensional scheme; complete intersection; Kähler different; Dedekind different; arithmetically Gorenstein scheme; Cayley-Bacharach scheme; Hilbert function Complete intersections, Modules of differentials, Linkage, complete intersections and determinantal ideals, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Characterizations of zero-dimensional complete intersections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this article, a noncommutative (regular projective) curve is a category \(\mathcal H\) having the same formal properties as the category \(\mathrm {coh}(X)\) of coherent sheaves on a (regular projective) curve \(X/k\). The formal properties is defined locally by Lenzing-Reiten, and globally by Stafford - van den Bergh: NC1: \(\mathcal H\) is a small, connected, abelian, and each object in \(\mathcal H\) is noetherian. NC2: \(\mathcal H\) is a \(k\)-category with finite dimensional \(\mathrm {Hom}\) and \(\mathrm {Ext}\)-spaces. NC3: There is an autoequivalence \(\tau\) on \(\mathcal H\), the \textit{Auslander-Reiten translation}, such that Serre duality \(\mathrm {Ext}^1_{\mathcal H}(X,Y)=\mathrm {DHom}_{\mathcal H}(Y,\tau X)\) holds, where \(D=\mathrm {Hom}_k(-,k)\). NC4: \(\mathcal H\) contains an object of infinite length.
By Serre duality it follows that \(\mathrm {Ext}^n_{\mathcal H}\) vanishes for all \(n\geq 2\), proving that \(\mathcal H\) is hereditary. \(\mathcal H_0\) denotes the Serre subcategory of \(\mathcal H\) consisting of objects of finite length, \(\mathcal H_+\) objects not containing a simple object. Then every indecomposable object belongs either to \(\mathcal H_+\) or \(\mathcal H_0\), and \(\mathcal H_0=\coprod_{x\in\mathbb X}\mathcal U_x\) for an index set \(\mathbb X\), where \(\mathcal U_x\) are connected, uniserial categories, called \textit{tubes}. By this, it is reasonable to write \(\mathcal H=\mathrm {coh}(\mathbb X)\), and the author add the additional assumption that \(\mathbb X\) consists of infinitely many points. Then \(\mathbb X\) is called a \textit{weighted noncommutative regular projective curve} over \(k\). Because \(\mathbb X\) is assumed to contain infinitely many points, it follows that for each \(x\in\mathbb X\), the number \(p(x)\) of isomorphism classes of simple modules in \(\mathcal U_x\) is finite, and for all but a finite set of \(x\in\mathbb X,\;p(x)=1\). The numbers \(p(x)>1\) are called the weights of \(\mathcal H\), and the corresponding points are called exceptional. A simple simple object \(S\) with \(\mathrm {Ext}^1(S,S)=0\) is called an \textit{exceptional simple sheaf}. An indecomposable object \(L\in\mathcal H\) is called a \textit{line bundle} if it becomes a simple object modulo \(\mathcal H_0\). If, in addition there is upto isomorphism precisely one simple sheaf \(S_x\) concentrated in \(x\) with \(\mathrm {Ext}^1(S_x,L)\neq 0,\) it is called \textit{special}. \(\mathcal H\) is called \textit{non-weighted} (homogeneous) if \(p(x)=1\) for all \(x\), which is equivalent to \(\mathrm {Ext}^1(S,S)\neq 0\) for each simple object \(S\).
The author proves that one can reduce to the non-weighted case: Let \(\mathcal H\) be a weighted noncommuative regular projective curve with the exceptional points given by \(x_1,\dots,x_t,\) with \(p(x_i)>1\). Choose for every \(i=1,\dots,t\) a simple sheaf \(S_i\) concentrated in \(x_i\), and let \(\mathscr{S}\) be the system \(\{\tau^i S_i|i=1,\dots,t;j=1,\dots,p_i-1\}.\) Then the right perpendicular category \(\mathcal H_{nw}=\mathscr{S}^\perp\subseteq\mathcal H\) is a full exact subcategory of \(\mathcal H\) which is a non-weighted noncommutative regular projective curve, and there is a special line bundle \(L\) in\(\mathcal H\).
Each weighted noncommutative regular projective curve \(\mathcal H\) over \(k\) is obtained from a non-weighted noncommutative regular projective curve \(\mathcal H_{nw}\) over \(k\) by insertion of weights into a finite number of points of \(\mathcal H_{nw}\). The authors always consider a pair \((\mathcal H,L)\), \(L\) a special line bundle considered as the structure sheaf. The quotient category \(\tilde H=\mathcal H/\mathcal H_0\) is semisimple with one simple object given by the class \(\tilde L\) pf \(L\) so that \(\tilde H=\mathrm {mod}(k(\mathcal{H}))\) for the skew field \(k(\mathcal H)=\mathrm {End}_{\tilde{\mathcal H}}(\tilde L)\), the \textit{function field}. Also, \(\mathcal H/\mathcal H_0\simeq\mathcal H_{nw}/(\mathcal H_{nw})_0\) implying that \(k(\mathcal H)\simeq k({\mathcal H}_{nw})\). The author proves that if \(\mathcal H\) is non-weighted, then it is uniquely determined by its function field.
The global skewness of \(\mathcal H\) is the number \(s(\mathcal H)=[k(\mathcal H):Z(k(\mathcal H))]^{1/2}\), and \(Z(k(\mathcal H))\simeq k(X)\) for a unique regular projective curve over \(k\), called the centre curve of \(\mathcal H\). In the main part of the text, \(\mathcal H\) is a noncommutative, non-weighted, regular projective curve over a perfect field \(k\), and \(S_x\) denotes the unique simple sheaf concentrated in \(x\).
The aim of the article is to give a detailed introduction, with examples, to noncommutative curves by the approach given with basis in the Auslander-Reiten translation \(\tau\) which is a global datum of the category \(\mathcal H\). The local properties of \(\tau\) is studied by looking into the explicit structure of the tubes \(\mathcal U_x\). The Auslander-Reiten translation \(\tau\) acts on each \(\mathcal U_x\) which is a hereditary category with with Serre duality, and is a basic, non-trivial example of a connected uniserial length category. Such categories where classified by their species by Gabriel. In the case of a homogeneous tube with one simple object \(S\), this species is the \(D-D\)-bimodule \(\mathrm {Ext}^1(S,S)\), \(D=\mathrm {End}(S)\), and these are classified explicitly. This determines the complete local rings as certain twisted power series rings. The core of the main results is stated verbatim as
Theorem. For each point \(x\in\mathbb X\) the full subcategory \(\mathcal U_x\) of skyskraper sheaves concentrated in \(x\) is equivalent to the category of finite length modules over the skew power series ring \(\mathrm {End}(S_x)[[T,\tau^-]]\). Here the twist \(\tau^-\), with \(Tf=\tau^-(f)T\) for all \(f\in\mathrm {End}(S_x)\), is given by the restriction of the inverse Auslander-Reiten translation \(\tau^-;\mathcal H\rightarrow\mathcal H\) to the simple object \(S_x\) concentrated in \(x\).
From this result, the restriction of \(\tau\) to \(\mathcal U_x\) is of order \(e_{\tau}(x)\), the \(\tau\) multiplicity in \(x\). The author study this multiplicity and proves that it has reasonable properties.
From the essential fact that each noncommutative regular projective curve is uniquely determined by its function field, many known results from the theory of orders follows. In particular, the \(\tau\)-multiplicities are just the ramification indices of \(\mathcal A\), the sheaf of \(\mathcal O_X\)-orders.
The author review facts on different and dualizing sheaves which follows after proving that \(\tau\in\mathrm {Pic}(\mathcal H)\). The author shows that \(\mathrm {Pic}(\mathcal H)\) is determined by \(\mathrm {Pic}(X)\), the Picard group over the centre curve \(X\).
The author defines the Euler characteristic and genus of a noncommutative regular projective curve, and proves that it becomes a Morita equivalence. Also, the elliptic case is studied as a particular case. Motivated by the representation theory of finite dimensional algebras, as characterized by admitting tilting objects, a detailed treatment of the genus \(0\) case is given. The main focus is on the ghost group \(\mathcal G(\mathcal H)\), the subgroup of \(\mathrm {Aut}(\mathcal H)\) given by those automorphisms fixing the structure sheaf \(L\) and all simple sheaves \(S_x,\;(x\in\mathbb X)\). The ghost group can be seen as a measure of the failure from the ground field to be algebraically closed, and is then used to study categories of finite dimensional modules.
The above results make possible the study of noncommutative regular projective curves over \(\mathbb R\), which also specializes to the classification of all genus zero and genus one Witt curves.
This is a very extensive article, and it should be mentioned that it also treats tubular curves, the Klein bottle, Fourier-Mukai partners, and that it gives formulas for the normalized orbifold Euler characteristic.
The treatment of noncommutative curves as schemes with coordinate rings and determined by their inclusions in the function field is very algebraic, and the article is a very nice entrance to noncommutative geometry. noncommutative regular projective curve; noncommutative function field; Auslander-Reiten translation; Picard-shift; ghost group; maximal order over a scheme; ramification; Witt curve; noncommutative elliptic curve; Klein bottle; Fourier-Mukai partner; weighted curve; orbifold Euler characteristic; noncommutative orbifold; tubular curve; finite dimensional algebra; Beilinson theorem Kussin, Dirk, Weighted noncommutative regular projective curves, J. Noncommut. Geom., 10, 4, 1465-1540, (2016) Noncommutative algebraic geometry, Algebraic functions and function fields in algebraic geometry, Arithmetic theory of algebraic function fields, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Abelian categories, Grothendieck categories, Elliptic curves, Orders in separable algebras, Klein surfaces Weighted noncommutative regular projective curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Bogomol'nyi-Prasad-Sommerfeld (BPS) states form representations of a supersymmetry algebra. This paper studies the BPS spectrum of a certain class of field theories with \(d=4 \mathcal N=2\) Poincaré supersymmetry. There has been recent progress in formulating algorithms for determining the BPS spectrum of these theories. One particular algorithm is based on the Seiberg-Witten curve. The relations of BPS states to various mathematical areas have enhanced the interest of mathematical physicists in recent years and so some reviews have appeared. The investigation of the BPS spectrum has led to a number of connections the Hitchin systems, integrable field theories, cluster algebras, and cluster varieties. In this paper, it is argued that spectral networks actually contain much more information than known until now. Section 2 reviews the concept of framed BPS states. In Section 3, the construction of a formal parallel transport is described and a definition of soliton paths is given. Section 4 provides applications and examples. In Section 5, \(m\)-herds are constructed and the refined soliton content of herds is studied. The analysis applies to \(m\)-herds with arbitrary positive integer \(m\). Section 6 provides extra remarks concerning Kac's theorem and the Poincaré polynomial stabilization. Connections to quiver representation theory and Chern-Simons theory are also discussed. supersymmetry; spectral networks; BPS spectrum; Seiberg-Witten curve; Hitchin system; spin systems; integrable field theories; soliton paths Lê, T.T.Q.: Quantum Teichmüller spaces and quantum trace map. ArXiv e-prints (Nov. 2015). arXiv:1511.06054 Supersymmetric field theories in quantum mechanics, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Finite-dimensional groups and algebras motivated by physics and their representations, Relationships between algebraic curves and integrable systems Spectral networks with spin | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 gives conditions such that the Birch and Swinnerton-Dyer conjecture holds for an elliptic curve over a function field in one variable over a finite field. More precisely let \(F/k\) be a function field of characteristic not 2, \(\infty\) a place of \(F\) with residue field \(k\) and \(E/F\) an elliptic curve with split multiplicative reduction at \(\infty\). Under this hypothesis, \(E\) has a Drinfeld-Heegner point \(y_0\) defined over the Hilbert class field \(K[0]\) of an imaginary quadratic extension \(K\) of \(F\). Denote \(P_0= \text{Tr}_{K[0]/ K} y_0\in E(K)\). Suppose that \(q\) is not a square, \(K\) is not obtained from \(F\) by a ground field extension and \(P_0\) has infinite order in \(E(K)\). Let \({\mathcal C}/k\) and \({\mathcal C}'/k\) be the minimal proper smooth models of the Néron models of \(E/F\) and \(E'/K= E\times_F K/K\), respectively. Then the author proves that \(E(F)\) is a finite abelian group and \(E(K)\) is an abelian group of rank one; the Tate and the Artin-Tate conjectures hold for the surfaces \({\mathcal C}/k\) and \({\mathcal C}'/k\) [\textit{J. T. Tate}, ``Algebraic cycles and poles of zeta functions'', in: Arithmetical algebraic Geom., Proc. Conf. Purdue Univ. 1963, 93-110 (1965; Zbl 0213.22804) and ``On the conjectures of Birch and Swinnerton-Dyer and a geometric analog'', in: Dix Exposés Cohomologie Schémas, Adv. Stud. Pure Math. 3, 189-214 (1968; Zbl 0199.55604)]; and the Birch and Swinnerton-Dyer conjecture holds for the elliptic curves \(E/F\) and \(E'/K\).
The author's proof involves the works by \textit{V. G. Drinfel'd} [``Elliptic modules'', Math. USSR, Sb. 23(1974), 561-592 (1976); translation from Mat. Sb., Nov. Ser. 94(136), 594-627 (1974; Zbl 0321.14014)]\ and \textit{V. A. Kolyvagin} [``Euler systems'', in: The Grothendieck Festschrift, Vol. II, Prog. Math. 87, 435-484 (1990; Zbl 0742.14017)]. The author's objective is to construct Euler systems in the cohomology of \(E/F\) by parametrizing \(E\) by Drinfeld modular curves and by obtaining analogues of Heegner points in the present context, the Drinfeld-Heegner points. He then proves the finiteness of the \(\ell\)-primary part of the Tate-Shafarevich group of \(E\), which due to results of Artin, Tate and Milne [\textit{J. S. Milne}, ``On a conjecture of Artin and Tate'', Ann. Math., II. Ser. 102, 517-533 (1975; Zbl 0343.14005)]\ is enough to obtain his theorem.
In particular this gives an algorithm to verify the above conjectures, once an imaginary quadratic extension \(K\) of \(F\) is given together with an elliptic curve \(E/F\) and the corresponding elliptic surface \({\mathcal E}/k\).
The existence of an algorithm for deciding whether the Tate-Shafarevich group of an elliptic curve over a function field over a finite field is finite had already been considered by Goldfeld and Szpiro without extra hypothesis [\textit{D. Goldfeld}, \textit{L. Szpiro}, ``Bounds for the order of the Tate-Shafarevich group'', Compos. Math. 97, 71-87 (1995)]. Their approach is more geometric and involves Szpiro's discriminant theorem [\textit{L. Szpiro}, ``Discriminant et conducteur des courbes elliptiques'' in: Les pinceaux de courbes elliptiques, Sémin., Paris 1988, Astérisque 183, 7-18 (1990; Zbl 0742.14026)]. elliptic surfaces; Birch and Swinnerton-Dyer conjecture; elliptic curve over a function field; finite field; Drinfeld-Heegner point; Néron models; Euler systems; Drinfeld modular curves; Tate-Shafarevich group M. L. Brown, On a conjecture of Tate for elliptic surfaces over finite fields, Proc. London Math. Soc. (3) 69 (1994), no. 3, 489 -- 514. Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Algebraic functions and function fields in algebraic geometry, Rational points, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Elliptic surfaces, elliptic or Calabi-Yau fibrations On a conjecture of Tate for elliptic surfaces over finite fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article can be considered as an appendix to \textit{A. Beilinson} and the author [Proc. Symp. Pure Math. 55, Pt. 2, 123-190 (1994; Zbl 0817.14014)]. There, a general theory of elliptic polylogarithms is given. Here, exact formulas for motivic elliptic polylogarithms, i.e., involving higher algebraic \(K\)-theory, are presented.
Let \(B\) be a connected scheme and consider an elliptic curve \(p:X\to B\) over \(B\), i.e., a flat family \(X\to B\) of relative dimension one with geometrical fibers of genus one, and a zero section \(0:B\to X\). Write \(U =X \backslash 0(B)\), \(p_U:U\to B\) for the restriction of \(p\) to \(U\), and \(j_U:U \hookrightarrow X\) for the open embedding. Also, let \(i_0:0(B) \hookrightarrow X\) be the closed embedding. \(p_1: U\times_BU\backslash \Delta\to U\) (where \(\Delta\) is the (relative) diagonal) is the projection on the first factor. \({\mathcal H}\) will denote the Hodge \((\ell\)-adic, motivic,\dots) sheaf \(R^1p_* (\mathbb{Q}(1))\) on \(B\). It can be considered as the sheaf of relative homologies. \(G^{ (1)}\) will denote the sheaf \(R^1p_{1*} (\mathbb{Q}(1))\) on \(U\). It can be extended to a sheaf, also denoted \(G^{(1)}\), on \(X\). One has a short exact sequence
\[
0\to p^*{\mathcal H}\to G^{(1)}\to \mathbb{Q} \to 0
\]
on \(X\), which splits over \(0(B)\). \(G^{(1)}\) comes equipped with a weight filtration, an action of \(\mathbb{Q}\) and a unipotent action of \({\mathcal H}\). \({\mathcal H}\) can be considered as the (abelian) fundamental group of \(X/B\). Write \(G^{(n)}\) for the symmetric product \(S^n (G^{(1)})= \text{Sym}^n(G^{(1)})\). The weight filtration on \(G^{(1)}\) induces one on \(G^{ (n)}\). Also, an element \(\ell\in{\mathcal H}\) acts as the exponential \(\exp(\ell)\) on \(G^{(n)}\). One defines the logarithmic sheaf \(G=\varprojlim(G^{(n)})\). It has a weight filtration with successive quotients \(Gr^W_{-i} (G)=W_{-i}/W_{-i-1}(G)\) equal to \(S^i(p^*{\mathcal H})\), \(i\geq 0\). The fundamental group \({\mathcal H}\) acts again on \(G\) by multiplication with the exponential. Let \(X^{(n)}= X\times_B\times \cdots \times_BX\) be the (relative) \(n\)-th power of \(X\). On \(X^{(n)}\) one can define a set of divisors \(D_i^{(n)}\), \(i=1, \dots,n+1\), and \(\Delta^{(n)}_{i,j}\), \(i,j=1,2, \dots, n+1\), \(i\neq j\). One defines \(U_0^{(n+1)}= X^{(n+1)} \left\backslash \bigcup^{n+1}_{i=1} D_i^{(n+1)} \right.\). Then there is a natural map \(\Sigma= \Sigma^{(n+1)}: U_0^{(n+1)} \to U\). Of particular importance are the varieties \(Y^{(n)}=U \times_BU_0^{(n+1)}\) with projection \(\pi: Y^{(n)} \to U\) defined by \(p_U\times \Sigma:U \times_BU_0^{(n+1)} \to B\times_BU =U\). The elliptic polylogarithm \({\mathcal P}\) is defined as the extension
\[
0\to j^*_UG(1) \to{\mathcal P} \to p^*_U {\mathcal H} \to 0
\]
determined by the map (cf. loc. cit.) \({\mathcal H} \to I= R^1p_{U*} (j^*_UG(1))\), where \(I\) is the augmentation ideal of the sheaf of symmetric algebras \(\sum S^j ({\mathcal H})\) of \({\mathcal H}\). Letting \({\mathcal P}_n= {\mathcal P}/W_{-n-3} ({\mathcal P})\) one has a short exact sequence
\[
0\to j^*_U G^{(n)} (1)\to {\mathcal P}_n \to p^*_U {\mathcal H}\to 0.
\]
Using \(R^{n+1} \pi_*(\mathbb{Q}(n+1)) =p^*_U {\mathcal H} \otimes j^*_U G^{(n)}\), a spectral sequence argument leads to the existence of a canonical map \(\alpha_n\): \(\text{Ext}^{n+2}_{Y^{(n)}} (\mathbb{Q},\mathbb{Q} (n+1))\to \text{Ext}^1_U (\mathbb{Q}, R^{n+1} \pi_*(\mathbb{Q} (n+1)))\). Let \({\mathcal P}^{(n)}\in \text{Ext}^1_U (p^*_U {\mathcal H} ,j^*_U G^{(n)} (1))= \text{Ext}^1_U (\mathbb{Q},p^*_U {\mathcal H} \otimes j^*_U G^{(n)})= \text{Ext}^1_U (\mathbb{Q}, R^{n+1} \pi_*(\mathbb{Q} (n+1)))\) be the element corresponding to \({\mathcal P}_n\). Then an element \({\mathcal P}_{\mathcal M}^{(n)} \in K_n(Y^{(n)})\) is constructed such that \(\alpha_n (r({\mathcal P}_{\mathcal M}^{(n)})) = {\mathcal P}^{(n)}\), where \(r\) is the regulator map from \(K\)-theory to the group of extensions \(\text{Ext}^*(\mathbb{Q}, \mathbb{Q}(*))\). In agreement with Beilinson's conjectures, the \({\mathcal P}^{(n)}_{\mathcal M}\) come from pairs \((Z_q^{(n)}, S_q^{(n)})\) of divisors \(Z^{(n)}_q\) on \(Y^{ (n)}\) together with elements \(S_q^{(n)}\) of Milnor's \(K_{n,\mathbb{Q}}\)-groups.
This program is realized for \(B=\text{Spec}(k)\), \(k\) a field. First, using Tate's normal form of the curve \(X \to \text{Spec}(k)\), \(n\)-th order elliptic Vandermonde functions \(W_n\) on \(X^{(n)}\) are constructed as well as their \(i\)-th partial derivatives \(W_{n;i}'\). The divisors of \(W_n\) and \(W_{n;i}'\) can be expressed in terms of the \(D_i^{(n)}\) and the \(\Delta_{i,j}^{(n)}\). One also defines functions \(F_i^{(n)}= W_{n;i}'/W_n\), \(i=1,2, \dots,n\), and \(F_{n+1}^{(n)} =(W_n)^{-2} \prod^n_{i=1} F_i^{(n)}\). For \(k=\mathbb{C}\) the \(W\)'s and the \(F\)'s can be expressed in terms of theta functions. With the \(F\)'s one defines ``roots of functions'' \(\Phi_i^{(n)} =F_i^{(n+1)} \Delta^{-(n+3)/12}\), where \(\Delta\) is the discriminant of \(X\). Besides, one explicitly constructs divisors \(Z_i^{(n)}\), \(i=1,2, \dots, n+2\), on \(Y^{(n)}\). The \(\Phi\)'s define elements of the \(K_{ 1,\mathbb{Q}}\) of the generic point of \(Z_i^{(n)}\). Again, using the \(\Phi\)'s, one constructs explicit symbols \(S_i^{(n)}\) on the \(Z_i^{(n)}\). Let \({\mathcal P}_{\mathcal M}^{(n)} =(Z_i^{(n)}\), \(S_i^{(n) })\). Then the \({\mathcal P}^{(n)}_{\mathcal M}\) have nice properties and, most importantly, their image in \(\text{Ext}^1_U (\mathbb{Q}, R^{n+1} \pi_*(\mathbb{Q} (n+1)))\) is equal to the elliptic polylogarithm \({\mathcal P}^{(n)}\). connected scheme; elliptic curve; zero section; sheaf of relative homologies; weight filtration; logarithmic sheaf; elliptic polylogarithm; spectral sequence; regulator map; Beilinson's conjectures; Tate's normal form; elliptic Vandermonde functions; theta functions DOI: 10.1007/BF02362334 Higher symbols, Milnor \(K\)-theory, Elliptic curves over global fields, Algebraic number theory: global fields, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(K\)-theory of schemes Elliptic polylogarithms in \(K\)-theory | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the preamble, the authors remark that the title of their book ``designates a wide research field'' and they ``concentrate mainly on the geometry of deformations and families of singular algebraic curves on algebraic surfaces defined over the complex or real field''. Several natural questions are in the focus of the investigation: what kind and how many of singularities can be on an algebraic curve of a given degree in the projective plane, which simultaneous deformations of singular points of a curve are possible if the curve varies in a given linear system on an algebraic surface, what are the geometric properties of equisingular families of curves, etc. \par The book under review consists of a preface, four chapters and an appendix written by O. Viro. Each chapter contains a section with highly interesting and useful historical notes and comments, a general overview of all discussed topics and bibliography. \par Chapter 1 presents the theory of zero-dimensional schemes associated with singular points of curves in the projective plane $\mathbb P^2$. It should be remarked that any such scheme is a union of the so-called zero-dimensional homogeneous singularities. First of all, the authors discuss the concept of a cluster for a zero-dimensional scheme, the notions of constellations, proximity, unloading, and other basic constructions associated with any set of infinitely near points (see, e.g., [\textit{E. Casas-Alvero}, Singularities of plane curves. Cambridge: Cambridge University Press (2000; Zbl 0967.14018)], [\textit{A. Campillo} et al., Math. Ann. 306, No. 1, 169--194 (1996; Zbl 0853.14002)]). Then they describe properties of Hilbert schemes of clusters, introduce several topological and analytical invariants related to the $H^1$-vanishing criteria (see [\textit{G.-M. Greuel} and \textit{U. Karras}, Compos. Math. 69, No. 1, 83--110 (1989; Zbl 0684.32015)]) for the ideal sheaves of zero-dimensional schemes on projective curves, which play an important role in the study of the fundamental geometric properties of families of singular algebraic curves on surfaces developed in Chapter 4. \par Chapter 2 is devoted to a detail overview of the basic principles of general global deformation theory; it is concerned mainly with those specific topics which are very helpful in the understanding of the theory of equisingular families of curves. Thus, the authors discuss Bézout's, Bertini's and Noether's theorems with applications, basic properties of polar and dual curves and equisingular families of singular algebraic varieties, Hesse problem for homogeneous polynomials, the notion of Hilbert functor and Hilbert schemes for singular hypersurfaces, Plücker, Riemann-Hurwitz and genus formulas, etc. Then they observe elements of the global theory of deformations of varieties with sections with applications to the theory equisingular families of singular algebraic varieties and hypersurfaces, describe the famous patchworking construction due to O. Viro and its modifications with the use of elements of toric geometry and the operation of gluing singular hypersurfaces. In addition, they also analyze in detail a series of concrete examples related to complete intersections, critical points of real polynomials, singular points of planar polynomial vector fields, applications of the patchworking in enumerative tropical geometry, and others. \par In Chapter 3 the authors describe several approaches to prove vanishing conditions of the first cohomology groups for ideal sheaves of zero-dimensional schemes on algebraic surfaces using Riemann-Roch and Kodaira vanishing theory, Reider-Bogomolov theory, the basic Horace method (see [\textit{A. Hirschowitz}, Manuscr. Math. 50, 337--388 (1985; Zbl 0571.14002)]) and its variations with applications to the case of generic zero-dimensional schemes in $\mathbb P^2$ (see [\textit{E. Shustin}, Trans. Am. Math. Soc. 356, No. 3, 953--985 (2004; Zbl 1044.14008)]), and so on. \par In Chapter 4, the authors answer the questions on the nonemptiness, dimension, smoothness, and irreducibility of equisingular families of curves in the projective plane and on algebraic surfaces. In fact, the main results of this chapter have been achieved by the authors in a series of earlier papers (see, e.g., [\textit{G.-M. Greuel} et al., J. Algebr. Geom. 9, No. 4, 663--710 (2000; Zbl 1037.14009)], [\textit{E. Shustin} and \textit{E. Westenberger}, J. Lond. Math. Soc., II. Ser. 70, No. 3, 609--624 (2004; Zbl 1075.14034)], [\textit{G.-M. Greuel} and \textit{C. Lossen}, NATO Sci. Ser. II, Math. Phys. Chem. 21, 159--192 (2001; Zbl 1065.14029)]). Moreover, some results about families of singular curves are supplied with natural generalizations to families of higher-dimensional projective hypersurfaces with isolated singularities. Finally, the authors observe tens ``of open problems and conjectures related to the geometry of equisingular families of curves and hypersurfaces as well as to the tropical enumerative geometry in which the patchworking construction appears as an important ingredient, providing a link between classical and tropical algebraic-geometric objects''. Among them the problem of quasi-projectivity of equisingular families of curves (the case of plane curves with nodes and cusps was clarified in [\textit{J. M. Wahl}, Am. J. Math. 96, 529--577 (1974; Zbl 0299.14008)]), the problem of existence of curves with prescribed specific singularities, sufficient conditions for the irreducibility of equisingular families of plane curves, the problem of enumeration of singular curves, and so on. Above all, the authors briefly discuss the questions about the existence, $T$-smoothness and irreducibility for equisingular and equianalytic families of reduced plane curves or hypersurfaces with isolated singularities defined over an algebraically closed field of positive characteristic (see [\textit{A. Campillo} et al., Compos. Math. 143, No. 4, 829--882 (2007; Zbl 1121.14003)]). \par The appendix, called ``Patchworking Real Algebraic Varieties'', is an authorized version English translation from Russian of O. Viro's unpublished D.Sc. dissertation; it was defended at Leningrad State University (Leningrad, USSR, 1983) (see summary in [\textit{O. Ya. Viro}, in: Topology conference, Proc., Collect. Rep., Leningrad 1982, 149--197 (1983; Zbl 0605.14021)], [\textit{O. Ya. Viro}, Leningr. Math. J. 1, No. 5, 1059--1134 (1990; Zbl 0732.14026); translation from Algebra Anal. 1, No. 5, 1--73 (1989)]), where the patchworking construction is described in full detail and its striking applications to Hilbert's 16th problem is presented. \par On the whole, the book is written in a very clear style, many topics are illustrated by various nice pictures, visual diagrams, etc. The main notions and theorems are followed by carefully chosen computational examples together with algorithms implemented in the computer algebra system SINGULAR [\textit{G. M. Greuel} et al., ACM Commun. Comput. Algebra 42, No. 3, 180--181 (2008; Zbl 1344.13002); ``Singular 3. A computer algebra system for polynomial computations'', Centre for Computer Algebra, University of Kaiserslautern (2005; \url{http://www.singular.uni-kl.de})]. As a result, this book is comprehensible, interesting and useful for graduate and post-graduate students; it is also very valuable for advanced researchers, lecturers, and practicians working in singularity theory, algebraic geometry, topology, combinatorics, tropical geometry and in other fields of mathematics and its applications. projective plane curves; plane curve singularities; nodes; cusps; homogeneous zero-dimensional singularities; fat points; infinitely near points; cluster schemes, constellations; Noether's formula, Newton diagrams; conductor scheme; normalization; Hilbert functor; global deformation theory; equisingular families; equisingular deformations with section; simultaneous deformations; Hilbert-Samuel polynomial; multiplicity; polar curves; dual curves; Hesse problem; Gordan-Noether theorem; joint versal deformations; resolution; blowing up; patchworking construction; hypersurfaces; Horace method; Castelnuovo function; vanishing criteria; toric geometry; tropical geometry Research exposition (monographs, survey articles) pertaining to algebraic geometry, Local theory in algebraic geometry, Real algebraic and real-analytic geometry, Complex singularities, Theory of singularities and catastrophe theory Singular algebraic curves. With an appendix by Oleg Viro | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 give a point-free definition of a Grothendieck scheme whose underlying topological space is spectral. Affine schemes aside, the prime examples are the
projective spectrum of a graded ring and the space of valuations corresponding to an abstract nonsingular curve. With the appropriate notion of a morphism between
spectral schemes, elementary proofs of the universal properties become possible. point-free topology; Grothendieck scheme; graded ring; valuations; nonsingular curve; spectral schemes; universal properties; projective spectrum Coquand T., Lombardi H., Schuster P.\(Spectral Schemes as Ringed Lattices\). Annals of Mathematics and Artificial Intelligence. 56, (2009), 339-360. Other constructive mathematics, Distributive lattices, Schemes and morphisms Spectral schemes as ringed lattices | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(F\) be a field of characteristic \(\neq 2\), \(X\) a projective quadric over \(F\) defined by a non-degenerate quadratic form \(q\), and \(F(X)\) its function field. Let \(H^i(F)\) be the \(i\)th Galois cohomology group with mod \(2\) coefficients. The unramified cohomology group \(H^i_{nr}(F(X)/F)\) is the kernel of the map \(H^i(F(X))\to\bigoplus_{x\in X^{(1)}}H^{i-1}(F_x)\) where \(F_x\) denotes the residue field at a codimension \(1\) point \(x\in X^{(1)}\). Similarly, on defines \(H^i_{nr}(F(X)/F, i-1)\) by replacing \(H^k(\cdot)\) (\(k=i,i-1\)) by \(H^k(\cdot ,k-1)\) in the above map. In the present paper, the authors continue their study of the kernel and the cokernel of the maps \(\eta^i_2\,:\,H^i(F(X))\to H^i_{nr}(F(X)/F)\) and \(\eta^i\,:\,H^i(F(X),i-1)\to H^i_{nr}(F(X)/F,i-1)\) induced by the usual restriction map which they (with M. Rost) began in [Am. J. Math. 120, No. 4, 841--891 (1998; Zbl 0913.11018)] where they showed various results for \(i\leq 4\).
In the present paper, they focus on the case \(i=4\) and they treat various cases not covered in their previous paper. They show first that the maps \(\text{Coker}(\eta^i_2)\to \text{Coker}(\eta^i)\) are surjective for all \(i\) provided \(F\) contains all \(2\)-primary roots of unity and they ask whether surjevtivity holds in general. It is proved that if \(\dim(X)\leq 2\), then \(\text{Ker}(\eta^4_2)=H^1F\cdot \text{Ker}(\eta^3_2)\) and that it is generated by symbols. For \(2\leq\dim(X)\leq 3\), it is shown that \(\text{Coker}(\eta^4)=0\) and they give various results on \(\text{Coker}(\eta^4)\), \(\text{Coker}(\eta^4_2)\) in the case \(\dim (X)=4\) where \(X\) comes from a Pfister neighbor or a so-called (virtual) Albert form.
Some of the proofs rely on the existence of certain spectral sequences obtained by the first author in [W. Raskind (ed.) et al., Algebraic \(K\)-theory, Seattle, WA, USA, 1997, Proc. Symp. Pure Math. 67, 149--174 (1999; Zbl 0954.19004)] which in turn depend on results of Voevodsky which require resolution of singularities and are therefore a priori only valid in characteristic \(0\) as is pointed out by the authors. A certain spectral sequence for the étale motivic cohomology of an affine quadric needed in one of the cases is constructed in an appendix. quadratic forms; function field of a quadric; spinor norm; Galois cohomology; unramified cohomology; étale cohomology; Chow group; spectral sequence B. Kahn and R. Sujatha, Unramified cohomology of quadrics II, Duke Mathematical Journal 106 (2001), 449--484. Algebraic theory of quadratic forms; Witt groups and rings, Quadratic forms over general fields, Galois cohomology, Motivic cohomology; motivic homotopy theory, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) Unramified cohomology of quadrics. II. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present quasi-periodic solutions in terms of Riemann theta functions of the Heisenberg ferromagnet hierarchy by using algebro-geometric method. Our main tools include algebraic curve and Riemann surface, polynomial recursive formulation and a special meromorphic function. Heisenberg ferromagnet hierarchy; spectral curve; Riemann theta function; quasiperiodic solution Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Theta functions and curves; Schottky problem Quasiperiodic solutions of the Heisenberg ferromagnet hierarchy | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\) be a smooth projective algebraic curve over \(\mathbb{C}\) with a choice of a theta characteristic \(K_{C}^{\frac{1}{2}}\). Let also \(\eta\) denote the tautological section of the pullback \(\pi^{*}K_{C}\) on \(T^{*}C\). For an \(\text{SL}(n, \mathbb{C})\)-Higgs bundle over \(C\) with a holomorphic Higgs field \(\Phi\), \textit{N. Hitchin} introduced [Duke Math. J. 54, 91--114 (1987; Zbl 0627.14024)] a smooth curve \(\Sigma \subset T^{*}C\) given by the equation \(\text{det}(\eta\text{Id} - \pi^{*}\Phi) = 0\); such curves are today called \textit{Hitchin spectral curves} and have been considered also for meromorphic Higgs fields, as well as in much broader contexts with a non-exhaustive list of applications.
In the setting of Hitchin spectral curves, the authors of the article under review provide in Section 2 a geometric framework for \textit{quantum curves}, which have first appeared in the physics literature. In particular, a quantum curve associated with the spectral curve \(\Sigma \subset T^{*}C\) of a holomorphic Higgs bundle over \(C\) is defined as a Rees \(\tilde{\mathcal{D}}_{C}\)-module whose semi-classical limit is the curve \(\Sigma\); here \(\tilde{\mathcal{D}}_{C}\) denotes the Rees ring of the sheaf of differential operators on an open subset of \(C\).
The authors extend their framework for meromorphic Higgs bundles too, as many important classical examples of differential equations are naturally defined over \(\mathbb{P}^{1}\) with regular and irregular singularities; the definition of quantum curves in this case involves the meromorphic extension of a Rees \(\tilde{\mathcal{D}}_{C}\)-module on \(C \setminus \text{supp}(D)\), where \(D\) is a fixed effective divisor on \(C\).
A deformation parameter that appears in the definition of Rees \(\mathcal{D}\)-modules is the \textit{Planck constant} \(\hbar\), which is a purely formal parameter for the asymptotic expansion in WKB analysis. However, the quantum curves constructed via quantization of Hitchin spectral curves are depending \textit{holomorphically} on \(\hbar\). In Section 3, the authors view this parameter from a rather geometric viewpoint, that is, as an element \(\hbar \in H^{1}(C, K_{C}) = \text{Ext}^{1}(K_{C}^{-\frac{1}{2}}, K_{C}^{\frac{1}{2}}) \cong \mathbb{C}\) and use the concept of \textit{opers}, as introduced by \textit{A. Beilinson} and \textit{V. Drinfeld} [``Opers'', Preprint, \url{arXiv:math/0501398}], in order to explain this holomorphic dependence. The main theorem of the article involves an explicit construction of an \(\hbar\)-family of \(\text{SL}(n, \mathbb{C})\)-opers over \(C\) and a unique quantization of the Hitchin spectral curve for a holomorphic or a meromorphic \(\text{SL}(n, \mathbb{C})\)-Higgs bundle. The Rees \(\mathcal{D}\)-module as the quantization result recovers the initial Hitchin spectral curve via the semi-classical limit. This biholomorphic quantization is also \(\mathbb{C}^{*}\)-equivariant and quite non-trivial; generalizations of this result for more general semisimple Lie groups can be also obtained using the method described in the paper.
From yet another point of view, \textit{D. Gaiotto} [``Opers and TBA'', Preprint, \url{arXiv:1403.6137}] conjectured that such a biholomorphic map would be canonically constructed through a scaling limit of connections built via the non-abelian Hodge correspondence. For a choice of a projective structure of \(C\) of genus \(g \ge 2\) coming from the Fuchsian uniformization, this conjecture has been verified [J. Differ. Geom. 117, No. 2, 223--253 (2021; Zbl 1458.53035)] and the limit oper of the latter article is, in fact, a connection in the \(\hbar\)-filtered extension appearing in the construction of the 1-parameter family of the authors. Thus, unlike the non-abelian Hodge correspondence, this point-by-point correspondence is biholomorphic.
In the sequel of the article, the authors deal with the problem of understanding the relation between the quantization mechanism described above and the one via topological recursion. In their preceding work [Lett. Math. Phys. 104, No. 6, 635--671 (2014; Zbl 1296.14026)], the authors have implemented PDE recursions of topological type that capture the local nature of the functions involved leading to an all-order WKB analysis of quantum curves for \(\text{SL}(2, \mathbb{C})\)-Higgs bundles. It is shown here that quantization of Hitchin spectral curves through this PDE recursion of topological type and the one via the construction of an \(\hbar\)-family of opers are equivalent for the case of holomorphic or meromorphic \(\text{SL}(2, \mathbb{C})\)-Higgs bundles. Simple examples of \(\text{SL}(2, \mathbb{C})\)-meromorphic Higgs bundles over \(C=\mathbb{P}^{1}\) for the Airy function are included to further illustrate the interplay between the various notions involved in this article. quantum curve; Hitchin spectral curve; Higgs field; Rees \(\mathcal{D}\)-module; opers; non-abelian Hodge correspondence; mirror symmetry; Airy function; quantum invariants; WKB approximation; topological recursion Families, moduli of curves (analytic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Topological field theories in quantum mechanics, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Rational and ruled surfaces, Classical hypergeometric functions, \({}_2F_1\), Bessel and Airy functions, cylinder functions, \({}_0F_1\), Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\), Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent), Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category Interplay between opers, quantum curves, WKB analysis, and Higgs bundles | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Recall that a linear system of hypersurfaces of some degree \(d\) in the projective space \(\mathbb{P}^n\) is called special if its dimension is larger than its expected dimension. For special linear systems \({\mathfrak L}(d, m_1,\dots,m_r)\) of hypersurfaces of degree \(d\) passing through a scheme \(Z=m_1p_1+\cdots+m_rp_r\) of fat points in general position in \(\mathbb{P}^2\), there exists a well-studied and partially proven characterization due to B. Harbourne and A. Hirschowitz [see \textit{B. Harbourne}, Can. Math. Soc. Conf. Proc. 6, 95--111 (1986; Zbl 0611.14002) and \textit{A. Hirschowitz}, J. Reine Angew. Math. 397, 208--213 (1989; Zbl 0686.14013)]. The authors study the analogous linear systems in \(\mathbb{P}^3\). Their main tool is the cubic Cremona transformation Cr :\((x_0:x_1: x_2:x_3)\mapsto(x_0^{-1}:x_1^{-1};x_1^{-1}:x_3^{-1})\). They describe the action of Cr on the Picard group of the blow-up \(X\) of \(\mathbb{P}^3\) at \(\{p_1,\dots,p_r\}\) and use it to bring the linear systems into a standard form. For linear systems in standard form, they present a conjectural characterization of the special ones. In [\textit{C. De Volder} and \textit{A. Laface}, J. Algebra 310, No. 1, 207--217 (2007; Zbl 1113.14036)], this conjecture has been verified for \(r\leq 8\) fat points. The authors also apply their conjecture to the ``homogeneous case'' \({\mathfrak L}(d,m,\dots,m)\) and provide further evidence for it in various other cases. Cremona transformation; virtual dimension; fat point scheme; linear systems Laface, A.; Ugaglia, L., On a class of special linear systems of \(\mathbb{P}^3\), Trans. Amer. Math. Soc., 358, 5485-5500, (2006), (electronic) Divisors, linear systems, invertible sheaves On a class of special linear systems of \(\mathbb{P}^3\) | 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 provide a coherent overview of a number of recent results obtained by the authors in the theory of schemes defined over the field with one element. Essentially, this theory encompasses the study of a functor which maps certain geometries including graphs to Deitmar constructible sets with additional structure, as such introducing a new zeta function for graphs. The functor is then used to determine the automorphism groups of the Deitmar constructible sets and their base extensions to fields.
For part I, see [\textit{K. Thas}, Proc. Japan Acad., Ser. A 90, No. 1, 21--26 (2014; Zbl 1329.14009)]. field with one element; Deitmar scheme; loose graph; zeta function; Grothendieck ring; automorphism group; functoriality Schemes and morphisms, Finite ground fields in algebraic geometry, Varieties over finite and local fields, Local ground fields in algebraic geometry, Combinatorial aspects of commutative algebra, Generalizations (algebraic spaces, stacks), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) The structure of Deitmar schemes. II: Zeta functions and automorphism groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper under review, the authors introduce a new class of algebraic varieties called the frieze varieties. The frieze variety is defined in an elementary recursive way by constructing a set of points in the affine space. From a more conceptual viewpoint, the coordinates of these points are specializations of cluster variables in the cluster algebra associated to the quiver. Note that each frieze variety is determined by an acyclic quiver.
It is well known that the acyclic quiver is representation finite if and only if its underlying graph is a Dynkin diagram of type \(\mathbb{A}\), \(\mathbb{D}\) or \(\mathbb{E}\), and it is tame if and only if the underlying graph is an affine Dynkin diagram of type \(\widetilde{\mathbb{A}}\), \(\widetilde{\mathbb{D}}\) or \(\widetilde{\mathbb{E}}\). All other acyclic quivers are wild.
In this paper, the authors give a new characterization of the finite-tame-wild trichotomy for acyclic quivers in terms of their frieze varieties.
More precisely, they prove that an acyclic quiver is representation finite, tame, or wild, respectively, if and only if the dimension of its frieze
variety is \(0\), \(1\), or \(\geq 2\), respectively.
Finally, let us mention that there are several characterizations of the finite-tame-wild trichotomy in the literature, however, it seems that the characterization given by the authors is the first one in terms of numerical invariants that are integers. frieze variety; representation type of quivers; cluster algebra; Coxeter matrix; spectral radius Representation type (finite, tame, wild, etc.) of associative algebras, Cluster algebras, Representations of quivers and partially ordered sets, Special varieties Frieze varieties: a characterization of the finite-tame-wild trichotomy for acyclic quivers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a perfect field of characteristic \(p>0\) with \(p\) odd, let \(W(k)\) denote the ring of Witt vectors over \(k\), and let \(W_n\) denote the group scheme of Witt vectors of finite length \(n\). The author studies the question, when a connected subgroup scheme of \(W_n\) lifts to \(W(k)\) in terms of Dieudonné modules. The Dieudonné ring \(E\) associated to \(k\) is the non-commutative ring \(E= W(k)[F,V]\) with relations \(FV=VF =p\), \(Fw = w^{\sigma}F\), \(wV = Vw^{\sigma}\) for \(w \in W(k)\), where \(\sigma\) raises each component of \(w\) to its \(p\)-th power. Under the anti-equivalence between affine commutative unipotent \(k\)-group schemes and certain modules over \(E\) [see \textit{M. Demazure} and \textit{P. Gabriel}, ``Groupes algébriques'', Tome I (1970; Zbl 0203.23401)] the connected subgroup schemes \(G\) of \(W_n\) correspond to cyclic Dieudonné modules, i.e., modules of the form \(E/I\) for some ideal \(I \subset E\). The author now characterizes the connected subgroup schemes \(G\) of \(W_n\), which lift to \(W(k)\), in terms of the structure of the associated cyclic Dieudonné modules. The main tool is the use of finite Honda systems, and is based on work of \textit{J.-M. Fontaine} [C. R. Acad. Sci., Paris, Sér. A 280, 1273-1276 (1975; Zbl 0331.14022)]. characteristic \(p\); Dieudonné modules; group scheme; Dieudonné ring; finite Honda systems Koch, Alan: Lifting Witt subgroups to characteristic zero. New York J. Math. 4, 127-136 (1998) Formal groups, \(p\)-divisible groups, Witt vectors and related rings, Local ground fields in algebraic geometry Lifting Witt subgroups to characteristic zero | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a group. A standard quadratic equation \(s=1\) over \(G\) is of one of the following forms: \(\prod_{i=1}^n[x_i,y_i]=1\), \(n>0\); \(\prod_{i=1}^n[x_i,y_i]\prod_{i=1}^mz_i^{-1}c_iz_id=1\), \(n,m\geq 0\), \(n+m\geq 1\); \(\prod_{i=1}^nx_i^2=1\), \(n>0\); \(\prod_{i=1}^nx_i^2\prod_{i=1}^mz_i^{-1}c_iz_id=1\), \(n,m\geq 0\), \(n+m\geq 1\).
The set of all solutions in \(G\) of the system of equations \(S=1\) is denoted as \(V_G(S)\).
The authors define a wide class of regular standard quadratic equations and of regular NTQ systems of equations over a free group.
The basic simplest form of the implicit function theorem over free groups is the following Theorem 3. Let \(S(X)=1\) be a regular standard quadratic equation over a non-Abelian free group \(F\) and let \(T(X,Y)=1\) be an equation over \(F\), \(|X|=m\), \(|Y|=n\). Suppose that for any solution \(U\in V_F(S)\) there exists a tuple of elements \(W\in F^n\) such that \(T(U,W)=1\). Then there exists a tuple of words \(P=P(X)=(p_1(X),\dots,p_n(X))\), with constants from \(F\), such that \(T(U,P(U))=1\) for any \(U\in V_F(S)\). Moreover, one can find a tuple \(P\) as above effectively.
The paper contains some other, more complicated, implicit function theorems for regular quadratic equations and regular NTQ systems over a free non-Abelian group. The language of algebraic varieties over groups (coordinate groups, Zariski topology and so on) is used. It is shown that the implicit function theorems are true only for varieties as above. From the algebraic geometrical point of view such results can be understood as lifting solutions of equations into generic points.
If one uses the model theoretic view-point one can see the existence of very simple Skolem functions for particular \(\forall\exists\)-formulas over free groups.
A new version of the Makanin-Razborov process for solving equations in free groups is presented, too.
In general, the results obtained in the paper under review are key results in the solution of the famous Tarski problems about the elementary theory of a free group. free groups; algebraic geometry; equations in free groups; Skolem functions; Tarski problems; implicit function theorems; decidability; regular quadratic equations; regular NTQ systems O. Kharlampovich and A. Myasnikov, ''Implicit Function Theorem over Free Groups,'' J. Algebra 290(1), 1--203 (2005). Word problems, other decision problems, connections with logic and automata (group-theoretic aspects), Free nonabelian groups, Noncommutative algebraic geometry, Decidability of theories and sets of sentences, Model-theoretic algebra Implicit function theorem over free groups. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The JMMS equations [cf. \textit{M. Jimbo, T. Miwa, Y. Mori, M. Sato}, Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Physica D 1, 80--158 (1980)] are studied using the geometry of the spectral curve of a pair of dual systems. It is shown that the equations can be represented as time-independent Hamiltonian flows on a Jacobian bundle. Isomonodromy; Spectral curve; Dual systems; Isomonodromic deformations; Duality; JMMS equations G. Sanguinetti and N. M. J. Woodhouse, The geometry of dual isomonodromic deformations, J. Geom. Phys., 52 (2004), 44--56. Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Relationships between algebraic curves and integrable systems, Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies The geometry of dual isomonodromic deformations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems See the review in Zbl 0591.58013. integrable Hamiltonian systems; Hamilton-Jacobi theory; algebraic- geometric spectral theory; hyperelliptic curves; Korteweg-de Vries equation B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, ''Integrable system. I,'' Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 4, 179--284 (1985). Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, KdV equations (Korteweg-de Vries equations), Curves in algebraic geometry, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Applications of PDEs on manifolds Integrable systems. I | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 look for the minimal possible Castelnuovo-Mumford regularity \(m_{p(z)}\) of schemes with a given Hilbert polynomial \(p(z)\) in characteristic \(0\), providing a sharp lower bound. Many authors have focused their efforts in finding bounds for the Castelnuovo-Mumford regularity \(\mathrm{reg}(X)\) of a scheme \(X\). A very famous upper bound is due to \textit{G. Gotzmann} [Math. Z. 158, 61--70 (1978; Zbl 0352.13009)] and coincides with the regularity of the saturated lexicographical ideal with a given Hilbert polynomial \(H(z)\). The authors involve the regularity \(\varrho\) of the Hilbert function \(H(z)\) of a scheme \(X\) and the general hyperplane section \(Z\) of \(X\), they obtain that if \(H(\rho -1) > p \,(\rho -1)\) then \(\mathrm{reg}(X) >\rho +1\), by using the relation \(\mathrm{reg}(X)=\max\{\mathrm{reg}(Z),\varrho+1\}\) proved in [\textit{F. Cioffi} et al., Collect. Math. 60, No. 1, 89--100 (2009; Zbl 1188.14020)]. Moreover the authors introduce a well-suited notion of minimal functions exploiting an idea of \textit{L. Roberts} [in: Curves Semin. at Queen's, Vol. 2, Kingston/ Can. 1981--82, Queen's Pap. Pure Appl. Math. 61, Exp. F, 21 p. (1982; Zbl 0593.13009)]. So they show that there is a scheme \(X\) having Hilbert polynomial \(p(z)\) and the minimal Hilbert function with the smallest possible \(\varrho\), which achieves the Castelnuovo-Mumford regularity \(m_{p(z)}\) and obtain a formula for \(m_{p(z)}\) depending on the Hilbert function of the hyperplane section. Their proofs are based on two new constructive methods,the first is called ``ideal graft'' of two schemes and the second ``expanded lifting''. Both these methods exploit the notion of growth-height lexicographic Borel sets introduced by \textit{D. Mall} [J. Pure Appl. Algebra 150, No. 2, 175--205 (2000; Zbl 0986.14002)], and developed also in [\textit{F. Cioffi} et al., Discrete Math. 311, No. 20, 2238--2252 (2011; Zbl 1243.14007)]. In an Appendix the authors give an algorithm and its implementation to compute \(m_{p(z)}\) and a Borel ideal defining a scheme with regularity \(m_{p(z)}\). Hilbert function; Hilbert polynomial; Castelnuovo-Mumford regularity of a projective scheme; regularity of a Hilbert function; Borel ideal Cioffi, F., Lella, P., Marinari, M.G., Roggero, M.: Minimal Castelnuovo-Mumford regularity fixing the Hilbert polynomial. arXiv:1307.2707 [math.AG] Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Syzygies, resolutions, complexes and commutative rings, Symbolic computation and algebraic computation, Calculation of integer sequences, Computational aspects in algebraic geometry Minimal Castelnuovo-Mumford regularity for a given Hilbert polynomial | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The reviewed paper is concerned with the problem of determining a complete set of invariants of a linear time-invariant system under the action of a full feedback group. Using tools from geometric invariant theory, the authors show that there exists a quasi-projective variety whose points parametrize the output feedback orbits in a unique way. If the McMillan degree \(n \geq mp\), \(mp\) being the product of the number of inputs and the number of outputs, then they prove that in the closure of every feedback orbit there is exactly one nondegenerate system. feedback invariants; autoregressive systems; geometric invariant theory; Grassmannian; quot scheme; feedback group; output feedback orbits; McMillan degree; nondegenerate system M. S. Ravi, J. Rosenthal, and U. Helmke, ''Output feedback invariants,'' Linear Algebra and Its Applications, 351--352, 623--637 (2002). Algebraic methods, Feedback control, Multivariable systems, multidimensional control systems, Group actions on varieties or schemes (quotients), Parametrization (Chow and Hilbert schemes) Output feedback invariants | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article under review deals with an important problem in semialgebraic geometry. Let us start with the fundamental definitions in this setting. A subset \(M \subset \mathbb{R}^m\) is basic semialgebraic if it can be expressed as \(\{x \in \mathbb{R}^m, f(x)=0, g_1(x)>0, \dots, g_{\ell}(x)>0\}\), where \(f, g_1, \dots, g_{\ell}\) are polynomials of \(\mathbb{R}[x_1, \dots, x_m]\). Then, a semialgebraic set is a finite union of basic semialgebraic sets. Given two semialgebraic sets \(M \subset \mathbb{R}^m\) and \(N \subset \mathbb{R}^n\), a continuous map \(f : M \rightarrow N\) is semialgebraic if its graph is a semialgebraic subset of \(\mathbb{R}^{m+n}\). Now, let \(\mathcal{S}(M)\) be the set of semialgebraic functions on \(M\). By means of the sum and product of functions, \(\mathcal{S}(M)\) is endowed with a structure of commutative \(\mathbb{R}\)-algebra with unit. Calling \(\mathcal{S}^*(M)\) the subset of \(\mathcal{S}(M)\) consisting of bounded functions, \(\mathcal{S}^*(M)\) is an \(\mathbb{R}\)-subalgebra of \(\mathcal{S}(M)\). Throughout the paper, the authors study jointly \(\mathcal{S}(M)\) and \(\mathcal{S}^*(M)\), and they call either of them \(\mathcal{S}^{\diamond} (M)\). Then, \(\text{Spec}^{\diamond} (M)\) is the Zariski spectrum of \(\mathcal{S}^{\diamond} (M)\) with the Zariski topology, and \(\beta^{\diamond}(M)\) its set of closed points. Call \(\texttt{r}_M : \text{Spec}^{\diamond} (M) \rightarrow \beta^{\diamond} (M)\) the natural retraction. Then, take a semialgebraic map \(\pi : M \rightarrow N\). It has associated a homomorphism of \(\mathbb{R}\)-algebras \(\varphi_{\pi}^{\diamond} : \mathcal{S}^{\diamond} (N) \rightarrow \mathcal{S}^{\diamond} (M)\) defined by \(g \mapsto g \circ \pi\). This determines two continuous morphisms, \(\text{Spec}^{\diamond}(\pi) : \text{Spec}^{\diamond} (M) \rightarrow \text{Spec}^{\diamond} (N)\), \(\mathfrak{p} \mapsto (\varphi_{\pi}^{\diamond})^{-1}(\mathfrak{p})\), and \(\beta^{\diamond} (\pi) = \texttt{r}_N \circ \text{Spec}^{\diamond}(\pi) |_{\beta^{\diamond}(M)} : \beta^{\diamond}(M) \rightarrow \text{Spec}^{\diamond} (N) \rightarrow \beta^{\diamond}(N)\).
A long list of papers by the authors of the present article during the last ten years studies the relationship between the above maps \(\pi\) and \(\text{Spec}^{\diamond}(\pi)\). Here, they consider the case in which \(\pi : M \rightarrow N\) is a semialgebraic branched covering. Consider a map \(\pi : X \rightarrow Y\). It is a finite quasi-covering if it is separated, open, closed, surjective, and its fibers are finite. Then, the branching locus of \(\pi\) is the set \(\mathcal{B}_{\pi}\) of points in \(X\) at which \(\pi\) is not a local homeomorphism. The ramification set of \(\pi\) is \(\pi (\mathcal{B}_{\pi}) = \mathcal{R}_{\pi}\), and the regular locus of \(\pi\), \(X_{\text{reg}}\), is \(X \setminus \pi^{-1}(R_{\pi})\).
The authors state Definition 2.11, according to which \(\pi\) is a branched covering if \(X_{\text{reg}}\) is dense in \(X\) and each \(y \in Y\) admits a so-called special neighborhood, which they also define in Section 2 of the paper. This specific definition of branched covering is made in the article in order to avoid anomalous cases as shown in Example 2.26. In these conditions, the main Theorem 1.1 of the paper is the following:
Let \(\pi : M \rightarrow N\) be a semialgebraic map. Then, \(\pi\) is a branched covering if and only if \(\text{Spec}^{\diamond}(\pi)\) is a branched covering, if and only if \(\beta^{\diamond} (\pi)\) is a branched covering. In that case, \(\mathcal{B}_{\text{Spec}^{\diamond}(\pi)} = \text{Cl}_{\text{Spec}^{\diamond}(M)}(\mathcal{B}_{\pi})\), \(\mathcal{B}_{\beta^{\diamond}(\pi)} = \text{Cl}_{\beta^{\diamond}(M)}(\mathcal{B}_{\pi})\), \(\mathcal{R}_{\text{Spec}^{\diamond}(\pi)} = \text{Cl}_{\text{Spec}^{\diamond}(N)}(\mathcal{R}_{\pi})\), \(\mathcal{R}_{\beta^{\diamond}(\pi)} = \text{Cl}_{\beta^{\diamond}(N)}(\mathcal{R}_{\pi})\), where \(\text{Cl}_X(A)\) stands for the closure of \(A\) in \(X\).
The other main goal of the article is to study the collapsing set of the spectral map \(\text{Spec}^{\diamond}(\pi)\) when \(\pi\) is a \(d\)-branched covering, that is to say, a branched covering such that the fibers of the points outside \(\mathcal{R}_{\pi}\) have constant cardinality \(d\). The collapsing set \(\mathcal{C}_{\pi}\) of \(\pi\) is defined as the set of points such that the fiber \(\pi^{-1}(\pi(x))\) is a singleton. Then, the authors study \(\mathcal{C}_{\text{Spec}^{\diamond}(\pi)}\) and \(\mathcal{C}_{\beta^{\diamond}(\pi)}\). In order to do that, a map \(\mu ^{\diamond} : \mathcal{S} ^{\diamond} (M) \rightarrow \mathcal{S} ^{\diamond} (N)\) is defined in Section 4 (Definition 4.1). The result is Theorem 1.2, according to which, given a semialgebraic \(d\)-branched covering \(\pi : M \rightarrow N\), then \(\mathcal{C}_{\text{Spec}^{\diamond}(\pi)}\) is the set of prime ideals of \(\mathcal{S}^{\diamond} (M)\) containing \(\text{ker}(\mu ^{\diamond})\), and it equals \(\text{Cl}_{\text{Spec}^{\diamond}(M)}(\mathcal{C}_{\pi})\), and \(\mathcal{C}_{\beta^{\diamond}(\pi)}\) is the set of maximal ideals of \(\mathcal{S}^{\diamond} (M)\) containing \(\text{ker}(\mu ^{\diamond})\), and it equals \(\text{Cl}_{\beta^{\diamond}(M)}(\mathcal{C}_{\pi}).\)
Section 2 of the paper is devoted to define and study branched coverings, and Section 3 to analyze the properties of spectral maps associated to quasi-finite coverings and branched coverings. Then, the proof of Theorem 1.2 is developed in Section 4, and that of Theorem 1.1 in Section 5. Finally, an example is detailed in an Appendix at the end of the paper. It is worth to note that the article is very carefully written, and its introduction is very enlightening both on the state-of-the-art, and on the purpose and procedures of the work. semialgebraic set; semialgebraic function; branched covering; branching locus; ramification set; ramification index; Zariski spectra; spectral map; collapsing set Semialgebraic sets and related spaces, 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 Spectral maps associated to semialgebraic branched coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The recent development in quantum field theory has led physicists to consider the so-called super theories. Physically, super theories emerged from the attempt to create appropriate counterparts of Fermions, whereas the mathematical framework focuses on the study of geometric objects associated with super commutative rings, together with the representations of their cohomological invariants in derived categories. The significance of super theories in physics has gained increasing evidence, to a large extent, by the epoch-making ideas and arguments of E. Witten. As for the mathematical framework of super (algebraic) geometry, the monography of \textit{Yu. I. Manin} on ``Gauge field theory and complex geometry'' (Moskau 1984; Zbl 0576.53002; English translation: 1988) is among the most valuable sources. In 1988, \textit{N. Kawamoto}, \textit{Y. Namikawa}, \textit{A. Tsuchiya} and \textit{Y. Yamada} gave a geometric realization of a conformal quantum field theory on so-called Virasoro uniformized Riemann surfaces [cf. Commun. Math. Phys. 116, No. 2, 247-308 (1988; Zbl 0648.35080)]. In their theory, the description of the moduli space of those Virasoro uniformized Riemann surfaces is of crucial importance, and this is basically achieved by using its embedding in Sato's universal Grassmann manifold and a suitable generalization of Hirota's tau function. This approach, in the non-super case, is the starting point of the present paper. The author's aim is to develop an analogous theory within the framework of (infinite-dimensional) super algebraic geometry. Based on the concept of an affine super scheme associated with a super commutative ring, the author introduces the notion of a ``family of Virasoro uniformized (compact) super Riemann surfaces'', and these super objects are then discussed extensively in the course of the paper. The author's main goal is to classify Virasoro uniformized super Riemann surfaces by the coefficients of a suitable analogue of the non-super tau function, or (equivalently) by suitable super versions of the classical theta functions of Jacobians of compact Riemann surfaces. This strategy is realized by constructing a ``super tau function'' as an infinite Beresinian (i.e., as a super determinant of an infinite matrix) and, in the sequel, by interpreting it as a sort of ``super theta function''. To this end, the concept of a ``super Jacobian'' (as a moduli space of line bundles on a super Riemann surface) is introduced, and then the super tau function is interpreted as a section of a line bundle over the super Jacobian. At the end of the paper, the super tau function (or super theta function, respectively) is further analyzed by complex-analytic methods, and this is the concluding step for proving the main results on the super tau function mentioned above.
The paper is written in an extremely careful, thorough and clear style. The construction of the super tau function (and the super theta function) is a highly important and welcome contribution to the development of super geometry and its applications in quantum field theory. -- As the author mentions in the preface of his paper, another theory of a super tau function calculus has been constructed by \textit{A. S. Schwarz} in 1989 [cf. ``Fermionic string and universal moduli space,'' Nuclear Physics, Particle Physics, B 317, No. 2, 323-343 (1989)]. The approach worked out in the present paper, however, seems to be more geometrical and, therefore, more convenient for the purpose of using the super tau function in the study of moduli spaces of line bundles over super Riemann surfaces. Hirota tau function; quantum field theory; super algebraic geometry; affine super scheme; Virasoro uniformized super Riemann surfaces; super tau function; super theta function Theta functions and abelian varieties, Virasoro and related algebras, Cohomology of Lie (super)algebras, Quantization in field theory; cohomological methods, Geometric quantization, Riemann surfaces; Weierstrass points; gap sequences On super theta 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 simple description of the extended Heisenberg double (cotangent bundle) of \(\text{SL}_q(n)\) in terms of a new set of generators, which derives \(\text{SL}_q(n)\)-type dynamical \(R\)-matrices, is given (\S3.3). Then solutions of an evolution operator for the model of the \(q\)-deformed isotropic top [\textit{A. Yu Alekseev} and \textit{L. D. Faddeev}, J. Math. Sci., New York 77, No.~3, 3137--3145 (1995); translation from Zap. Nauchn. Semin. POMI 200, 3--16 (1992; Zbl 0835.17008), \url{arxiv:hep-th/9406196}; hereafter refered to as [1]] are calculated. The evolution operator is not uniquely defined and two expressions are given, one by the Riemann theta function, and the other by an almost free motion operator in terms of the logarithm of the spectral variables. They are related by a modular functional equation for the Riemann theta function (\S4).
As preliminaries, \S2 explains \(R\)-matrices; an element \({\mathcal R}\in{\mathfrak A}\otimes{\mathfrak A}\) such that \({\mathcal R}\Delta(x)= \Delta^{op}(x){\mathcal R}\), \({\mathfrak A}\) is a unital Hopf algebra, \(\Delta\) is its coproduct and \(\Delta^{op}(x)= x_2\otimes x_1\) if \(\Delta(x)= x_1\otimes x_2\). After dealing with the \(R\)-matrix representation of the braid group and the Hecke type \(R\)-matrix, a \(\text{GL}_q(n)\)-type \(R\)-matrix \(R\) is defined if it satisfies
\[
A^{(n)}\Biggl({q^n\over n_q} I- R_n\Biggr) A^{(n)}= 0,\quad \text{rk\,}A^{(n)}= 1.
\]
Here \(R\) is assumed to satisfy characteristic identity \((R- qI)(R+ q^{-1}I)= 0\), where \((q^i- q^{-i}(q- q^{-1})\neq 0\), \(2\leq i\leq k\), and \(A^{(k)}\) is the \(k\)-antisymmetrizer (Def. 2.7).
In \S3, as algebras of quantized functions over the matrix semigroup and the matrix group, the RTT algebra \({\mathfrak F}[R]\) and its extension \({\mathfrak F}{\mathfrak G}[R]\) are defined [Def. 3.1. cf. \textit{V. G. Drinfel'd}, J. Math. Sci. 41, 898--915 (1988; Zbl 0641.16006)]. The determinant of the matrix \(T\) is defined by \(\text{Tr}_{(1,\dots, n)}(A^{(n)}T_1\cdots T_n)\) (\S3, (3.1.6)). Then, the \(\text{GL}_q(n)\)-type RTT algebra \({\mathfrak F}_{\text{GL}_q(n)}[R]\) and the \(\text{SL}_q(n)\)-type RTT algebra \({\mathfrak F}_{\text{SL}_q(n)}[R]\) are defined. Quantum right invariant vector fields are defined as elements of the reflection equation algebra (RE algebra) \({\mathfrak L}{\mathfrak G}[R]\) [Def. 3.7., \textit{P. P. Kulish} and \textit{E. K. Skylyanin}, J. Phys. A, Math. Gen. 25, No. 22, 5963--5975 (1992; Zbl 0774.17019)]. In the Hecke type RE algebra, elementary symmetric functions can be defined. Then, Cayley-Hamilton and Newton identities in the \(\text{GL}_q(n)\) type and Hecke type RE algebras are described (Th.3.13). By these identities, characteristic polynomials of the \(\text{GL}_q(n)\) type RE algebras are described and semisimple spectral completions of \(\text{GL}_q(n)\) and \(\text{SL}_q(n)\) type RE algebras are defined by using characteristic polynomials (Def. 3.15). After these preparations, an algebra of quantized differential operators over the matrix group (Heisenberg dual(HD)) \({\mathfrak D}{\mathfrak G}[R,\gamma]\) are defined to be the algebra generated by the components of the matrices \(T\), whose components generate \({\mathfrak F}{\mathfrak G}[R]\), and \(L\), whose components generate \({\mathfrak L}{\mathfrak G}[R]\) with the relation
\[
\gamma^2 T_1 L_2= R_{12} L_1 R_{12} T_1
\]
[Def. 3.19., cf. \textit{M. A. Semenov-Tyan-Shanskij}, Theor. Math. Phys. 93, No.~2, 1292--1307 (1992); translation from Teor. Mat. Fiz. 93, No. 2, 302--329 (1992; Zbl 0834.22019)]. The authors remark that the Heisenberg dual is closely related to a smash product [cf. \textit{S. Montgomery}, Hopf algebras and their actions on rings, Regional Conference Series in Mathematics. 82. Providence, RI: American Mathematical Society (AMS) (1993; Zbl 0793.16029)]. After studying relations in the Hecke type HD algebra \({\mathfrak D}{\mathfrak G}[R,\gamma]\), the \(\text{GL}_q(n)\)-type HD algebra \({\mathfrak D}_{\text{GL}_q(n)}[R,\gamma]\) is defined by the relations
\[
\gamma^{2n}L(\text{det}_RT)^{-1}= q^2(\text{det}_RT)^{-1}(O_r LO^{_1}_R),\quad \gamma^{2n}(a_n)^{-1} T= q^2 T(a_n)^{-1}.
\]
\({\mathfrak D}_{\text{SL}_q(n)}[R]\) is similarly defined taking quotient by relations \(\text{det}_RT= 1\) and \(a_n= q^{-1}I\) (Prop.3.24). The elements of the characteristic subalgebra of a \(\text{GL}_q(n)\) type HD algebra are shown to satisfy the commutation relation
\[
\gamma^{2nk}\text{det}_R T\text{\,ch}(x^{(k)})= g^{2k}ch(x^{(k)})\text{det}_RT
\]
(Cor.3.26). Then a (semisimple) spectral completion of the \(\text{GL}_q(n)\) (or \(\text{SL}_q(n)\)) type HD algebra \(\overline{{\mathfrak D}}_{\text{GL}_q(n)}[R,\gamma]\) by the algebra of polynomials in mutually commuting indeterminants \(\mu^\pm_\alpha\) is defiend by the Weyl relations
\[
\gamma^2(P^\beta T) \mu_\alpha= q^{2\delta_{\alpha\beta}} \mu_\alpha(P^\beta T)
\]
(Th.3.27). In \(\overline D_{\text{GL}_q(n)}[R,\gamma]\), the permutation relation
\[
\gamma^{2n}\text{det}_R T\mu_\alpha= q^2 \mu_\alpha et_RT
\]
(Cor. 3.31) is shown. These are the first main results in this paper. In Def. 3.19, a HD algebra is constructed by quantum right invariant vector fields. Corresponding constructions by quantum left invariant vector fields and explicit relations between the spectra of left and right invariant vector fields are given in subsection 3.4 (detailed calculations are given in Appendix B). \S3 is concluded by showing that the dynamical \(R\)-matrix, which was used in the construction of the HD algebra in [\textit{A. Yu. Alekseev} and \textit{L. D. Faddeev}, Commun. Math. Phys. 141, No.~2, 413--422 (1991; Zbl 0767.17024), hereafter refered to as [2]], satisfies the dynamical Yang-Baxter equation (Cor.3.37)
\[
R(\mu)^{12} R(\nabla^1(\mu))^{23} R(\mu)^{12}= R(\nabla^1(\mu))^{23} R(\mu)^{12} R(\nabla^1(\mu))^{23},
\]
where \(\nabla^1\) is a diagonal shift operator (Cor. 3.37). The authors say dynamical solutions of the Yang-Baxter equation are calculated by solving a system of three linear equations.
In [1],[2], a discrete time evolution (\(q\)-deformed quantum isotropic top)
\[
\begin{aligned} \theta^k:{\mathfrak D}{\mathfrak G}[R,\gamma] &\to {\mathfrak D}{\mathfrak G}[R,\gamma],\\ \theta^k(T,L) & = (T(k), L(k)),\;T(k)= L^kT,\;L(k)= L,\end{aligned}
\]
is introduced. \(\theta\) agrees with the defining relations of \({\mathfrak D}{\mathfrak G}[R,\gamma]\) and is consistent with the reduction condition in the definition of \(\text{SL}_q(n)\) (Prop.4.1). To solve this equation, the authors impose the ansatz
\[
T(k+ 1)= LT(k)= (ga_n)^{1/n}\Theta T(k)\Theta^{-1}, \Theta\in\overline{{\mathfrak C}{\mathfrak h}}[R].
\]
If \(|q|< 1\), by setting \(\Theta(\mu_n)= \sum\{\vec k\in\mathbb{Z}^{n-1} c(\vec k) \mu^{k_1}_1\cdots \mu^{k_{n-1}}_{n-1}\), by the above ansatz, a solution \(\Theta^{(1)}\) is obtained to be
\[
\Theta^{(1)}(\mu_a)= \theta(\vec z,\Omega).
\]
Here, \(\theta(\vec z,\Omega)\) is a Riemann theta function and the parametrization is taken as \(q= \exp(2\pi i\tau)\), \(q^{1/n}\mu_\alpha= \exp(2\pi iz_\alpha)\) and
\[
\sum^{n- 1}_{\alpha=1} z_\alpha= 0,\quad \Omega= {2\tau\over n} A^*_{\alpha\beta}= 2\tau\Biggl(\delta_{\alpha\beta}- {1\over n}\Biggr).
\]
As for arbitrary \(q\), assuming \(L\), \(M\) and \(T\) are diagonalized as
\[
L= gU\,DU^{-1},\quad M= \gamma^2 q^{-1}V\, DV^{-1},\quad T= U^{-1}QV^{-1},
\]
the anzats reduces to \(qDQ= \Theta Q\Theta^{-1}\). Hence,
\[
\Theta^{(2)}(z_\alpha)= \exp\Biggl(-{\pi i\over 2\tau} \sum^n_{\beta= 1} z^2_\beta\Biggr)
\]
is shown to be a solution of the evolution equation. The authors stress that the logarithmic change of variables \(\mu_\alpha\to z_\alpha\) is inevitable in this case. By a functional equation for the Riemann theta function, relation
\[
\Theta^{(2)}(\vec z)= {1\over \sqrt{n}}\Biggl({2\tau\over i}\Biggr)^{(n-1)/2}{\theta(\vec z,\Omega)\over\theta(\Omega^{-1}\vec z,-\Omega^{(1)}}.
\]
between \(\Theta^{(1)}= \theta\) and \(\Theta^{(2)}\) is given. These are the second main results of this paper.
In the ntroduction, the authors say that to formulate a problem of spectral extension for the Heisenberg double over orthogonal and symplectic quantum groups and over quantum linear subgroups, and the extension of a modular double construction [\textit{L. D. Faddeev}, in: Conférence Moshé Flato 1999: Quantization, deformation, and symmetries, Dijon, France, September 5--8, 1999. Volume I. Dordrecht: Kluwer Academic Publishers. Math. Phys. Stud. 21, 149--156 (2000; Zbl 1071.81533)] are further problems. The authors also mention an observation that a ribbon element serve a \(q\)-top evolution on the smash product algebra of a ribbon Hopf algebra with its dual Hopf algebra (\S4, example 4.3), which could open a way for the spectral extension of a quantum Hopf algebra. quantum group; \(R\)-matrix; Heisenberg double; spectral completion; quantum top; Riemann theta function Isaev, A. P.; Pyatov, P., Spectral extension of the quantum group cotangent bundle, Comm. Math. Phys., 288, 1137-1179, (2009) Geometry of quantum groups, Ring-theoretic aspects of quantum groups, Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups (quantized function algebras) and their representations, Theta functions and abelian varieties Spectral extension of the quantum group 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 We discuss the Lie-Poisson group structure associated to splittings of the loop group \(LGL(N,\mathbb{C})\), due to Sklyanin. Concentrating on the finite-dimensional leaves of the associated Poisson structure, we show that the geometry of the leaves is intimately related to a complex algebraic ruled surface with a \(\mathbb{C}^*\)-invariant Poisson structure. In particular, Sklyanin's Lie-Poisson structure admits a suitable abelianisation once one passes to an appropriate spectral curve. The Sklyanin structure is then equivalent to one considered by Mukai, Tyurin and Bottacin on a moduli space of sheaves on the Poisson surface. The abelianization procedure gives rise to natural Darboux coordinates for these leaves, as well as separation of variables for the integrable Hamiltonian systems associated to invariant functions on the group. Hilbert schemes; spectral curves; integrable systems on loop algebras and loop groups; Lie-Poisson group structure; abelianisation; separation of variables; integrable Hamiltonian systems Hurtubise, J.; Markman, E., Surfaces and the Sklyanin bracket, Commun. Math. Phys., 230, 485-502, (2002) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Parametrization (Chow and Hilbert schemes), Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Surfaces and the Sklyanin bracket | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Algebraic knots are known to be iterated torus knots and to admit \(L\)-space surgeries. However, \textit{M. Hedden} [Int. Math. Res. Not. 2009, No. 12, 2248--2274 (2009; Zbl 1172.57008)] proved that there are iterated torus knots that admit \(L\)-space surgeries but are not algebraic. We present an infinite family of such examples, with the additional property that no nontrivial linear combination of knots in this family is concordant to a linear combination of algebraic knots. The proof uses the Ozsváth-Stipsicz-Szabó Upsilon function, cf. [\textit{P. S. Ozsváth} et al., Adv. Math. 315, 366--426 (2017; Zbl 1383.57020)], and also introduces a new invariant of \(L\)-space knots, the formal semigroup. iterated torus knots; \(L\)-space surgery; Upsilon function; formal semigroup Invariants of knots and \(3\)-manifolds, Knots and links in the 3-sphere, Singularities in algebraic geometry Semigroups of \(L\)-space knots and nonalgebraic iterated torus knots | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present a Lie algebra theoretical schema leading to integrable systems, based on the Kostant-Kirillov coadjoint action. Many problems on Kostant-Kirillov coadjoint orbits in subalgebras of infinite dimensional Lie algebras (Kac-Moody Lie algebras) yield large classes of extended Lax pairs. A general statement leading to such situations is given by the Adler-Kostant-Symes theorem and the van Moerbeke-Mumford linearization method provides an algebraic map from the complex invariant manifolds of these systems to the Jacobi variety (or some subabelian variety of it) of the spectral curve. The complex flows generated by the constants of the motion are straight line motions on these varieties. We study the isospectral deformation of periodic Jacobi matrices and general difference operators from an algebraic geometrical point of view and their relation with the Kac-Moody extension of some algebras. We present in detail the Griffith's aproach and his cohomological interpretation of linearization test for solving integrable systems without reference to Kac-Moody algebras. We discuss several examples of integrable systems of relevance in mathematical physics. spectral theory; integrable systems; Lie algebras; Jacobians; Prym varieties Lesfari, A., Théorie spectrale et problèmes non-linéaires, Surv. Math. Appl., 5, 141-180, (2010) Applications of Lie algebras and superalgebras to integrable systems, Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics, Relationships between algebraic curves and integrable systems, Jacobians, Prym varieties Spectral theory and nonlinear problems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems See the review in Zbl 0664.58007. integrable Hamiltonian systems; Hamilton-Jacobi theory; spectral theory; Korteweg-de Vries equation; hyperellipitic curve Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Curves in algebraic geometry, Applications of PDEs on manifolds Integrable systems. II | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \(\mathcal M\)ultivariate \(\mathcal Q\)uadratic public key schemes have been suggested as early as 1985 by Matsumoto and Imai as an alternative for the RSA scheme. Since then, several schemes have been proposed, for example hidden field equations, unbalanced oil and vinegar schemes, and stepwise triangular schemes. All these schemes have a rather large key space for a secure choice of parameters. Surprisingly, the question of equivalent keys has not been discussed in the open literature until recently. In this article, we show that for all basic classes mentioned above, it is possible to reduce the private - and hence the public - key space by several orders of magnitude, i.e. the size of the set of possible private and hence public keys can be reduced. For the Matsumoto-Imai scheme, we are even able to show that the reductions we found are the only ones possible, i.e. that these reductions are tight. While the theorems developed in this article are of independent interest themselves as they broaden our understanding of \(\mathcal M\)ultivariate \(\mathcal Q\)uadratic public key systems, we see applications of our results both in cryptanalysis and in memory efficient implementations of \(\mathcal {MQ}\)-schemes. \(\mathcal M\)ultivariate \(\mathcal Q\)uadratic polynomials; public key signature; hidden field equations; Matsumoto-Imai scheme A; C\(^*\); unbalanced oil and vinegar; stepwise triangular systems; equivalent keys; post-quantum cryptography Wolf, C.; Preneel, B., Equivalent keys in multivariate quadratic public key systems, J. Math. Cryptol., 4, 375-415, (2011) Applications to coding theory and cryptography of arithmetic geometry, Cryptography, Finite ground fields in algebraic geometry, Algebraic coding theory; cryptography (number-theoretic aspects), Enumerative problems (combinatorial problems) in algebraic geometry Equivalent keys in \(\mathcal M\)ultivariate \(\mathcal Q\)uadratic public key 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 In this paper one gives the solution to the classical Waring problem for forms, namely to determine \(g(d, r)=\) the smallest length of a decomposition of a general form of degree \(d\) in \(r\) variables as a sum of \(d\)-th powers of linear forms.
The main tools are a result of J. Alexander and A. Hirschowitz about the Hilbert function of the ideal of functions vanishing to order two at a generic set of points of the projective space \(\mathbb{P}^{r-1}\) and a theorem of \textit{J. Emsalem} and \textit{A. Iarrobino} [Inverse system of a symbolic power. I, J. Algebra 174, 1080-1090 (1995; Zbl 0842.14002)].
One studies also other related lengths for a form \(f\), different in general from the above \(g(d, r)\), named smoothable length and scheme length. One proposes also new ``Waring problems'' with respect to these lengths. Macaulay inverse system; classical Waring problem for forms; Hilbert function; smoothable length; scheme length Iarrobino, A, Inverse system of a symbolic power. II. the Waring problem for forms, J. Algebra, 174, 1091-1110, (1995) Forms of degree higher than two, Relevant commutative algebra, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Waring's problem and variants Inverse system of a symbolic power. II: The Waring problem for forms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a closed reduced zero-dimensional subscheme contained in a cubic surface \(F\) of \(\mathbb{P}^3\). In this paper we relate the behaviour of the first difference \(H(X,-)\) of its Hilbert function with geometric properties of linear systems of surfaces on which \(X\) lies. In particular we prove, using linear series, that if \(F\) is irreducible then \(H(X,n)\) is of ``non increasing type''. The main result says that if \(H(X,n+1)\geq H(X,n)+1\) then the surfaces of degree \(\leq n+1\) have a fixed component of degree two or one. zero-dimensional subscheme of cubic surface; Hilbert function; linear systems Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On the Hilbert function of zero-dimensional subschemes contained in a cubic surface of \(\mathbb{P}^3\) | 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 The purpose of our work is to apply \textit{H. Flaschka}'s techniques [Nonlinear integrable systems - classical theory and quantum theory, Proc. RIMS Symp., Kyoto 1981, 219-240 (1983; Zbl 0551.58017)] to operators of order \(n\geq 2\). We define higher Neumann systems whose theory is closely tied to the spectral theory of linear differential operators of order n. \textit{C. Tomai} [''The Boussinesq equation'', Ph. D. Thesis, Courant Inst. (1981)], using scattering theory, obtained some of our \(n=3\) formulas. higher Neumann systems; spectral theory of linear differential operators of order n; scattering theory DOI: 10.1090/S0273-0979-1986-15444-3 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Ergodic theorems, spectral theory, Markov operators, Jacobians, Prym varieties Generalizations of the Neumann system | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is a survey of the author's results covering the following problems: conditions for the rationality of the Reidemeister zeta functions for a group endomorphism and for a continuous map of a compact polyhedron; explicit formulas for Reidemeister zeta functions; the radius of convergence for the Nielsen zeta function of a surface homeomorphism and its connection to the dynamics. Reidemeister zeta functions; group endomorphism; continuous map; radius of convergence; Nielsen zeta function Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Zeta-functions of Reidemeister and Nielsen | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R\) be a complete discrete valuation ring of mixed characteristic, \(k\) its residue field of characteristic \(p\), \(\pi\) a uniformizing element, \(e\) the absolute ramification index. Let \(a\) be a positive integer, \(\overline R=R/\pi^ aR\). Let \(G\) be \(a\) a finite commutative \(p\)-group scheme over \(R\); here ``\(p\)-group'' means that \(G\) is killed by a certain power of \(p\). Let \(G\) be a lifting of \(\overline G\) to a commutative finite \(p\)-group over \(R\). When \(a=1\) (i.e., \(\overline R=k)\) and \(e=1\) (i.e., \(\overline R=W(k))\) the classification of the liftings was done by \textit{J. H. Fontaine} [``Groups \(p\)-divisibles sur les corps loceaux'', Astérisque 47-48 (1977; Zbl 0377.14009); see also \textit{A. Badra}, Thèse de 3ème cycle (Université de Rennes 1979)] not only for finite \(p\)-groups but also for \(p\)-divisible groups in terms of the so called Honda systems \((M,L)\). Here \(M\) is the Dieudonné module of \(\overline G\) and \(L\) is an \(R\)-submodule of \(M\) satisfying certain conditions imposed by the action of Frobenius on \(M\). The author develops a rather involved theory of Honda systems in the general case and uses it for the classification of all liftings \(G\) of \(\overline G\) up to an isomorphism. As a corollary, he obtains that, when \(p\geq 5\) and \(e\geq 2\) every commutative finite \(p\)-group scheme over \(k\) can be lifted to \(R\). deformations; \(p\)-group scheme; Honda systems Group schemes, Formal groups, \(p\)-divisible groups, Formal methods and deformations in algebraic geometry Finite group schemes over a discrete valuation ring and associated Honda 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 A three-parameters family of special double confluent Heun equations, with real parameters \(l\), \(\lambda\), \(\mu\), is considered. The study of the real part of the spectral curve leads to applications to model of Josephson junction which is a family of dynamical systems on 2-torus depending on parameters \((B, A, \omega)\), where \(\omega\) is called the frequency. The authors provide an approach to study the boundaries of the phase-lock areas in \(\mathbb R^2(B, A)\) and their solutions, as \(\omega\) decreases to 0.They prove the irreducibility of the complex spectral curve \(\Gamma_l\) for every \(l\in\mathbb N\). They calculate its genus for \(l\le 20\) and present a conjecture on general genus formula. They apply the irreducibility result to the complexified boundaries of the phase-lock areas of model of Josephson junction. They show as well that its complexification is a complex analytic subset consisting of just four two-dimensional irreducible components, and describe them. They prove that the spectral curve has no real ovals. Finally, they present a Monotonicity Conjecture on the evolution of the phase-lock area portraits, as \(\omega\) decreases, and a partial positive result towards its confirmation. model of Josephson junction; dynamical systems on torus; rotation number function; phase-lock area; double confluent Heun equations Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, Plane and space curves, Rotation numbers and vectors, Junctions On spectral curves and complexified boundaries of the phase-lock areas in a model of Josephson junction | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 some properties of families of smooth curves which are the normalizations of plane curves of degree \(n\) with \(s\) ordinary singular points of multiplicity \(m_1,..., m_s\) at fixed points \(P_1,...,P_s\) and with \(c\) nodes as other singularities. The family of such curves is denoted \(W(n,m_i,P_i,c)\) and the family of its normalizations \(NW(n,m_i,P_i,c)\). The authors study the set of nodes of curves in \(NW(n,m_i,P_i,c)\). Particularly, they show that for \(n\) large enough, in all its principal components (= components containing deformations of arrangements of \(n\) lines intersecting in the points \(P_i\) with multiplicities \(m_i\)), the monodromy of nodes is the full symmetric group, and there exists a component whose general curve has nodes of maximal rank. They also show the nonexistence of some linear series on curves in \(NW(n,m_i,P_i,c)\) under certain numerical constraints on the data. singular points with high multiplicity; linear systems; Hilbert scheme; normalizations of plane curves; principal components; arrangements; monodromy of nodes Singularities of curves, local rings, Divisors, linear systems, invertible sheaves, Families, moduli of curves (algebraic) On plane curves with several singular points with high multiplicity | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Schlesinger equations \(S_{(n,m)}\) appear in the theory of isomonodromic deformations of Fuchsian systems. The monodromy matrices of the Fuchsian system
\[
\frac{d}{dz}=\sum_{i=1}^n \frac{A_k(u)}{z-u_k}\Phi,\qquad z\in \mathbb C \setminus\{u_1,\ldots,u_n\},
\]
do not depend on \(u=(u_1,\dots,u_n)\) if the \(m \times m\) matrices \(A_i(u)\) satisfy the equations
\[
\begin{aligned} \frac{\partial}{\partial u_j} A_i&= \frac{[A_i,A_j]}{u_i-u_j}, \quad i\neq j, \\ \frac{\partial}{\partial u_i} A_i&= - \sum_{i \neq j}\frac{[A_i,A_j]}{u_i-u_j}. \end{aligned}
\]
After a short review of the results characterizing the close connection of Schlesinger equations with Painlevé's and Garnier's equations, the authors note that the problem of construction and classification of solutions to the Painlevé equations remains open (for \(P_{\text{IV}}\)) although there are many results based on the theory of symmetries of the Painlevé equations. The last results give a geometric approach to study monodromy data of Fuchsian equations.
The object of the paper under review is the investigation of the symmetries of Schlesinger equations, i.e., of birational transformations acting in the space of Fuchsian systems that map solutions to solutions. One class of such symmetries, so called gauge transformations, is well known. Using as example the works [\textit{B. Malgrange}, Mathématique et physique, Sémin. Éc. Norm. Supér., Paris 1979--1982, Prog. Math. 37, 401--426 (1983; Zbl 0528.32017), \textit{Yu. I. Manin}, Sixth Painlevé equation, universal elliptic curve, and mirror of \(\mathbb{P}^2\). In: Khovanskij, A. (ed.) et al., Geometry of differential equations. Dedicated to V. I. Arnold on the occasion of his 60th birthday. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 186(39), 131--151 (1998; Zbl 0948.14025), \textit{D. Arinkin} and \textit{s. Lysenko}, Int. Math. Res. Not. 1997, No. 19, 983--999 (1997l Zbl 0918.14015), \textit{K. Okamoto} and \textit{H.Kimura}, Q. J. Math., Oxf. (2) 37, 61--80 (1986; Zbl 0597.35114), \textit{T. Tsuda}, Universal characters and integrable systems, PhD thesis, Tokyo Graduate School of Mathematics (2003)] where the authors used the Hamiltonian formulation of the corresponding equations and counting that some of their symmetries were generalized to the Schlesinger equations \(S_{(n,2)}\), the authors of the reviewed paper present a canonical Hamiltonian formulation of the Schlesinger equation \(S_{(n,m)}\) for all \(n, m\).
The first result is the Hamiltonian system in canonical form:
\[
\frac{\partial q_i}{\partial u_k}=\frac{\partial {\mathcal H}_k} {\partial p_i},\qquad \frac{\partial p_i}{\partial u_k}=-\frac{\partial {\mathcal H}_k} {\partial q_i},
\]
where \(q_1,\ldots,q_g\) are connected with apparent singularities arising when the Fuchsian system is transformed into the ordinary Fuchsian differential equation
\[
y^{(m)}=\sum_{i=1} ^{m-1} d_l(z)y^{(i)},
\]
and the corresponding \(p_i\) coordinates (conjugated momenta) are
\[
p_i=\text{Res}_{z=q_i}(d_{m-2}(z)+1/2 d_{m-1}(z)^2), \quad i=1,\ldots,g.
\]
The corresponding Hamiltonian is
\[
\mathcal H=\mathcal H_k(q,p,u)=-\text{Res}_{z=u_k}(d_{m-2}(z)+1/2 d_{m-1}(z)^2), \quad k=1,\ldots,n.
\]
Rational Darboux coordinated means that the elementary symmetric functions \(\sigma_1(q),\ldots,\sigma_g(q)\) and \(\sigma_1(p),\dots,\sigma_g(p)\) are rational functions of the coefficients of the system and the poles \(u_1,\ldots,u_n\). Moreover, there exist rational functions \(A_i=A_i(q,p)\), \(i=1,\ldots,n\), symmetric in \((q_1,p_1),\ldots,(q_g,p_g)\) with coefficients depending on \(u_1,\ldots,u_n\) and on the eigenvalues of the matrices \(A_i,\quad i=1,\ldots,n,\infty.\) All other Fuchsian systems with the same poles \(u_1,\ldots,u_n\) are obtained by simultaneous diagonal conjugation.
The authors note that the situation is more complex in the nongeneric case. The other result of the paper is the comparison of the isomonodromic Darboux coordinates with those obtained in the framework of the theory of algebro-geometrically integrable systems, so called spectral Darboux coordinates.
The structure of the paper is the following. In Section 2, the known relationship between the Schlesinger equations and isomonodromic deformations of Fuchsian systems are presented. In Section 3, the Hamiltonian formulation of Schlesinger equations is given. Besides, the formula for symplectic structure of Schlesinger equations is considered. This formula recently found by I. Krichever is useful for calculations with isomonodromic coordinates. In Section 4, the construction of the isomonodromic Darboux coordinates is given and a birational isomorphism between the space of Fuchsian systems considered modulo conjugations and the space of special Fuchsian differential equations is established. Section 5 is devoted to the semiclassical asymptotics of the isomonodromic Darboux coordinates via spectral Darboux coordinates (necessary material concerning Darboux coordinates is given in the appendix). In conclusion the results are applied to constructing nontrivial symmetries of Schlesinger equations. Schlesinger equations; Fuchsian systems; Darboux coordinates; Painlevé equations; Hamiltonian structure; apparent singularities; spectral coordinates; canonical transformations B. Dubrovin and M. Mazzocco, ''Canonical structure and symmetries of the Schlesinger equations,'' math.DG/0311261 (2003). Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation), Relationships between algebraic curves and integrable systems, Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain, Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Canonical structure and symmetries of the Schlesinger equations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this monograph is to establish a link between the theory of algebraic differential equations (ADE) with no movable singularity and the Galois theory of ADE's. In the one-dimensional case a modern treatment of the first mentioned theory was given by \textit{M. Matsuda} [First order algebraic differential equations (Lect. Notes Math. 804) (1980; Zbl 0447.12014)], while the second mentioned theory was given its final form in papers of Kolchin [e.g. \textit{E. R. Kolchin}, ''Differential algebra and algebraic groups'' (1973; Zbl 0264.12102)].
After giving same background in chapter I, in chapter II the author generalizes most of Matsuda's results to the higher dimensional case. Let K be a field of characteristic zero, \(\Delta\) a - not necessarily finite - set of differential operators of K, \(F| K\) a function field. If \(\Delta\) is finite then \(F| K\) has a \(\Delta\)-model; in general a \(\Delta\)-function field in said to have no movable singularity if it has a projective \(\Delta\)-model. \(F| K\) is called split if \(F=K(F^{\Delta})\). Such function fields have no movable singularity.
The two important resuls of chapter II are the following: Let K be an algebraically closed \(\Delta\)-field and V a projective variety over K. Then \(K^{\Delta}\) is a field of definition of V (theorem 1.1). From Kolchin's theory the author uses the concepts of weak and strong normality for \(\Delta\)-fields \(F| K:\) \(F| K\) is called weakly normal if \(F^{Gal_{\Delta}(F| K)}=K\) and strongly normal if in addition for any K-isomorphism \(\sigma\) of F into a \(\Delta\)-extension E of F both extensions \(F\cdot \sigma F/F\) and \(F\cdot \sigma F/\sigma F\) are split. Using theorem 1.1 the author shows (theorem 2.1 and corollary 2.2): If \(F| K\) has no movable singularity then there exists a \(\Delta\)-field extension \(K_ 1| K\) with \(K_ 1^{\Delta}| K^{\Delta}\) algebraic such that \(Q(F\otimes_ K K_ 1)/K_ 1\) is split. If moreover \(\Delta\) is finite and the operators in \(\Delta\) are pairwise commuting and K is algebraically closed then \(K_ 1| K\) can be chosen as a strongly normal extension.
In chapter III there are given the main results concerning the connections between the properties of being weakly normal (WN), strongly normal (SN) and having no movable singularity (NMS). Let \(\Delta\) be finite and the operators in \(\Delta\) pairwise commuting. Suppose K is algebraically closed and \(F^{\Delta}=K^{\Delta}\). Then \(F| K\) has no movable singularity iff there exists a Picard-Vessiot extension \(E| F\) such that \(E| K\) is strongly normal (theorem 2.1). Using this result, in theorem 3.1 it is proven that \(F| K\) is (SN) iff it is (WN) and (NMS). Here the (NMS)-condition can be removed in the following cases: \((1)\quad tr.\deg. F| K=1;\) (2) \(tr.\deg. F| K=2\) and \(\kappa(F| K)\geq 0\) (\(\kappa\) is the Kodaira dimension) (3) \(tr.\deg. F| K=q(F| K)\) and \(\kappa(F| K)\geq 0\) \((q(F| K)\) is the irregularity of \(K_ aF/K_ a\) where \(K_ a\) is an algebraic closure of K).
In chapter IV the author gives many comments showing that Kolchin's theory and the theory developed in this monograph have applications to classical situations. Strongly normal function fields lead to differential systems which can be linearized by means of abelian functions and \(\theta\)-functions - so this linearization property will also hold for function fields with no movable singularity. The author states that the algebraically complete integrable Hamiltonian systems - they can be integrated by \(\theta\)-functions - fit into his theory of function fields having (NMS). He shows this in the case of the Euler equations describing the movement of a rigid body with fixed gravity center. differential function field; algebraic differential equations; ADE; Galois theory; no movable singularity; Picard-Vessiot extension; Kolchin's theory; normal function fields; linearization property; complete integrable Hamiltonian systems; Euler equations Buium, A.: ''Differential function fields and moduli of algebraic varieties'', Lecture notes in math. 1226 (1986) Differential algebra, Algebraic functions and function fields in algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Abstract differential equations, Hamilton's equations, Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to field theory Differential function fields and moduli of algebraic varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main result of this paper concerns the positivity of the Hodge bundles of abelian varieties over global function fields. As applications, we obtain some partial results on the Tate-Shafarevich group and the Tate conjecture of surfaces over finite fields. Hodge bundle; ample vector bundle; function field; height; abelian scheme; Néron model; purely inseparable points; torsors; Tate-Shafarevich group; BSD conjecture; Tate conjecture Positive characteristic ground fields in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Global ground fields in algebraic geometry Positivity of Hodge bundles of abelian varieties over some function fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We describe a class of nonlinear transformations acting on many variables. These transformations have their origin in the theory of quantum integrability: they appear in the description of the symmetries of the Yang-Baxter equations and their higher dimensional generalizations. They are generated by involutions and act as birational mappings on various projective spaces. We present numerous figures, showing successive iterations of these mappings. The existence of algebraic invariants explains the aspect of these figures. We also study deformations of our transformations. iterated mappings; dynamical systems; Coxeter groups; birational transformations; Cremona transformations; inversion relations; Yang- Baxter equations; automorphisms of algebraic varieties; elliptic curves; resonant tori; Plücker variables; quantum integrability; involutions; birational mappings; deformations M.P. Bellon, J.-M. Maillard, C.-M. Viallet, Dynamical systems from quantum integrability, in: J.-M. Maillard (Ed.), Proceedings of the Conference Yang--Baxter Equations in Paris, World Scientific, Singapore, 1993, pp. 95--124, Int. J. Mod. Phys. Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems, Rational and birational maps, Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics, Automorphisms of surfaces and higher-dimensional varieties, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act Dynamical systems from quantum integrability | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this article, the authors prove a ''good'' solution to the Euler-Cramer problem (ECP). The problem is to find ''good'' conditions on a set of points Z in \({\mathbb{P}}^ 2\), which will ensure that Z is a (scheme- theoretic) complete intersection. The condition is in terms of the Cayley-Bacharach property. Let Z be a finite set of points in \({\mathbb{P}}^ 2\). Let \(\alpha =\alpha (Z)\) be the smallest degree of curves containing Z, and \(\delta=\delta(Z)=\) degree of Z=cardinality of Z. Then Z is a complete intersection if and only if \(\alpha^{-1}\delta \in {\mathbb{Z}}\) and Z has \(CB(\alpha +\alpha^{-1}\delta -3).\) The proofs are elementary and in spite of the technical appearance quite lucid. The main tool is the Hilbert function. Less satisfactory results are proved in the case when Z is not reduced. scheme-theoretic complete intersection; Euler-Cramer problem; Cayley- Bacharach property; Hilbert function E. Davis and P. Maroscia, Complete intersections in \( {{\mathbf{P}}^2}\), Proc. Conf. on Complete Intersections (Acireale, June 1983), Lecture Notes in Math., (in press). Complete intersections, Projective techniques in algebraic geometry Complete intersections in \({\mathbb{P}}^ 2:\) Cayley-Bacharach characterizations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 is concerned with the quaternionic KP hierarchy, which is equivalent to the Davey-Stewartson II hierarchy. This article studies its relationship with the theory of conformally immersed tori in the 4-sphere via quaternionic holomorphic geometry. The Sato-Segal-Wilson construction of KP solutions is adapted to this setting and the connection with quaternionic holomorphic curves is made. Then the author compares three different notions of ``spectral curve'': the QKP spectral curve; the Floquet multiplier spectral curve for the related Dirac operator; and the curve parameterising Darboux transforms of a conformal 2-torus in the 4-sphere. integrable systems; conformally immersed tori; quaternionic holomorphic curves; spectral curves McIntosh, I.: The quaternionic KP hierarchy and conformally immersed 2-tori in the 4-sphere. http://arxiv.org/abs/0902.3598 KdV equations (Korteweg-de Vries equations), Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relationships between algebraic curves and integrable systems, Conformal differential geometry, Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) The quaternionic KP hierarchy and conformally immersed 2-tori in the 4-sphere | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0551.00002.]
This note mostly summarizes known results on compressed algebras and on the punctual Hilbert scheme \({\mathcal H}=Hilb^ n({\mathbb{P}}^ r)\) (length n zero-dimensional subschemes of \({\mathbb{P}}^ r)\). If \(r<3\) or \(n<8\) we have \({\mathcal H}=\bar U\), where \({\mathcal U}={\mathcal U}(n,r)\) parametrizes non- singular subschemes of \({\mathcal H}\), i.e. sets of n distinct points in \({\mathbb{P}}^ r\). In general \({\mathcal H}\) may have several components. The author uses compressed algebras [cf. \textit{A. Iarrobino}, Trans. Am. Math. Soc. 285, 337-378 (1984; Zbl 0548.13009)] to show the existence of additional components of \({\mathcal H}\). A new result that is proved shows that compressed Gorenstein algebras have termwise maximal Hilbert function among Artin algebras of the same embedding dimension and the same socle degree. local Hilbert scheme; compressed algebras; punctual Hilbert scheme; maximal Hilbert function Iarrobino, A.: Compressed algebras and components of the punctual hubert scheme. Springer lecture notes #1124, 146-165 (1985) Parametrization (Chow and Hilbert schemes), Other special types of modules and ideals in commutative rings, Multiplicity theory and related topics Compressed algebras and components of the punctual Hilbert scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author defines the following function of sequences of matrices, which he calls deter-cube (to suggest something like `cubic determinant'). He states his definition in an informal way. A standard formal version of that definition is as follows. Given an \(n\times n\) matrix M, let \(M_{ij}\) be the (n-1)\(\times (n-1)\) matrix obtained from M deleting the ith row and the jth column. Let \(s=(M^{(1)},...,M^{(n)})\) be a sequence of n matrices of order n and let \(m_{ij}^{(k)}\) be the (i,j)-entry of \(M^{(k)}\). The deter-cube, say det(s), of s is defined inductively as follows: if \(n=1\), then the sequence s is just a number, m say, and we let \(\det (s)=m\) in this case, just as for determinants; if \(n>1\), then we set
\[
\det (M^{(1)},...,M^{(n)})=\sum^{n}_{i,j=1}(-1)^{i+j} m^{(1)}_{i,j} \det (M_{ij}^{(2)},...,M_{ij}^{(n)}).
\]
Properties similar to those of the usual determinant function hold for this deter-cube function. The author applies it to the study of functional dependences between linearly independent systems of quadrics, focusing on some examples. Actually, his method seems to work fairly well. function of sequences of matrices; deter-cube; cubic determinant; linearly independent systems of quadrics Determinants, permanents, traces, other special matrix functions, Projective analytic geometry, Projective techniques in algebraic geometry On the deter-cube and some of its applications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We offer short and conceptual re-proofs of some conjectures of Voevodsky's on the slice filtration. The original proofs were due to Marc Levine using the homotopy coniveau tower. Our new proofs use very different methods, namely, recent development in motivic infinite loop space theory together with the birational geometry of Hilbert schemes. algebraic \(K\)-theory; motivic cohomology; motivic spectral sequence; framed correspondences; Hilbert scheme Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Motivic cohomology; motivic homotopy theory Voevodsky's slice conjectures via Hilbert schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper the author computes the filling radius with rational coefficients of the complex projective \(n\)-space as \({1\over 2} \arccos(- {1\over 3})\) by a straightforward homological calculation using the Serre-spectral sequence and the Schubert calculus. He also computes the integer filling radius of the complex projective 2-space as \({1\over 2}\arccos(-{1\over 3})\) again and exhibits a torsion obstruction to filling complex projective 3-space. Serre-spectral sequence; Schubert calculus; filling radius Katz, M.: The rational filling radius of complex projective space. Topology appl. 42, 201-215 (1991) Differential geometry of symmetric spaces, Global differential geometry of Hermitian and Kählerian manifolds, Grassmannians, Schubert varieties, flag manifolds The rational filling radius of complex 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 \(u_{1},\dots, u_{k-1}, z \in {\mathbb C}^{g}\) and let \(D\) be a linear differential operator with respect to \(z\) with constant coefficients. Then \(D\) determines a \(k-\)linear operator \(\text{D}\) by the formula
\[
\text{D}(f_{1},\dots, f_{k})(u_{1},..., u_{k-1}) = D[f_{1}(u_{1}+z)\dots f_{k-1}(u_{k-1}+z) f_{k}((u_{1}+\cdots +u_{k-1}) - z)] \mid_{z=0}.
\]
An equation of the form \(\text{D}(f_{1},\dots , f_{k}) = 0\) is called a \(k-\)linear equation. When \(k=2,\) this construction yields familiar bilinear operators and Hirota's equations, which have important applications in the theory of integrable systems. The equation \(\text{D}(f_{1}, f_{2}, f_{3})(u,v)=0\) reduces to a polynomial relation between the logarithmic derivatives of the functions \(f_{1}(u), f_{2}(v)\) and \(f_{3}(u+v).\) The authors obtained these relations for functions on the Jacobian varieties of plane algebraic curves. Consider the operator
\[
Q(D)(f_{1}, f_{2}, f_{3})(u, v) = \frac{\text{D}(f_{1}, f_{2}, f_{3})(u, v)}{f_{1}(u), f_{2}(v), f_{3}(u + v)}.
\]
Theorem 3. Let \(\sigma\) be the sigma-function associated with the plane algebraic curve
\[
V = \{(x, y) \in {\mathbb C}^{2} : y^{n} - x^{s} - \Sigma_{ns-in-js}>0,\;\lambda_{ns-in-js} x^{i} y_{j} = 0\},
\]
where \((n, s) = 1.\) Then
\[
\sigma(u) \sigma(v) \sigma(u + v) R_{3g}(x, y) = c \psi^{-3}(x, y) \sigma(u + A(x, y)) \sigma(v + A(x, y)) \sigma(u + v - A(x, y)),
\]
where \(R_{3g}(x, y)\) is an entire function on \(V\) yielding the addition law on \(\text{Sym}^{g}(V),\) \(c\) is a constant, \(A : V \rightarrow \text{Sym}^{g}(V)\) is a Abel map and
\[
\psi(x, y) = \exp \Biggl\{- \int_{\infty}^{(x, y)} \langle A^{*}(x^{'}, y^{'}), \,d A(x^{'}, y^{'}) \rangle\Biggr\}.
\]
Here \(A^{*}\) is the map associated with \(A\) given by abelian integrals of the second kind.
Theorem 4. The smallest order of a linear differential operator \(D\) satisfying \(Q(D)(\sigma, \sigma, \sigma)(u, v) = 0\) is equal to \(g + 1,\) where \(g = \frac{(n - 1)(s - 1)}{2}\) is the genus of the curve \(V.\) linear differential operator; integrable systems; the sigma-function; Abel map Relationships between algebraic curves and integrable systems, Jacobians, Prym varieties, Generalized hypergeometric series, \({}_pF_q\), Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Trilinear functional equations. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We prove that the zero-dimensional superficial schemes \(Z\subset \mathbb {P}^3\) such that \(\deg (Z)\leq 3m\) and \(h^1(\mathcal {I}_Z(m))=0\) are only the expected ones (``superficial'' means that the Zariski tangent spaces have dimension \(\leq 2\)). postulation; zero-dimensional scheme; Hilbert function Projective techniques in algebraic geometry Superficial defective subschemes of \(\mathbb {P}^3\) | 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 Let \(W=m_1P_1+\cdots+m_sP_s\) be a scheme of fat points in \(\mathbb P^n_K\), where \(K\) is a field of characteristic zero.
The authors deal with the problem of computing the Hilbert polynomial of the modules of the Kähler differential \(k\)-forms of the coordinate ring of \(W\), \(\Omega^{k}_{R_{W/K}}\). After introducing notation and basic facts in Section 2, they show in Theorem 3.7 that the Hilbert polynomial of \(\Omega^{n+1}_{R_{W/K}}\) is \(\sum_j\binom{m_j+n-2}{n},\) i.e., it is the Hilbert polynomial of the coordinate ring of the fat points scheme \((m_1-1)P_1+\cdots+(m_s-1)P_s\).
This answers positively to a conjecture the authors stated in a previous paper, see Conjecture 5.7 in [\textit{M. Kreuzer} et al., J. Algebra 501, 255--284 (2018; Zbl 1388.13051)].
Moreover, making use of Theorem 3.7, the authors compute the Hilbert polynomial of the modules of the Kähler differential of a scheme of fat points in \(\mathbb P^2\), see Proposition 4.1. Hilbert function; fat point scheme; regularity index; Kähler differential module Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Modules of differentials, Cycles and subschemes Hilbert polynomials of Kähler differential modules for fat point schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In an earlier paper [\textit{R. Ragusa} and \textit{G. Zappalà}, Beitr. Algebra Geom.\ 44, No.1, 285--302 (2003; Zbl 1033.13004)], the authors introduced the notion of ``partial intersection schemes'' in projective space \(\mathbb P^r\). These schemes are \(c\)-codimensional, reduced, arithmetically Cohen-Macaulay unions of linear varieties, obtained by starting with a partially ordered subset of \(\mathbb N^c\) and carrying out a certain technical procedure. As the authors point out in the current paper (remark 1.7), their configurations are precisely the pseudo-liftings of Artinian monomial ideals, a special case of a construction by \textit{J. C. Migliore} and \textit{U. Nagel} [Commun.\ Algebra 28, No. 12, 5679--5701(2000; Zbl 1003.13005)]. Nevertheless, the authors' combinatorial approach provides a fresh and useful way of viewing these objects.
They first show that partial intersection schemes are not necessarily in the linkage class of a complete intersection (i.e.\ they are not necessarily licci). Then they give a large class of partial intersection schemes that nonetheless are licci. They complete the picture by showing that every partial intersection is in the Gorenstein linkage class of a complete intersection (i.e.\ glicci). This latter result had been shown earlier from the point of view of pseudo-liftings [\textit{J. C. Migliore} and \textit{U. Nagel}, Compos. Math.\ 133, No. 1, 25--36 (2002; Zbl 1047.14034)].
The last part of the paper gives interesting bounds and connections between the first and last graded Betti numbers of partial intersections, especially in codimension 3. A nice statement (among others) is that if \(X\) is a 3-codimensional partial intersection having \(p\) minimal last syzygies, and if \(\nu (I_X)\) is the number of minimal generators, then \(\lceil {{p+5} \over 2} \rceil \leq \nu(I_X) \leq 2p+1\). The authors also show that all possibilities in this range can occur. As they point out, such a bound is impossible in the case of arithmetically Cohen-Macaulay subschemes in general. Hilbert function; Betti numbers; liaison; arithmetically Cohen-Macaulay scheme; partial intersection schemes Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Syzygies, resolutions, complexes and commutative rings, Linkage On some properties of partial intersection 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 Building on their previous work, the authors compute the zeta-function of a monoidal scheme of finite type over \(\mathbb{F}_1\), the ``field of one element''; it turns out to be a rational function, satisfying a suitable functional equation. They then go on to study the \(\mathbb{F}_1\) zeta-function of a split reductive group over \(\mathbb{Z}\) (cf. the work of \textit{O. Lorscheid} [C. R., Math., Acad. Sci. Paris 348, No. 21--22, 1143--1146 (2010; Zbl 1221.11149)] for similar calculations). Some more general zeta-functions are defined and studied via an appropriate regularization procedure. field of one element; zeta-function; monoidal scheme; reductive group; regularized determinant; counting functions Deitmar, A.; Koyama, S.; Kurokawa, N., Counting and zeta functions over \(\mathbb{F}_1\), Abhandlungen math. sem. univ. hambg., 85, 59-71, (2015) Varieties over finite and local fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry, Linear algebraic groups over finite fields, Schemes and morphisms, Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) Counting and zeta functions over \({\mathbb F}_1\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We compute the Fourier coefficients of the weight one modular form \(\eta (z)\eta (2z)\eta (3z)/\eta (6z)\) in terms of the number of representations of an integer as a sum of two squares. We deduce a relation between this modular form and translates of the modular form \(\eta (z)^4/\eta (2z)^2\). In the last section we use our main result to give an elementary proof of an identity by \textit{V. G. Kac} [Adv. Math. 30, 85--136 (1978; Zbl 0391.17010)]. Dedekind eta function; eta products; Fourier coefficient; punctual Hilbert scheme Fourier coefficients of automorphic forms, Dedekind eta function, Dedekind sums, Parametrization (Chow and Hilbert schemes), Finite ground fields in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry The Fourier expansion of \(\eta (z)\eta (2z)\eta (3z)/\eta (6z)x\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper we develop a general method to prove independence of algebraic monodromy groups in compatible systems of representations, and we apply it to deduce independence results for compatible systems both in automorphic and in positive characteristic settings. In the abstract case, we prove an independence result for compatible systems of Lie-irreducible representations, from which we deduce an independence result for compatible systems admitting what we call a Lie-irreducible decomposition. In the case of geometric compatible systems of Galois representations arising from certain classes of automorphic forms, we prove the existence of a Lie-irreducible decomposition. From this we deduce an independence result. We conclude with the case of compatible systems of Galois representations over global function fields, for which we prove the existence of a Lie-irreducible decomposition, and we deduce an independence result. From this we also deduce an independence result for compatible systems of lisse sheaves on normal varieties over finite fields. compatible systems of Galois representations; independence of algebraic monodromy groups; automorphic compatible systems; compatible systems over global function fields Galois representations, Representation-theoretic methods; automorphic representations over local and global fields, Langlands-Weil conjectures, nonabelian class field theory, Structure of families (Picard-Lefschetz, monodromy, etc.), Positive characteristic ground fields in algebraic geometry Independence of algebraic monodromy groups in compatible 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 is a survey of results about bounds for the regularity index of fat points in \(\mathbb{P}^ n\) [mainly by \textit{M. V. Catalisano}, \textit{G. Valla} and the author, Proc. Am. Math. Soc. 118, No. 3, 717-724 (1993; Zbl 0787.14030), and \textit{G. Valla} and the author, Math. Z. 219, No. 2, 187-201 (1995)]. The question is the following: Let \(X = \{P_ 1, \dots, P_ s\}\) be a set of points in \(\mathbb{P}^ n_ k\), where \(k = \overline k\) and \(\text{char} k = 0\), and let \(p_ i\) be the homogeneous ideal, in \(R = k[x_ 0, \dots, x_ n]\), associated to \(P_ i\); consider for \(m_ 1 \geq \cdots \geq m_ s \geq 0\), the scheme \(Z\) associated to the ideal \(I = {\mathfrak p}_ 1^{m_ 1} \cap \cdots \cap {\mathfrak p}_ s^{m_ s}\); then, for every \(t \geq 0\), the vector space \(I_ t\) represents the space of hypersurfaces of \(\mathbb{P}^ n\) having a point of multiplicity at least \(m_ i\) at each \({\mathfrak p}_ i\). The multiplicity of \(Z\) is \(e(Z) = \sum_ i {m_ i + n - 1 \choose n}\), and its index of regularity \(r(Z)\) is the least integer \(g\) for which \(h_ A(t) = e(Z)\), where \(h_ A(t)\) is the Hilbert function of the coordinate ring \(A = R/I\) of \(Z\); in other words, \(r(Z)\) is the least integer \(t\) for which \(Z\) imposes \(e(Z)\) independent conditions on hypersurfaces of degree \(t\). Even for \(n = 2\), it is quite a hard problem to determine the value of \(r(Z)\) for a generic choice of \(X\) (it will only depend on the \(m_ i\)'s).
A (sharp) bound for \(r(Z)\), for any choice of \(X\) in general position (i.e. no \(n + 1\) points on a hyperplane), can be found thanks to an approach described in the paper under review: consider the ideal \(J = {\mathfrak p}_ 1^{m_ 1} \cap \cdots \cap {\mathfrak p}_{s - 1}^{m_{s - 1}}\), then \(r(Z) = \max \{m_ s - 1, r(R/J), r(R/J + {\mathfrak p}_ s^{m_ s})\}\). One can use this in order to work by induction (since \(r(Z)\) is known for small \(s)\), noting that \(r(R/(J + {\mathfrak p}_ s^{m_ s}))\) is the least \(t\) for which \([R/(J + {\mathfrak p}_ s^{m_ s})]_ t = 0\), and constructing a form \(F\) of degree \(t\) as small as possible with \(F \in J_ t\) and \(F \notin I_ t\). The bound which is found is a generalization of B. Segre's bound for \(n = 2\): \(r(Z) \leq \max \{m_ 1 + m_ 2 - 1, [(\sum m_ i + n - 2)/n]\}\).
The bound is sharp since it is attained if \(X\) lies on a rational normal curve; this is an ``if and only if'' when \(s \geq 2n + 3\) and \(m_ s\) is large enough. -- When \(X\) has the uniform position property, i.e. every two subsets of \(X\) with same cardinality have the same Hilbert function, better bounds can be found. linear systems; rational normal curves; bounds for the regularity index of fat points; Hilbert function of the coordinate ring Trung, N. V.: An algebraic approach to the regularity index of fat points in P2, Kodai math. J. 17, No. 3, 382-389 (1994) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series An algebraic approach to the regularity index of fat points in \(P^ n\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0741.00019.]
This is a survey article, and at the same time an interesting mathematical essay around the theme of hypergeometric functions, or the hyperplane configurations. It reveals how they are related to so many different topics of mathematics, old and new, such as Beta function, polylogarithms, Bloch-Wigner function, special value of Dedekind zeta- function, local systems, Gauss-Manin connection, cohomology of logarithmic differentials, determinant of matrix of period integrals, monodromy representation of Knizhnik-Zamolodchikov equation, braid groups, conformal field theory, quantum groups.
An elaborated reference list, to which the author himself made important contributions, is an important guide to further study on these subjects. hypergeometric functions; hyperplane; Beta function; polylogarithms; Bloch-Wigner function; local systems; Gauss-Manin connection; logarithmic differentials; monodromy; braid groups; quantum groups A. Varchenko, Multidimensional hypergeometric functions in conformal field theory, algebraic \(K\)-theory, algebraic geometry , Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, pp. 281-300. Research exposition (monographs, survey articles) pertaining to special functions, Generalized hypergeometric series, \({}_pF_q\), Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions), Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Multidimensional hypergeometric functions in conformal field theory, algebraic \(K\)-theory, 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 [For the entire collection see Zbl 0575.00008.]
Soient \(p_ 1,...,p_ n\) des points distincts du plan projectif, situés sur C= droite ou conique lisse. En étudiant les systèmes linéaires sur l'éclaté du plan en les points \(p_ 1,...,p_ n\), l'A. obtient un algorithme qui calcule la dimension du système linéaire des courbes de degré d passant par les points \(p_ 1,...,p_ n\) avec multiplicité \(m_ 1,...,m_ n\). Le résultat dépend seulement des entiers \(m_ 1,...,m_ n\). Pour C= droite, c'est là un résultat de \textit{E. D. Davis} et \textit{A. V. Geramita} [in Curves Semin. at Queen's, Vol. 3, Kingston/Can. 1983, Queen's Pap. Pure Appl. Math. 67, Exposé H (1984; Zbl 0597.13014)]. Avec certaines restrictions, il y a aussi des résultats dans le cas \(C= cubique\) intègre [l'A., Trans. Am. Math. Soc. 289, 213-226 (1985; Zbl 0609.14004)]. Ceci permet ensuite, pour des 0-cycles effectifs \(\sum m_ ip_ i du\) type ci-dessus, de déterminer en quel degré la ''fonction de Hilbert'' du 0-cycle se met à être égale au polynôme de Hilbert (qui est ici une constante). L'A. détermine enfin tous les 0-cycles du type ci-dessus dont la fonction de Hilbert est ''générique''. dimension of linear systems of curves; Hilbert polynomial; generic Hilbert function; rational surfaces; zero-cycles B. Harbourne, \textit{The geometry of rational surfaces and Hilbert functions of points in the} \textit{plane}, in: Proceedings of the 1984 Vancouver Conference in Algebraic Geometry, CMS Conf. Proc. 6, Amer. Math. Soc., Providence, RI, 1986, 95--111. [30] B. Harbourne, \textit{Free resolutions of fat point ideals on }P2, J. Pure Appl. Algebra 125 (1998), 213--234. Divisors, linear systems, invertible sheaves, Algebraic cycles, Rational and unirational varieties The geometry of rational surfaces and Hilbert functions of points in the plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(p\) be a smooth point of the integral projective variety \(X\). For all \(m>0\) let \(mp\) denote the closed subscheme of \(X\) with \((\mathcal {I}_p)^m\) as its ideal sheaf. A fat point scheme of \(X\) is a finite union of different fat points \(m_ip_i\). Key geometric problems were solved computing \(h^0(X,\mathcal{I}_{m_1p_1\cup \cdots \cup m_rp_r}\otimes L)\) for certain line bundles \(L\) on \(X\) and for general \((p_1,\dots ,p_r)\in X^r\). Since [\textit{A. Hirschowitz}, Manuscrip. Math. 50, 337--388 (1985; Zbl 0571.14002)] it was used taking a flat limit of fat point schemes and using the semicontinuity theorem for cohomology. Soon, \textit{J. Roé} [Trans. Amer. Math. Soc. 366, No. 2, 857--874 (2014; Zbl 1291.14019)] found stronger tools, essentially virtual limits, instead of a scheme which is a flat limit. But the problem to compute the flat limits, at least in some cases, is very interesting and this is the content of the paper under review.This is done for small multiplicities and small dimension. This is important for interpolation in \(\mathbb {P}^n\) and it gives some cases, \(r\le 15\) and \(d\ge 3m\) of a conjecture in [\textit{A. Laface} and \textit{L. Ugaglia}, Trans. Am. Math. Soc. 358, No. 12, 5485--5500 (2006; Zbl 1160.14003)] giving the exact value of \(h^0(\mathbb {P}^3,\mathcal {I}_Z(d))\), where \(Z\) is a general union of \(r\) fat points of multiplicity \(m\). complex algebraic geometry; fat points scheme; collisions of fat points; degeneration of linear systems; interpolation theory; Laface-Ugaglia conjecture Secant varieties, tensor rank, varieties of sums of powers, Projective techniques in algebraic geometry Collisions of fat points and applications to interpolation 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 gives the definition of the algebraic connective \(K\)-theory for quasi-projective schemes of finite type over the base field \(k\) as the \(D^2_{p,q}\) groups of the Brown-Gersten-Quillen spectral sequence. In topological case the connective \(K\)-theory shares many properties of singular homology and topological \(K\)-groups. The author shows that many properties of Chow groups and algebraic \(K\)-groups are shared by the connective algebraic \(K\)- groups. The author also proves that the Quillen \(K\)-groups are localizations of the connective \(K\)-groups at a distinguished element \(\beta\). connective \(K\)-theory; quasi-projective scheme; Brown-Gersten-Quillen spectral sequence Shuang Cai, Algebraic connective \?-theory and the niveau filtration, J. Pure Appl. Algebra 212 (2008), no. 7, 1695 -- 1715. Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory of schemes, Algebraic cycles Algebraic connective \(K\)-theory and the niveau 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 Multiparametric families of hypergeometric \(\tau\)-functions of KP or Toda type serve as generating functions for weighted Hurwitz numbers, providing weighted enumerations of branched covers of the Riemann sphere. The aim of this work is to provide an accessible introduction that summarizes the main results in this direction. All relevant constructions are indicated and the principal results stated. Some proofs are provided in detail; others are only sketched. The results include explicit parametrization of both the classical and quantum spectral curves; the system of recursion relations satisfied by dual pairs of adapted bases for the underlying space of formal power series in a spectral parameter; construction of suitably defined multipair correlators and multicurrent correlators, which serve as generating functions for weighted Hurwitz numbers; expressions for all relevant quantities as fermionic vacuum state expectation values; a finite rank expression for the Fredholm integral kernel associated with the pair correlator, analogous to the Christoffel-Darboux kernel appearing in the theory of orthogonal polynomials; and, finally, the topological recursion relations satisfied by the multiform weighted Hurwitz number generators. Hurwitz number; hypergeometric \(\tau\)-functions; Baker function; Christoffel-Darboux relation; spectral curve Alexandrov, A., Chapuy, G., Eynard, B., Harnad, J.: Weighted Hurwitz numbers and topological recursion: an overview. arXiv:1610.09408 Coverings of curves, fundamental group, Compact Riemann surfaces and uniformization, Exact enumeration problems, generating functions, Generalized hypergeometric series, \({}_pF_q\), Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Relationships between surfaces, higher-dimensional varieties, and physics, Other hypergeometric functions and integrals in several variables, Low-dimensional topology of special (e.g., branched) coverings Weighted Hurwitz numbers and topological recursion: an overview | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 algorithms for computing the cube of an ideal in an imaginary quadratic number field or function field. In addition to a version that computes a non-reduced output, we present a variation based on Shanks' NUCOMP algorithm that computes a reduced output and keeps the sizes of the intermediate operands small. Extensive numerical results are included demonstrating that in many cases our formulas, when combined with double base chains using binary and ternary exponents, lead to faster exponentiation. quadratic fields; quadratic function fields; hyperelliptic curves; ideal arithmetic; double base number systems IMBERT, L.-JACOBSON, M. J., JR.-SCHMIDT, A.: Fast ideal cubing in imaginary quadratic number and function fields, Adv. Math. Communications 4 (2010), 237-260. Cryptography, Special algebraic curves and curves of low genus, Computational aspects of algebraic curves Fast ideal cubing in imaginary quadratic number and function fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The possible Hilbert functions of reduced arithmetically Cohen-Macaulay (ACM) subschemes of projective space of any codimension is well known. What can be the Hilbert function of a reduced t and irreducible ACM subvariety of projective space? In codimension two this is again well-known. Answering this question for reduced and irreducible ACM subvarieties in codimension \(\geq 3\) is an open question of great interest. There is a strong connection with the Hilbert function of points in uniform position, due to Harris, but in codimension \(\geq 3\) it is not even known if the answers are the same, never mind a complete classification result in either case.
In codimension \(c\), the standard determinantal subschemes (i.e.\ the subschemes of codimension \(c\) defined by the maximal minors of a \(t \times (t+c-1)\) homogeneous matrix) are all ACM, and when \(c=2\) the notions of ACM and ``standard determinantal'' coincide. The authors here first prove some folklore facts about the Hilbert function of standard determinantal schemes of codimension \(c\), both in the general case and in the irreducible case.
The possible Hilbert functions of arithmetically Gorenstein (AG) subschemes of projective space are also not known, except in codimension \(\leq 3\); much less is it known what are the Hilbert functions of reduced, irreducible AG subschemes. A nice class of AG subschemes was studied by \textit{Kleppe} et al. [Gorenstein liaison, complete intersection liaison invariants and unobstructedness. Mem. Am. Math. Soc. 154 (2001; Zbl 1005.14018)], namely those that are twisted anticanonical divisors on an ACM subscheme satisfying certain mild local conditions. The authors of the present paper study when such divisors on a reduced, irreducible ACM subscheme \(S\) can again be reduced and irreducible, and then they describe the Hilbert functions that occur in terms of the Hilbert function of \(S\). Then as an application, they consider the case where \(S\) is a reduced, irreducible standard determinantal subscheme, and combine their folklore results with the latter results to obtain a large class of Hilbert functions that arise for reduced, irreducible AG subschemes of any codimension. Hilbert function; standard determinantal scheme; degree matrix; irreducible arithmetically Gorenstein scheme; divisor; regularity Budur, N.; Casanellas, M.; Gorla, E.: Hilbert functions of irreducible arithmetically Gorenstein schemes. J. algebra 272, No. 1, 292-310 (2004) Determinantal varieties, Divisors, linear systems, invertible sheaves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Hilbert functions of irreducible arithmetically Gorenstein schemes. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let R be a Noetherian ring. An R-sequence \(x_ 1,...,x_ n\) is built by always choosing \(x_ i\) to be not in the union of the primes in Ass R/(x\({}_ 1,...,x_{i-1})R\). In the terminology of this paper, \(A(I)=Ass R/I\) is a grade scheme and its associated grade function is \(f(I)=the\) (classical) grade of I. If instead, we take \(A(I)=\{P| P\quad is\quad an\) essential prime (respectively, asymptotic prime) of \(1\}\), and construct a sequence in the same way, we get the essential (respectively asymptotic) grade function. This paper studies the abstract nature of such grade schemes and their associated grade functions, in particular, finding which properties of the above examples are abstract in nature, and which are indigeneous to the given example.
The main theorem says that if f is a function from the set of all ideals in all localizations of R, to the set of nonnegative integers, then f is the grade function of some grade scheme A(I) if and only if f satisfies (i) f(I\({}_ S)=\min \{f(P_ P)| P_ S\) is
\[
a prime
\]
containing \(I/_ S\}\), (ii) f(P\({}_ P)\leq height P\) for all \(P\in Spec R\), and \((iii)\quad if\quad (Q,U)\) is a conforming pair in R, and if \(f(P_ P)\leq n\) for all \(P\in U\), then \(f(Q_ Q)\leq n-1\). - Here, (Q,U) is a conforming pair if \(Q\in Spec R\) and U is an infinite set of primes, each properly containing Q, such that if W is any infinite subset of U, then \(\cap \{P\in W\}=Q\). asymptotic grade; essential grade; Noetherian ring; grade function; grade scheme Commutative ring extensions and related topics, Commutative Noetherian rings and modules, Relevant commutative algebra Grade schemes and grade functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(Z\) be a union of \(r\) generic fat points with multiplicities \(m_1\), \dots,~\(m_r\) in \(\mathbb P^2_k\). One conjectured that the Hilbert function of~\(Z\) is
\[
H_Z(d) = \min\left( \binom{d+1}{2}, \sum_{i=1}^r \binom{m_i}{2} \right).
\]
If \(r \leq 9\), then this equation was proved and used for giving a counterexample to Hilbert's fourteenth problem by Nagata. In the present paper, the author proves the equation when the characteristic of~\(k\) is zero, \(r\) is a power of~\(4\) and \(m_1 = \dots = m_r\). Hilbert function; dimension of linear system; zero-dimensional scheme; Hilbert's fourteenth problem; generic fat points Evain, L., La fonction de Hilbert de la réunion de \(4^h\) gros points génériques de \(\mathbb P^2\) de même multiplicité, J. Algebraic Geom., 8, 4, 787-796, (1999) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series The Hilbert function of the union of \(4^h\) generic fat points of \(\mathbb{P}^2\) of the same multiplicity | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author initiates the application of the geometric Langlands program to the case of mirabolic vector bundles on a curve \(X\): that is, the author constructed an equivalence between the derived category of twisted \(D\)-modules on the moduli stack of vector bundles with mirabolic structure and the derived category of quasicoherent sheaves on a moduli stack of mirabolic local systems on \(X\). When \(X\) has the genus 1, this equivalence generically solves (in the sense of noncommutative geometry) the quantum Calogero-Moser system. integration by inverse spectral and scattering methods; algebraic geometry; mechanics of particles and systems Nevins, T., Mirabolic Langlands duality and the quantum Calogero-Moser system, Transform. Groups, 14, 931-983, (2009) Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems, Geometric Langlands program (algebro-geometric aspects), Relationships between algebraic curves and integrable systems, Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Mirabolic Langlands duality and the quantum Calogero-Moser system | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems 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 For a scheme \(X\) over a field of characteristic \(p\), \textit{L. Illusie} [Ann. Sci. Éc. Norm. Supér. (4) 12, 501--661 (1979; Zbl 0436.14007)] defined the so called de Rham Witt complex, a sheaf complex on \(X\), functorial in \(X\), which, if \(X\) is smooth and the base is perfect, computes the crystalline cohomology of \(X\) as its hypercohomology. Illusie's construction has subsequently been generalized into various directions; in particular, \textit{A. Langer} and \textit{T. Zink} [J. Inst. Math. Jussieu 3, No. 2, 231--314 (2004; Zbl 1100.14506)] constructed a relative de Rham Witt complex for morphisms \(X\to S\) where \(p\) is nilpotent on the \({\mathbb Z}_{(p)}\)-scheme \(S\).
In the present paper, this is generalized to fine log schemes \(X\) over fine log schemes \(S\) over \({\mathbb Z}_{(p)}\). Following the approach of Langer and Zink, the (log) de Rham Witt complex is constructed as the initial object of a certain category of log \(F-V\)-complexes. It is then shown that its hypercohomology computes the relative log crystalline cohomology of \(X\to S\) in cases where \(X/S\) is a relative semistable log scheme, or where \(X/S\) is a log scheme associated with a smooth scheme with a normal crossings divisor. Next, in these situations, generalizing constructions of \textit{A. Mokrane} [Duke Math. J. 72, No. 2, 301--337 (1993; Zbl 0834.14010)] and \textit{C. Nakayama} [Am. J. Math. 122, No. 4, 721--733 (2000; Zbl 1033.14012)], a weight spectral sequence for the crystalline cohomology is constructed. It is shown to degenerate modulo torsion at \(E_2\) when the base scheme is the spectrum of a (not necessarily perfect) field.
As already indicated, the approach taken here is inspired by the one suggested by Langer and Zink and thus differs from e.g. the one taken by \textit{O. Hyodo} and \textit{K. Kato} [in: Périodes \(p\)-adiques. Séminaire du Bures-sur-Yvette, France, 1988. Paris: Société Mathématique de France. 221--268 (1994; Zbl 0852.14004)] (resp. by Mokrane for the spectral sequence) in a more restricted log scheme setting. The key technical ingredient is the identification of a certain explicit bases of the (log) de Rham Witt complex in certain explicit cases; its elements are called log basic Witt differentials.
Finally, overconvergent de Rham Witt complexes for log schemes are introduced, generalizing the overconvergent de Rham Witt complexes for usual schemes introduced by Davis, Langer and Zink [\textit{C. Davis} et al., Ann. Sci. Éc. Norm. Supér. (4) 44, No. 2, 197--262 (2011; Zbl 1236.14025)]. More precisely, for a log scheme \(X\) arising from a smooth scheme together with a normal crossings divisor \(D\), an overconvergent de Rham Witt complex is constructed. It is shown that its hypercohomology computes the rigid cohomology of the open complement \(X-D\). De Rham Witt complex; crystalline cohomology; log scheme; weight spectral sequence; log basic Witt differentials; overconvergent de Rham Witt complex; rigid cohomology \(p\)-adic cohomology, crystalline cohomology, de Rham cohomology and algebraic geometry On relative and overconvergent de Rham-Witt cohomology for log schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a closed oriented surface, of genus \(g\geq 2\), and let \(K_X\) be its canonical line bundle (the holomorphic cotangent bundle of \(X\)). Given any reductive complex Lie group \(G\), a holomorphic Higgs \(G\)-bundle over \(X\) is a pair \((P,\theta)\) where \(P\) is a holomorphic principal \(G\)-bundle over \(X\) and \(\theta\) is a global holomorphic section of \(\mathrm{ad}(P)\otimes K_X\), where \(\mathrm{ad}(P)=P\times_G\mathfrak{g}\) is the adjoint bundle, \(G\) acting on its Lie algebra \(\mathfrak{g}\) by the adjoint respresentation. The section \(\theta\) is usually called the Higgs field.
Let \(\omega_{\mathrm{sy}}\) denote the standard symplectic form on \(\mathbb{C}^{2n}\). Let \(\mathrm{Gp}(2n,\mathbb{C})\) be the group of automorphisms \(T\in\mathrm{GL}(2n,\mathbb{C})\) of \(\mathbb{C}^{2n}\) such that there is some constant \(c_T\in\mathbb{C}\setminus\{0\}\) satisfying the identity \(\omega_{\mathrm{sy}}(T(v),T(w))=c_T\omega_{\mathrm{sy}}(v,w)\) for all \(v,w\in\mathbb{C}^{2n}\). Applying the above definition of Higgs \(G\)-bundle to the case of \(G=\mathrm{Gp}(2n,\mathbb{C})\), one concludes that, in terms of vector bundles, a Higgs \(\mathrm{Gp}(2n,\mathbb{C})\)-bundle over \(X\) is a quadruple \(((E,L,\varphi),\theta)\) where:
(1) \(E\) is a holomorphic vector bundle of rank \(2n\) over \(X\);
(2) \(L\) is a holomorphic line bundle over \(X\);
(3) \(\varphi\) is an \(L\)-valued symplectic form on \(E\) i.e. \(\varphi:E\otimes E\to L\) is a homomorphism which is symplectic on each fiber of \(E\);
(4) \(\theta\) is a holomorphic global section of \((\mathrm{Sym}^2(E)\oplus\mathrm{Id}_E)\otimes L^*\otimes K_X\).
The existence of the symplectic form \(\varphi\) implies that \(\mathrm{det}(E)\cong L^n\), so that the degree of \(E\) is a multiple of \(n\). Let \(d\) be a multiple of \(n\) and let \(\mathcal{M}_H(d)\) (resp. \(\mathcal{M}^s_H(d)\)) be the moduli space of semistable (resp. stable) Higgs \(\mathrm{Gp}(2n,\mathbb{C})\)-bundles over \(X\) such that \(\mathrm{deg}(E)=d\). Let also \(\mathcal{M}(d)\) (resp. \(\mathcal{M}^s(d)\)) be the moduli space of semistable (resp. stable) \(\mathrm{Gp}(2n,\mathbb{C})\)-bundles over \(X\) such that \(\mathrm{deg}(E)=d\). Then the total space of the cotangent bundle \(T^*\mathcal{M}^s(d)\) is embedded in \(\mathcal{M}_H(d)\) and, from the work of \textit{N. J. Hitchin} [Duke Math. J. 55, 91--114 (1987; Zbl 0627.14024)], this gives rise to a symplectic form \(\Omega_{\mathrm{Higgs}}\) on \(\mathcal{M}_H(d)\), extending the Liouville symplectic form on \(T^*\mathcal{M}^s(d)\).
On the other hand, if \(K_X\) denotes also the total space of the canonical bundle, let \(\Omega_{\mathrm{Li}}\) denote the Liouville symplectic form on \(K_X\). Then, if \(\mathrm{Hilb}^k(K_X)\) is the Hilbert scheme that parametrizes the \(0\)-dimensional subschemes of \(K_X\) of length \(k\), \(\Omega_{\mathrm{Li}}\) defines [see \textit{A. Beauville}, J. Diff. Geom. 18, 755--782 (1983; Zbl 0537.53056)] a holomorphic symplectic form on \(\mathrm{Hilb}^k(K_X)\), denoted by \(\Omega_{\mathrm{Li}}^k\).
The main result of this paper is a comparison between the symplectic forms \(\Omega_{\mathrm{Higgs}}\) and \(\Omega_{\mathrm{Li}}^k\).
This is done by considering the moduli space \(\mathcal{M}_T\) of objects of the form \((((E,L,\varphi),\theta),\sigma)\) where \(((E,L,\varphi),\theta)\) is a stable Higgs \(\mathrm{Gp}(2n,\mathbb{C})\)-bundle over \(X\), and \(\sigma\) is an element of the projective space \(\mathbb{P}H^0(X,E)\). Let
\[
\psi:\mathcal{M}_T\longrightarrow\mathcal{M}_H^s(d)
\]
defined by \(\psi(((E,L,\varphi),\theta),\sigma)=((E,L,\varphi),\theta)\).
There is a so-called Hitchin map \(\mathcal{H}:\mathcal{M}_H(d)\to\bigoplus_{i=0}^n H^0(X,K_X^{2i})\) which sends \(((E,L,\varphi),\theta)\) to the coefficents of the caracteristic polynomial of \(\theta\). Moreover, given \(\omega\in\bigoplus_{i=0}^n H^0(X,K_X^{2i})\), one can construct a curve \(Y_\omega\) inside the total space of \(K_X\) which is generically smooth, and such that the natural projection \(\phi_\omega:Y_\omega\to X\) is a finite map. See \textit{N. J. Hitchin} [Duke Math. J. 55, 91--114 (1987; Zbl 0627.14024)] and \textit{A. Beauville, M. S. Narasimhan} and \textit{S. Ramanan} [J. Reine Angew. Math. 398, 169--179 (1989; Zbl 0666.14015)] for the general theory on the Hitchin map. Furthermore, given \(((E,L,\varphi),\theta)\in\mathcal{H}^{-1}(\omega)\), there is a unique (up to isomorphism) torsion-free sheaf \(\mathcal{F}\) on \(Y_\omega\) such that \((\phi_\omega)_*\mathcal{F}=E\). The degree of such \(\mathcal{F}\) is \(d+2n(2n-1)(g-1)\). Since \(\phi_\omega\) is finite, one concludes that \(H^0(X,E)\cong H^0(Y_\omega,\mathcal{F})\). So, define the morphism
\[
\beta:\mathcal{M}_T\longrightarrow\mathrm{Hilb}^{d+2n(2n-1)(g-1)}(K_X)
\]
such that \(\beta(((E,L,\varphi),\theta),\sigma)=\mathrm{div}(\tilde\sigma)\), where \(\tilde\sigma\) is the section corresponding to \(\sigma\) under the isomorphism \(H^0(X,E)\cong H^0(Y_\omega,\mathcal{F})\).
The main theorem of this paper, states that
\[
\psi^*\Omega_{\mathrm{Higgs}}=\beta^*\Omega^{d+2n(2n-1)(g-1)}_{\mathrm{Li}}
\]
on \(\mathcal{M}_T\). Riemann surface; vector bundles; moduli spaces; Higgs Gp(2n; C)-bundles; Hilbert scheme; spectral curve; symplectic form Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, Global differential geometry of Hermitian and Kählerian manifolds, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) On moduli space of Higgs Gp\((2n, \mathbb C)\)-bundles over a Riemann surface | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author proves that if \(S\) is a Poisson surface, i.e. a smooth projective surface with a Poisson structure, then the Hilbert scheme of points of \(S\) has a natural Poisson structure, induced by the one of \(S\). This generalizes previous results obtained by \textit{A. Beauville} [J. Differ. Geom. 18, 755-782 (1983; Zbl 0537.53056)] and \textit{S. Mukai} [Invent. Math. 77, 101-116 (1984; Zbl 0565.14002)] in the symplectic case, i.e. when \(S\) is an Abelian or K3 surface. Finally, the author applies his results to give some examples of integrable Hamiltonian systems naturally defined on these Hilbert schemes. In the case \(S= \mathbb{P}^2\) he obtains a large class of integrable systems, which includes the ones studied by \textit{P. Vanhaecke} [Prog. Math. 145, 187-212 (1997; Zbl 0873.58038)]. Poisson surface; Hilbert scheme; Poisson structure; integrable Hamiltonian systems; Hilbert schemes F. BOTTACIN, Poisson structures on Hilbert schemes of points of a surface and integrable systems, Manuscripta Math., 97 (1998), pp. 517-527. Zbl0945.53049 MR1660136 Poisson manifolds; Poisson groupoids and algebroids, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Parametrization (Chow and Hilbert schemes), Special surfaces Poisson structures on Hilbert schemes of points of a surface and integrable 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 From author's abstract: ``Let \(f=(f_ 1,...,f_ k): ({\mathbb{C}}^{n+k},0)\to ({\mathbb{C}}^ k,0)\) be a germ of an analytic mapping such that \(V=\{z\in {\mathbb{C}}^{n+k}\); \(f_ 1(z)=...=f_ k(z)=0\}\) is nondegenerate complete intersection variety with an isolated singularity at the origin. We give a formula for the principal zeta-function of the monodromy of the Milnor fibration. As a corollary, we obtain a formula for the zeta-function of iterated hyperplane sections of a Milnor fibration of a nondegenerate analytic function.'' complete intersection; isolated singularity; zeta-function; monodromy; Milnor fibration; iterated hyperplane sections M. Oka, ''Principal Zeta-Function of Non-degenerate Complete Intersection Singularity,'' J. Fac. Sci. Univ. Tokyo, Sect. IA 37, 11--32 (1990). Local complex singularities, Complete intersections, Milnor fibration; relations with knot theory Principal zeta-function of non-degenerate complete intersection singularity | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For arithmetically Cohen-Macaulay (aCM) subschemes of \(\mathbb{P}^r\) of codimension \(c= 2\), the possible sets of graded Betti numbers have been completely classified [ see \textit{G. Campanella}, J. Algebra 101, 47--60 (1986; Zbl 0609.13001) and \textit{R. Maggioni} and \textit{A. Ragusa}, Matematiche 42, 195--209 (1987; Zbl 0701.14030)]. For aCM schemes of codimension \(c\geq 3\) with fixed Hilbert function, there is still a maximum for the graded Betti numbers, but not necessarily a minimum.
The authors develop a construction which allows them to obtain a large part of the possible sets of graded Betti numbers of aCM schemes with fixed Hilbert function. Their machinery is based on the concept of partial intersection subschemes of \(\mathbb{P}^r\). Those schemes are reduced, aCM, and unions of linear varieties similar to those used by \textit{J. Migliore} and \textit{U. Nagel} [ Commun. Algebra 28, No. 12, 5679--5701 (2000; Zbl 1003.13005)] and more general than the \(k\)-configurations used by \textit{A. V. Geramita}, \textit{T. Harima} and \textit{Y. S. Shin} [ Adv. Math. 152, No. 1, 78--119 (2000; Zbl 0965.13011)]. In codimension \(c=3\), the authors succeed in computing all graded Betti numbers in terms of certain combinatorial data used to construct the partial intersection scheme. In general codimensions \(c\geq 3\), they determine the Hilbert function, the degrees of the minimal generators of the vanishing ideal, and the degrees of the last syzygies in terms of those data. graded Betti number; arithmetically Cohen-Macaulay scheme; partial intersection subscheme; Hilbert function; generators of the vanishing ideal; syzygies Ragusa, A.; Zappalà, G.: Partial intersection and graded Betti numbers. Beitr. algebra geom. 44, No. 1, 285-302 (2003) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Partial intersections and graded Betti 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 In this paper we introduce the concept of a sign function of an imaginary quadratic field. As is proved in this article this concept is very helpful in the study of some arithmetical problems. Classically by sign we mean the extension to \(\mathbb R^\times\) of the continuous homomorphism \(s:\mathbb Q^\times\to \{-1,1\}\) satisfying \(s(-1)=-1\). Here \(A^\times\) is the multiplicative group of the ring \(A\). In 1985 \textit{D. R. Hayes} introduced the concept of a sign function of a global function field and used this notion to normalize Drinfeld modules of rank one [cf. Compos. Math. 55, 209--239 (1985; Zbl 0569.12008)]. The torsion points of these modules have many important arithmetical properties. They are essential in the construction of Stickelberger elements, Stark units, Euler systems, groups of cyclotomic units in characteristic \(p\), etc. To recall this definition we let \(K\) be a global function field. We denote by \(\infty\) a fixed place of \(K\), and by \(\widehat{K}\) the completion of \(K\) at \(\infty\). Let us also denote by \(\mathbb F_q\) both the finite field of \(q\) elements and the constant field of \(\widehat{K}\). Then a sign function, with respect to \((K,\infty)\), is a continuous homomorphism \(s:\widehat{K}^\times\to\mathbb F_q^\times\) satisfying \(s(a)=a\) for all \(a\in\mathbb F_q^\times\).
Our definition of a sign function in the case of an imaginary quadratic field \(k\subset\mathbb C\) is as follows. Let \(H\subset\mathbb C\) be the Hilbert class field of \(k\). Then a sign function of \(k\) is a surjective group homomorphism \(s:\Lambda_6\to\mu_H\) satisfying \(s({\mathcal U}_{12})=1\) and such that \(s(\xi)=\xi\) for all \(\xi\in\mu_k\) (see the notation below). As one may check the homomorphism \(\widetilde{\kappa}:\Lambda_6\to\mu_H\) induced by \(\kappa^{-1}\), where \(\kappa\) is the character defined by \textit{F. Hajir} and \textit{F. R. Villegas} [Duke Math. J. 90, No.~3, 495--521 (1997; Zbl 0898.11025)] satisfies all these properties. Hence \(\widetilde{\kappa}\) is a sign function of \(k\).
In Section 2 we associate to each couple \((s,{\mathfrak m})\), where \(s\) is a sign function and \({\mathfrak m}\) is a nonzero integral ideal of \(k\) prime to 6, a finite abelian extension \(k_{{\mathfrak m},s}\subset\mathbb C\) of \(k\). The field \(k_{{\mathfrak m},s}\) is well described by class field theory. In particular \(k_{{\mathfrak m},s}\) contains the ray class field modulo \({\mathfrak m}\), which we denote by \(k_{{\mathfrak m}}\). The extension \(k_{{\mathfrak m},s}/k_{{\mathfrak m}}\) is cyclic of degree \(w_H\) (resp. \(w_H/w_k\)) if \({\mathfrak m}\neq(1)\) (resp. \({\mathfrak m}=(1)\)). The properties of the ramification in the extension \(k_{{\mathfrak m},s}/H_s\), where \(H_s= k_{(1),s}\) lead us to consider \(k_{{\mathfrak m},s}\) as the analog of a cyclotomic number field or a cyclotomic function field as well.
In Section 3 we associate to each integral ideal \({\mathfrak c}\) of \(k\) prime to \(6N({\mathfrak m})\) an algebraic integer \(\Gamma_{{\mathfrak m}}({\mathfrak c})\), which is a root of the Ramachandra invariant. The construction of \(\Gamma_{{\mathfrak m}}({\mathfrak c})\) involves the Klein function and the eta function of Hajir-Villegas. First we describe the Galois action on \(\Gamma_{{\mathfrak m}}({\mathfrak c})\). This is essentially done by using the Shimura reciprocity law. In particular we prove that \(\Gamma_{{\mathfrak m}}({\mathfrak c})\in k_{m,s}\), where \(m= n({\mathfrak m})\) and \(s\) is the sign. Then we describe the behavior under the norm map of a certain power of \(\Gamma_{{\mathfrak m}}(1)\). In this we use the distribution law of the Siegel function stated in [\textit{D. S. Kubert}, Invent. Math. 117, No. 2, 227--273 (1994; Zbl 0834.14016), \S2]. The result we get is a refinement of the well-known Theorem 2 of \textit{G. Robert} [Bull. Soc. Math. Fr., Suppl., Mém. 36, 77 p. (1973; Zbl 0314.12006)] that gives the norm formulas satisfied by the Ramachandra invariants.
In Section 4 we define the level \({\mathfrak m}\) universal ordinary \(s\)-distribution \(U_s({\mathfrak m})\), in the spirit of those considered in Kubert, Anderson and Yin. We give the structure of \(U_s({\mathfrak m})\) as an abelian group and compute the Tate cohomology groups \(\widehat{H}^n(J,U_s({\mathfrak m}))\), where \(J=\text{Gal}(k_{{\mathfrak m},s} /k_{{\mathfrak m}})\). In this we follow the method of \textit{Y. Ouyang} [Proc. Am. Math. Soc. 130, No.~8, 2203--2213 (2002; Zbl 0997.11089)], which essentially uses Anderson's resolution and related spectral sequences. These cohomology groups naturally appear in many settings. See for instance Anderson's theory of epsilon extensions and its analog for function fields [\textit{G. W. Anderson}, Duke Math. J. 114, No.~3, 439--475 (2002; Zbl 1056.11060) and \textit{S. Bae} and \textit{L. Yin}, Manuscr. Math. 110, No.~3, 313--324 (2003; Zbl 1098.11058)]. Let us remark that \(U_s({\mathfrak m})\) is naturally a \(\text{Gal}(k_{{\mathfrak m}/s}/k)\)-module. Its Galois module structure is closely related to a certain group of elliptic units. This connexion will be made clear in a forthcoming paper in which we extend some results of Ouyang's paper (loc. cit.) to our case and use them to improve Theorem B of [\textit{H. Oukhaba}, Compos. Math. 137, No.~1, 1--22 (2003; Zbl 1045.11043)]. sign function; narrow ray class field; Shimura reciprocity law; ordinary \(s\)-distributions; Anderson's resolution; spectral sequences Oukhaba H. (2005). Sign functions of imaginary quadratic fields and applications. Ann. Inst. Fourier 55(3):753--772 Elliptic and modular units, Complex multiplication and abelian varieties, Other number fields Sign functions of imaginary quadratic fields and applications. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The ring \(G_{m,n}\) of m generic \(n\times n\) matrices over some algebraically closed field k of characteristic 0 is well-known to possess a classical division algebra of fractions \(D_{m,n}\). Since the center \(K_{m,n}\) of \(D_{m,n}\) is known to be purely transcendental over \(K_{2,n}\), the problem of (dis)proving that \(K_{m,n}\) is purely transcendental over k reduces to the case \(m=2.\)
The author presents the following geometric interpretation of \(K_{2,n}\). For any pair of positive integers (n,d), denote by \(Q_{n,d}\) the variety parametrizing the couples (C,D), where \(C\subseteq {\mathbb{P}}^ 2_ k\) is a curve of degree n and where D is a divisor of degree d of C \((Q_{n,d}\) is essentially the Picard scheme over a generic plane curve). The main result of the paper then says that \(K_{2,n}\) is the function field of \(Q_{n,n(n-1)/2}\). As an application, it is then shown that \(K_{2,3}\) is rational, by proving that the variety \(Q_{3,3}\) is rational. (The rationality of \(K_{2,n}\) has been proved by E. Formanek for \(n=2,3,4\).) generic n\(\times n\) matrices; classical division algebra of fractions; center; purely transcendental; Picard scheme; generic plane curve; function field; variety; rationality Van den Bergh M., The center of the generic division algebra, J. Algebra, 1989, 127(1), 106--126 Infinite-dimensional and general division rings, Center, normalizer (invariant elements) (associative rings and algebras), Transcendental field extensions, Algebraic functions and function fields in algebraic geometry, Picard schemes, higher Jacobians The center of the generic division algebra | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of the article is to provide an elementary introduction to polyhedral methods for understanding and solving systems of multivariate polynomial equations. The author does not assume a background in algebraic geometry. His treatment culminates at a self-contained proof of an extended version of Bernstein's Theorem relating volumes of polytopes with the number of connected components of the complex zero set of a system of polynomial equations. For low dimensions the author also gives a complexity bound, namely for computing mixed area and thereby the generic number of complex roots of two polynomial equations in the plane. systems of multivariate polynomial equations; Bernstein's Theorem; Kushnirenko's Theorem; A-discriminant; resultant; mixed subdivision; mixed volume; toric varieties; amoeba theory; polyhedral homotopy. J.M. Rojas, Why polyhedra matter in non-linear equation solving, in Proceedings of the Conference on Algebraic Geometry and Geometric Modelling, Vilnius, Lithuania, 29 July--2 August 2002. Contemp. Math., vol. 334 (AMS, Providence, 2003), pp. 293--320. Computational aspects of field theory and polynomials, Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Numerical computation of solutions to systems of equations, Symbolic computation and algebraic computation Why polyhedra matter in nonlinear equation solving | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Following the approach in the book [Commutative algebra. With a view toward algebraic geometry. Berlin: Springer-Verlag (1995; Zbl 0819.13001)], by \textit{D. Eisenbud}, where the author describes the generic initial ideal by means of a suitable total order on the terms of an exterior power, we introduce first the \textit{generic initial extensor} of a subset of a Grassmannian and then the \textit{double-generic initial ideal} of a so-called \textit{GL-stable subset} of a Hilbert scheme. We discuss the features of these new notions and introduce also a partial order which gives another useful description of them. The double-generic initial ideals turn out to be the appropriate points to understand some geometric properties of a Hilbert scheme: they provide a necessary condition for a Borel ideal to correspond to a point of a given irreducible component, lower bounds for the number of irreducible components in a Hilbert scheme and the maximal Hilbert function in every irreducible component. Moreover, we prove that every isolated component having a smooth double-generic initial ideal is rational. As a by-product, we prove that the Cohen-Macaulay locus of the Hilbert scheme parameterizing subschemes of codimension 2 is the union of open subsets isomorphic to affine spaces. This improves results by \textit{J. Fogarty} [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)] and \textit{R. Treger} [J. Algebra 125, No. 1, 58--65 (1989; Zbl 0705.14047)]. generic initial ideal; extensor; Hilbert scheme; irreducible component; maximal Hilbert function; rationality Parametrization (Chow and Hilbert schemes), Computational aspects of higher-dimensional varieties, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Rationality questions in algebraic geometry, Exterior algebra, Grassmann algebras Double-generic initial ideal and Hilbert scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a finite group scheme, the subadditive functions on finite-dimensional representations are studied. It is shown that the projective variety of the cohomology ring can be recovered from the equivalence classes of subadditive functions. Using Crawley-Boevey's correspondence between subadditive functions and endofinite modules, we obtain an equivalence relation on the set of point modules introduced in our joint work with \textit{S. B. Iyengar} and \textit{J. Pevtsova} [J. Am. Math. Soc. 31, No. 1, 265--302 (2018; Zbl 1486.16011)]. This corresponds to the equivalence relation on \(\pi \)-points introduced by \textit{E. M. Friedlander} and \textit{J. Pevtsova} [Duke Math. J. 139, No. 2, 317--368 (2007; Zbl 1128.20031)]. subadditive function; endofinite module; stable module category; finite group scheme Group schemes, Cohomology of groups, Representations of associative Artinian rings, Modular representations and characters, Cohomology theory for linear algebraic groups The variety of subadditive functions for finite group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems An analogue of the correspondence between \(GL(k)\)-conjugacy classes of matricial polynomials and line bundles is given for \(K\)-conjugacy classes, where \(K \subset GL(k)\) is one of the following: maximal parabolic, maximal torus, \(GL(k - 1)\) embedded diagonally. The generalised Legendre transform construction of hyperkähler metrics is studied further, showing that many known hyperkähler metrics (including the ones on coadjoint orbits) arise in this way, and giving a large class of new (pseudo-)hyperkähler metrics, analogous to monopole metrics. spectral curves; hyperkähler metrics; integrable systems; magnetic monopoles; spectral curves; hyperkähler metrics; integrable systems; magnetic monopoles Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Twistor methods in differential geometry, Relationships between algebraic curves and integrable systems Line bundles on spectral curves and the generalised Legendre transform construction of hyperkähler metrics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems People who love Jacobi matrices will be excited as well as puzzled by the presented construction. And there definitely exist quite a few, even if only for the close connection between Jacobi matrices and physics. For the rest of the population, the text is probably too thick. Still, its reading is of its own reward: its relationship to the classical moment problem offers the key sample reason, and for me it was its character of the generalization of the old relationship of Jacobi matrices to continued fractions and the so-called Titchmarsh-Weyl \(m\)-functions.
An arbitrary prescribed function \(f(z)\) is assumed equal to the \((i,j)\)'s Green's function of some Jacobi matrix \(J\) which is to be re-constructed. The core of the paper lies in the proposal of a geometric arrangement of the information about the solution set: In an overall setting using the language of orthogonal polynomials and probability distribution functions (p.d.f.s), the first surprise comes at the very start: a non-diagonal generalization of the problem is considered -- for the first time, as far as I know (and the author claims). In fact, the absolute value of the difference of indices \(i\) and \(j\) is an invariant of the solution set, and it makes the difference if it vanishes (the solution -- re-derived here -- was already known) or not (the results are new and qualitatively different).
In order to factor the map of \(J\) on \(f\), the paper describes, roughly speaking, a bijective parametrization of p.d.f.s and subdeterminants \(q\) of \(J\). This characterizes the solution set, in geometric language, as a fibration over a (connected) coordinate base, and opens the way towards constructions via the intermediate, so called auxiliary polynomial of a solution. In this formulation, the inverse problem is reduced to the construction of the fibres which makes use of the properties of roots. As a result, every output of the construction is shown to be a solution (the existence of which is assumed), and one becomes free to generate formulas for solutions.
People who still hesitate may be definitely persuaded and impressed by an explicit and illustrative example on p. 4746. Jacobi matrices; Titchmarsh-Weyl \(m\)-function; orthogonal polynomials; inverse spectral theory; Green function; probability distributions; factorization of the inverse problem; parametrization of the fibres; singular solutions; explicit formulas DOI: 10.1090/S0002-9947-02-03078-7 Jacobi (tridiagonal) operators (matrices) and generalizations, Inverse problems in linear algebra, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Inverse problems for functional-differential equations Inverse spectral theory of finite Jacobi matrices | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main goal of this article under review is to extend results on the Cayley-Bacharach properties of \(K\)-rational points in \(\mathbb P^n_k\) to arbitrary 0-dimensional subschemes over an arbitrary field \(K\) using the technique of liaison theory.
The Cayley-Bacharach property of a set of \(K\)-points \(Z\) in \(\mathbb P^n_k\) is well-studied.
Among others cited in the paper, [\textit{P. Griffiths} and \textit{J. Harris}, Ann. Math. (2) 108, 461--505 (1978; Zbl 0423.14001); \textit{D. Eisenbud} et al., Bull. Am. Math. Soc., New Ser. 33, No. 3, 295--324 (1996; Zbl 0871.14024)] are important references.
We say \(Z \subset \mathbb{P}^n\) satisfies the Cayley-Bacharach property with respect to the linear system \(|\mathcal O(l)|\) (abbreviated \(CBP(l)\)) if whenever a divisor \(D\) in \(|\mathcal O(l)|\) contains a co-length one subscheme of \(Z\), it must contain all of \(Z\).
This definition is subtly extended for arbitrary \(0\)-dimensional subschemes \(Z\) over an arbitrary field \(K\) using the notion of ``maximal \(p_j\) subschemes''.
The regularity index \(r_Z\) of \(Z\) is the minimal degree after which the Hilbert function and the Hilbert polynomial of \(Z\) coincide.
The scheme \(Z\) is said to be a Cayley-Bacharach scheme if it satisfies \(CBP(r_Z-1)\), which is the highest possible degree \(d\) such that \(Z\) can satisfy \(CBP(d)\).
It is known that for a set of \(K\)-points \(Z\), being a Cayley-Bacharach scheme is equivalent to the condition b) where a generic element of the least degree in the ideal of its link in a complete intersection does not vanish anywhere on \(Z\).
The main result of this paper is Theorem 3.5 which characterizes the property of being a Cayley-Bacharach scheme in terms of different conditions on its link.
The authors show that condition b) is only sufficient but necessary for \(Z\) to be a Cayley-Bacharach scheme in the general setting.
Last but not least, the authors characterize the Cayley-Bacharach property
of degree \(d\) using the canonical module of \(Z\), and use this result to give an equivalent condition for \(Z\) to be arithmetic Gorenstein in terms of the Hilbert function of the Dedekind different of \(Z\). zero-dimensional scheme; Cayley-Bacharach property; Hilbert function; liaison theory; Dedekind different Linkage, complete intersections and determinantal ideals, Linkage, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Projective techniques in algebraic geometry An application of liaison theory to zero-dimensional schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors compute the spectral action on noncommutative tori with the help of zeta functions. We briefly recall the important notion of spectral action in noncommutative geometry. Roughly speaking, for a given spectral triple \(( \mathcal{A},\mathcal{H},\mathcal{D}) \) where \( \mathcal{A}\) is an algebra acting on a Hilbert space \(\mathcal{H}\) and \( \mathcal{D}\) is a Dirac-like operator, the spectral action \(\mathcal{S} ( \mathcal{D}_{A},\Phi ,\Lambda ) :=\text{Tr}( \Phi ( \mathcal{D}_{A}/\Lambda ) ) \) of the covariant Dirac operator \( \mathcal{D}_{A}:=\mathcal{D}+A+\varepsilon JAJ^{-1}\) for a 1-form \( A=\sum_{i}a_{i}[ \mathcal{D},b_{i}] \) with \(a_{i},b_{i}\in \mathcal{A}\), an even positive cut-off function \(\Phi \), and the mass scale \( \Lambda \), counts the spectral values of \(| \mathcal{D}_{A}| \) less than \(\Lambda \), where \(J\) is a real structure with \(J\mathcal{D} =\varepsilon \mathcal{D}J\) for \(\varepsilon \in \{ \pm 1\} \).
The smooth noncommutative \(n\)-torus \(C^{\infty }( \mathbb{T}_{\Theta }^{n}) \) determined by a nonzero skew-symmetric matrix \(\Theta \in M_{n}( \mathbb{R}) \) consists of \(\sum_{k\in \mathbb{Z}^{n}}a_{k}U_{k}\) with a trace \(\tau :\sum_{k\in \mathbb{Z}^{n}}a_{k}U_{k}\mapsto a_{0}\), where \( k\mapsto a_{k}\) is a Schwartz function on \(\mathbb{Z}^{n}\) and \(U_{k}:=e^{- \frac{i}{2}k\cdot \chi k}u_{1}^{k_{1}}\cdots u_{n}^{k_{n}}\) for unitary generators \(u_{i}\), \(1\leq i\leq n\), satisfying \(u_{i}u_{j}=e^{i\Theta _{ij}}u_{j}u_{i}\) and the upper triangular restriction \(\chi \) of \(\Theta \). A matrix \(\Theta \in M_{n}( \mathbb{R}) \) is called diophantine if \( v:=^{t}\Theta ( u) \) is a diophantine vector for some \(u\in \mathbb{Z }^{n}\), i.e., there are \(\delta ,c>0\) such that \(| w\cdot v-m| \geq c| w| ^{-\delta }\) for all \(0\neq w\in \mathbb{Z}^{n}\) and \( m\in \mathbb{Z}\).
For \(\mathcal{A}=C^{\infty }(\mathbb{T}_{\Theta}^{n})\)
with \(\frac{1}{2\pi}\Theta\) real skew-symmetric diophantine
and a selfadjoint 1-form \(A\), the authors show that
\[
\begin{aligned}
\mathcal{S}(\mathcal{D}_{A},\Phi,\Lambda) &=
\begin{cases} 4\pi\Phi_{2}\Lambda^{2}+\mathcal{O}(\Lambda^{-2}) \quad&\text{for }n=2,\\
8\pi^{2}\Phi_{4}\Lambda^{4}-\frac{4\pi ^{2}}{3}\Phi(0)
\tau(F_{\mu\nu}F^{\mu\nu})+\mathcal{O}(\Lambda^{-2}) \quad&\text{for }n=4,\end{cases}\\
\text{and in general,} &=\sum_{k=0}^{n}\Phi _{n-k}c_{n-k}( A) \Lambda
^{n-k}+\mathcal{O}( \Lambda ^{-1}) \\ &\qquad\qquad\text{with }c_{n-2}( A) =0=c_{n-k}( A)
\text{ for }k\text{ odd}, \\ &\qquad\qquad\qquad\text{where }\Phi _{k}:=\frac{1}{2} \int_{0}^{\infty }\Phi ( t) t^{\frac{k}{2}-1}\,dt.\end{aligned}
\]
Furthermore, when \(n\) is odd, \(\zeta _{D_{A}}(
0) -\zeta _{D}( 0) =0 \) for \(D:=\mathcal{D}+P_{0}\) and
\(D_{A}:=\mathcal{D}_{A}+P_{A}\), where \(P_{0} \) and \(P_{A}\) are the
projections onto \(\ker ( \mathcal{D}) \) and \(\ker ( \mathcal{D}_{A}) \),
respectively, and \(\zeta_{X}( s) :=\text{Tr}(| X|^{-s})\). spectral action; spectral triple; noncommutative torus; zeta function; holomorphic continuation; Diophantine number; noncommutative integration; residue; covariant Dirac operator; noncommutative geometry D. Essouabri, B. Iochum, C. Levy, and A. Sitarz, Spectral action on noncommutative torus. J. Noncommut. Geom. 2 (2008), 53-123. Noncommutative differential geometry, Noncommutative geometry (à la Connes), Topological algebras of operators, Noncommutative function spaces, Noncommutative measure and integration, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Spectral action on noncommutative torus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Fix a linear subspace \(V\subseteq\mathbb{P}^n\) and a linearly independent set \(S\subset V\). Let \(Z_{S,V}\subset V\) or \(Z_{s,r}\) with \(r:=\dim(V)\) and \(s=\sharp(S)\), be the zero-dimensional subscheme of \(V\) union of all double points \(2p,p\in S\), of \(V\) (not of \(\mathbb{P}^n\) if \(n>r\)). We study the Hilbert function of \(Z_{S,V}\) and of general unions in \(\mathbb{P}^n\) of these schemes. In characteristic 0 we determine the Hilbert function of general unions of \(Z_{2,1}\) (easy), of \(Z_{2,2}\) and, if \(n=3\), general unions of schemes \(Z_{3,2}\) and \(Z_{2,2}\). zero-dimensional scheme; Hilbert function; postulation Projective techniques in algebraic geometry Zero-dimensional subschemes of projective spaces related to double points of linear subspaces and to fattening directions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this paper is to study the rational Picard group of the projectivized moduli space \(P\overline{\mathcal{M}}_g^{(n)}\) of holomorphic \(n\)-differentials on complex genus \(g\) stable curves. The authors define \(n-1\) natural classes in this Picard group that they call Prym-Tyurin classes. They express these classes as linear combinations of boundary divisors and the divisor of \(n\)-differentials with a double zero. Two different proofs of this result are given, using two alternative approaches: an analytic approach that involves the Bergman tau function and its vanishing divisor and an algebro-geometric approach that involves cohomological computations on the universal curve. This paper is organized as follows: The first chapter is an introduction to the subject. The chapter 2 deals with the space of admissible \(n\)-differentials. The chapter 3 is devoted to the Bergman tau function and Hodge class on \(P\overline{\mathcal{M}}_g^{(n)}\). In chapter 4 the authors explain an alternative computation of \(\delta_{\operatorname{deg}}\) in \(\mathrm{Pic}(P\overline{\mathcal{M}}_g^{(n)})\). Chapter 5 deals with Prym-Tyurin differentials on \(\widehat{C}\) and holomorphic \(n\)-differentials on on the base Riemann surface \(C\). moduli space of curves; \(n\)-differentials; cyclic covers; Bergman tau function; integrable systems Families, moduli of curves (analytic), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Relationships between algebraic curves and integrable systems, Differentials on Riemann surfaces, Picard groups Tau functions, Prym-Tyurin classes and loci of degenerate differentials | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(G_ T\) be the family of all zero-dimensional homogeneous ideals of \(\mathbb{C} [x,y]\) that have a fixed Hilbert function \(T\). \(G_ T\) is a smooth irreducible algebraic variety and a closed subscheme of the Hilbert scheme parametrizing the zero-dimensional subschemes of the complex plane supported by the origin [\textit{A. A. Iarrobino}, ``Punctual Hilbert schemes'', Mem. Am. Math. Soc. 188 (1977; Zbl 0355.14001)]. There is a cellular decomposition of \(G_ T\) introduced by \textit{L. Göttsche} [Manuscr. Math. 66, No. 3, 253-259 (1990; Zbl 0714.14004)]. The author recovers such a cellular decomposition using ``escaliers'', and studies incidence between cells. Hilbert function; Hilbert scheme Yameogo, J.: Décomposition cellulaire de variétés paramétrant des idéaux homogènes de c\?x,y\?. Incidence des cellules. I. Compositio math. 90, No. 1, 81-98 (1994) Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Formal neighborhoods in algebraic geometry Cellular decomposition of varieties parametrizing homogeneous ideals of \(\mathbb{C}[[x,y]]\). Incidence of cells. I | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 For an arithmetically Cohen-Macaulay closed subscheme \(X\) in \(\mathbb{P}^n\) with the saturated defining ideal \(I_X \subset k[x_0, \dots, x_n]\), let \(\text{gin}(I_X)\) be the generic initial ideal under the reverse lexicographic order. In this article, we find the minimal system of generators of \(\text{gin}(I_X)\) in terms of characters \(f_i\)'s, which will be defined in (3.1). Then we obtain the formula for \(\deg(X)\) in terms of characters. For a curve \(C\) in \(\mathbb{P}^n\), we get the formula for the arithmetic genus in terms of characters of a general hyperplane section and the number of sporadic zeros.
As an application, we give a new proof of the following upper bound on the regularity given by \textit{J. Ahn} and \textit{J. C. Migliore} [J. Pure Appl. Algebra 209, No. 2, 337--360 (2007; Zbl 1128.13011)] and \textit{U. Nagel} [Math. Nachr. 142, 27--43 (1989; Zbl 0688.13008)]:
If \(X \subset \mathbb{P}^n\) is an arithmetically Cohen-Macaulay closed subscheme, then
\[
\text{reg}(I_X)\leq \deg(X)-{{s-1+\text{codim}(X)}\choose{s-1}} +s,
\]
where \(s = \{ l | (I_X)l_l \neq 0\}\). Furthermore we give an equivalent condition for an arithmetically Cohen-Macaulay closed subscheme \(X\subset \mathbb{P}^n\) to satisfy the equality from the view point of the generic initial ideal of \(I_X\), this shows that the upper bound is sharp. arithmetically Cohen-Macaulay scheme; arithmetic genus; Borel-fixed monomial ideal; degree; generic initial idea; Hilbert function; minimal system of generators; regularity Cho, H.M.; Cho, Y.H.; Park, J.P., Generic initial ideals of arithmetically Cohen-Macaulay projective subschemes, Commun. Algebra, 35, 2281-2297, (2007) Syzygies, resolutions, complexes and commutative rings, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Generic initial ideals of arithmetically Cohen-Macaulay projective subschemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems There is an extensive research on the Hilbert scheme of points on a variety, and this article gives a deformation theoretic study of such in some particular cases.
For a variety \(V\) with \(\dim V\geq 3\), the first complication is that the Hilbert scheme \(V^{[n]}\) of \(n\) points in \(V\) is not smooth unless \(n=2,3\). This article focuses on the case \(n=2\), in which \(V^{[2]}\) is called a Hilbert square, and generalize the surface case to higher dimensions.
\textit{N. Hitchin} [Mosc. Math. J. 12, No. 3, 567--591 (2012; Zbl 1267.32010)] showed that for every smooth projective surface \(S\) with \(H^1(S,\mathcal O_S)=0\) there is a short exact sequence \[ 0\rightarrow H^1(S,T_S)\rightarrow H^1(S^{[n]},T_{S^{[n]}})\rightarrow H^0(S,\omega_S^\vee)\rightarrow 0 \] which allows him to interpret the deformation theory of \(S^{[n]}\) in terms of the deformation theory and the Poisson geometry of \(S.\)
The main idea in this article comes from the observation that the above sequence can be recovered more abstractly using derived categories and Hochschild cohomology. This follows from a recent result due to [\textit{A. Krug} and \textit{P. Sosna}, Sel. Math., New Ser. 21, No. 4, 1339--1360 (2015; Zbl 1331.18015)] that says that for a smooth projective surface \(S\) with \(H^i(S,\mathcal O_S)=0,\;i\geq 1\), i.e. \(\mathcal O_S\) is exceptional, the Fourier-Mukai functor \(\Phi_{\mathcal I}:\mathbf{D}^b(S)\rightarrow\mathbf{D}^b(S^{[n]})\) is fully faithful. Here, \(\mathcal I\) is the ideal sheaf of the universal family \(Z\hookrightarrow S\times S^{[n]}\). It then follows that there is an isomorphism \(\operatorname{HH}^\bullet(S)\cong\operatorname{Ext}^\bullet_{S\times S^{[n]}}(\mathcal I,\mathcal I).\) The article explicitly explains that the degeneracy of the relative local-to-global Ext spectral sequence for the second projection \(S\times S^{[n]}\) computing the right-hand side of the isomorphism recovers the exact sequence above. Thus Hitchin's deformation-theoretic results for surfaces have an analogue in higher dimensions, and the authors generalize the results of Krug-Sosna. The first main result in this direction says that for a smooth projective variety \(X\) with \(\dim X\geq 2\) and exceptional \(\mathcal O_X,\) then \(\Phi_{\mathcal I}:\mathbf D^b(X)\rightarrow\mathbf D^b(X^{[2]})\) is fully faithful. While Krug and Sosna relied on the Bridgeland-King-Reid-Haiman equivalence, the generalisation relies on an explicit geometry of \(X^{[2]}\) which is given in any dimension. More recently, it was proved that \(\Phi_{\mathcal I}\) is faithful for all \(X\) and \(n,\) subject to a numerical condition which is satisfied when \(n=2,\) thus the main enhancement in this paper is the proof of the fullness.
Using a spectral sequence argument as in the case of surfaces, the authors obtain generalizations of the first main result, and relate the deformation theories of \(X\) and \(X^{[2]}.\) The second main result deals with the degeneration of the spectral sequences in question to the generalised situation. This result says that for a smooth projective variety of \(\dim X\geq 2\) with exceptional \(\mathcal O_X,\) the relative local to global \(\operatorname{Ext}\) spectral sequence \[ E^{i,j}_2=H^i(X^{[2]},\mathcal Ext^j_{p_X^{[2]}}(\mathcal I,\mathcal I))\Rightarrow\operatorname{Ext}^{i+j}_{X\times X^{[2]}}(\mathcal I,\mathcal I) \] degenerates at the \(E_2\)-page. Moreover, the abutment is the Hochschild cohomology of \(X\) such that the induced filtration coincides with the filtration associated to the Hochschild-Kostant-Rosenberg decomposition up to a degree shift.
The authors then study deformation theory of \(X^{[2]}\) for \(\dim X\geq 3,\) obtaining a result saying that for a smooth projective variety of \(\dim X\geq 3\) with exceptional \(\mathcal O_X,\) there are isomorphisms \(H^i(X,T_X)\overset\sim{\rightarrow} H^i(X^{[2]},T_{X^{[2]}})\) for all \(i\geq 0.\)
The last result says that rigidity of \(X\) implies rigidity of \(X^{[2]}\) which contradicts the surface case, e.g., for \(S=\mathbb P^2,\) \(S\) is rigid while \(S^{[n]}\) is not for \(n\geq 2.\) This is very interesting as it proves that these additional deformations are associated to noncommutative deformations of \(S\).
The article is very well written, and gives good examples and important results. It proves the need for noncommutative deformations to study Hilbert schemes of points in the general situation. One should also notice the computational technique presented, giving obstruction theory by the embeddings of the diagonal. This gives a direct way to compute the obstructions by spectral sequences, and to prove the connection to noncommuative deformation theory. Hilbert squares; derived categories; deformations; Hilbert scheme of points; Hilbert square; exceptional coordinate sheaf; Fourier-Mukai functor; Hochschild cohomology; relative local-to-global Ext spectral sequence; faithful; full and faithfull; relative local to global \(\operatorname{Ext}\) spectral sequence; noncommuative deformation theory Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, moduli, classification: algebraic theory Hilbert squares: derived categories and deformations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The theory of Kac-Moody Lie algebras has undergone a tremendous development since they were introduced more than 40 years ago. These algebras generalize the finite-dimensional semisimple Lie algebras, and they have close and important connections to several areas in mathematics, and also in mathematical physics. Today the topic has become a standard tool in mathematics. The purpose of the present book is to present the algebro-geometric and topological aspects of Kac-Moody theory in characteristic \(0\).
A lot of different topics are treated in this monumental work. Many of the topics are brought together for the first time in book form. The emphasis is on the study of Kac-Moody groups and their flag varieties. Among the main topics are, the Weyl-Kac character formula, a detailed construction of Kac-Moody groups and their flag varieties; the Demazure character formula, a study of the geometry of Schubert varieties, including normality and Cohen-Macaulayness; the Borel-Weil-Bott theorem; the Bernstein-Gelfand-Gelfand resolution; the Kempf resolution; conjugacy theorems for the Cartan subalgebras and invariance of the generalized Cartan matrix; determination of the defining ideal of the flag varieties via the Plücker relations; a study of the \(T\)-equivariant and singular cohomology of the flag varieties via the nil-Hecke ring, including positivity results for the cup product, and various criteria for the smoothness and rational smoothness of points on Schubert varieties. A particular chapter is devoted to the realization of affine Kac-Moody algebras and the corresponding groups, as well as their flag varieties.
There are several sections introducing the necessary background material, like ind-varieties; pro-groups, pro-Lie algebras; Tits systems; the nil-Hecke ring; Coxeter groups; the Bernstein-Gelfand-Gelfand \(\mathcal O\) category; Lie algebra homology; as well as appendices on results from algebraic geometry; local cohomology; results from topology; relative homological algebra; introduction to spectral sequences.
Although the book is devoted to the general Kac-Moody theory, many of the topics of the book will be useful for those only interested in the finite-dimensional case.
The book is self contained, but is on the level of advanced graduate students. Most statements, definitions and proofs are short, and there are few examples. Moreover many results are given as exercises, some of these are on the difficult side. For the motivated reader who is willing to spend considerable time on the material, the book can be a gold mine. Chapter by chapter the contents is:
I. Kac-Moody algebras; Basic theory
This section contains the definition of Kac-Moody algebras, root space decompositions and the Weyl groups associated to Kac-Moody algebras. Moreover the dominant chamber and the Tits cone are introduced, and the invariant bilinear form and the Casimir operator are treated.
II. Representation theory of Kac-Moody algebras
Here the Bernstein-Gelfand-Gelfand \(\mathcal O\) category is introduced, the Weyl-Kac character formula is proved, and the important Shapovalov bilinear form is given.
III. Lie algebra homology and cohomology
Results of Konstant-Garland-Lepowsky are proved and the Laplacian is introduced and used in calculations.
IV. An introduction to ind-varieties and pro-groups
Ind-varieties, ind-groups and their Lie algebras are defined and the smoothness of ind-varieties discussed. Pro-groups and pro-Lie algebras are introduced.
V. Tits systems; Basic theory
The basic theory of Tits systems and refined Tits systems is presented.
VI. Kac-Moody groups; Basic theory
The definitions of Kac-Moody groups and the related parabolic subgroups are given. Moreover, the representations of Kac-Moody groups are treated.
VII. Generalized flag varieties of Kac-Moody groups
The ind-variety structure of generalized flag varieties is given, and line bundles on these varieties are studied. Moreover the group \(\mathcal G^{\text{min}}\), defined by Kac-Peterson, and the group \(\mathcal U^-\) are studied.
VIII. Demazure and Weyl-Kac character formulas
The normality of Schubert varieties is proved and the Demazure Character formula proved. Moreover extensions of the Weyl-Kac character formula and the Borel-Weil-Bott theorem are given.
IX. Bernstein-Gelfand-Gelfand and Kempf resolutions
The Bernstein-Gelfand-Gelfand resolution and the dual Kempf resolution are obtained in the general Kac-Moody situation.
X. Defining equations of \(\mathcal G/\mathcal P\) and conjugacy theorems
The quadratic relations defining \(\mathcal G/\mathcal P\) in the canonical projective embedding are given, and the conjugacy theorems for Lie algebras, and groups, are proved.
XI. Topology of Kac-Moody groups and their flag varieties
The nil-Hecke ring is introduced, and the \(T\)-equivariant cohomology of \(\mathcal G/\mathcal B\) is determined. Some positivity results of cup products are given, and the degeneracy of the Leray-Serre spectral sequence for the fibration \(\mathcal G^{\text{min}} \to \mathcal G^{\text{min}}/T\) is proved.
XII. Smoothness and rational smoothness of Schubert varieties
The singular locus and rational smoothness of Schubert varieties are studied.
XIII. An introduction to affine Kac-Moody Lie algebras and groups
The affine Kac-Moody Lie algebras and groups are introduced and studied. Kac-Moody Lie algebras; Kac-Moody groups; representation theory; flag varieties; semisimple simply-connected algebraic groups; parabolic subgroups; Weyl-Kac character formula; \(\mathfrak n\)-homology; ind-varieties; pro-groups; pro-Lie algebras; Tits systems; Demazure character formula; Schubert varieties; Cohen-Macaulay varieties; Borel-Weil-Bott theorem; Bernstein-Gelfand-Gelfand resolution; Kempf resolution; Cartan subalgebras; generalized Cartan matrix; Plücker relations; nil-Hecke rings; cup products; Leray-Serre spectral sequence; smoothness of Schubert varieties; affine Kac-Moody algebras Kumar, S., Kac-Moody groups, their flag varieties and representation theory, 204, (2002), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc., Boston, MA Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Loop groups and related constructions, group-theoretic treatment, Research exposition (monographs, survey articles) pertaining to topological groups, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Grassmannians, Schubert varieties, flag manifolds, Rational and unirational varieties Kac-Moody groups, their flag varieties and representation 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 This amazing book provides, in only 229 pages, an introduction to a large part of modern number theory -- from A. Weil's proof of the Riemann hypothesis for zeta functions of curves over finite fields to the theory of modular forms -- both in the classical and adelic languages. It includes most of the necessary background needed to derive estimates of many sorts of exponential sums which arise in the spectra of explicit Ramanujan graphs constructed as Cayley graphs of various finite groups. In particular, there is a discussion of the finite upper half plane graphs considered by the reviewer and co-workers in a series of papers. See the reviewer [Survey of spectra of Laplacians on finite symmetric spaces, Exp. Math. 5, No. 1, 15-32 (1996)] for a survey. The author provides a different method from that of N. Katz to show that these finite upper half plane graphs are Ramanujan. No \(\ell\)-adic cohomology is required. However, she does provide an introduction to this subject in Chapter 2.
The book begins with a chapter on finite fields. Then Chapter 2 shows how Weil arrived at the Weil conjectures (proved by P. Deligne) on zeta functions for projective varieties over finite fields starting with the estimation of the number of solutions to polynomial equations over finite fields using Jacobi sums. Chapter 3 concerns valuations of function fields over finite fields, completions, the adeles and the ideles. Chapters 4 and 5 concern zeta and \(L\)-functions of idele class characters of function fields. Here one finds proofs of the Riemann-Roch theorem and the Riemann hypothesis for the zeta function of a nonsingular projective curve over a finite field. Chapter 6 uses the preceding results plus a bit of class field theory from \textit{A. Weil} [Basic number theory, Springer-Verlag, Berlin (1973; Zbl 0267.12001)] to estimate character sums.
Chapter 7 sketches the classical theory of modular forms for congruence subgroups of \(SL(2, \mathbb{Z})\), including the theory of Hecke operators and \(L\)-functions attached to modular forms. Some of the background for the work of A. Wiles and R. Taylor proving Fermat's Last Theorem can be found here.
Chapter 8 gives the adelic representation theoretic version of the theory of modular forms. Work of H. Jacquet, R. Langlands, S. Gelbart, \dots including converse theorems, strong multiplicity one theorems, and the correspondence between automorphic representations of quaternion groups and admissible cuspidal representations of the adelic \(GL(2)\) are discussed.
Chapter 9 applies the preceding work to a part of graph theory which is of interest in computer science. A connected \(k\)-regular graph \(X\) is Ramanujan if for any eigenvalue \(\lambda\) of the adjacency matrix with \(|\lambda |\neq k\), we have \(|\lambda |\leq 2\sqrt {k-1}\). This bound implies the graph is a best possible expander among \(k\)-regular graphs as the number of vertices goes to infinity by a theorem of Alon and Boppana for which the author gives two proofs. It also implies that the simple random walk on \(X\) converges extremely rapidly to uniform. Some references for Ramanujan graphs are \textit{A. Lubotzky} [Discrete groups, expanding graphs, and invariant measures, Birkhäuser, Prog. Math. 125 (1994; Zbl 0826.22012)] and \textit{P. Sarnak} [Some applications of modular forms, Cambridge Tracts Math. 99 (1990; Zbl 0721.11015)].
The author allows her graphs to be directed if the adjacency matrix is diagonalizable by unitary matrices. It had been known for a long time that Ramanujan graphs exist, but G. Margulis and, independently, A. Lubotzky, R. Phillips and P. Sarnak constructed the first explicit examples in 1988 using quaternion groups. The author considers these constructions as well as others constructed using finite abelian groups by Fan Chung and herself, independently. Her earlier character sum estimates are necessary for the estimates of the eigenvalues of adjacency operators. Similarly these estimates are needed to show that the finite upper half plane graphs mentioned above are Ramanujan.
Finally she gives a proof due to herself and K. Feng of a result of B. D. McKay. It supposes one is given a sequence \(X_m\) of connected \(k\)-regular graphs with \(|X_m|\) going to infinity with \(m\), such that the number of primitive cycles (no backtracking and containing no proper subcycle) in \(X_m\) has moderate growth. Then the distribution of eigenvalues approaches the Wigner semi-circle distribution (alias the Sato-Tate distribution) as \(m\to \infty\). This is then applied to obtain a result of Feng and the author estimating the growth of the dimension of cusp forms of weight 2 for \(\Gamma_0 (N)\) which are eigenfunctions of the Hecke operator \(T_p\) with integral eigenvalues.
It is very interesting that number theory leads to graph theory and then back to new results in number theory, showing how fruitful this approach has become. In short, this book is highly recommended to those with an interest in some of the deepest and most beautiful results in modern number theory as well as the applications to the spectral theory of graphs. It is not an easy book for a reader without any familiarity with the modern adelic point of view. However, the reader will certainly be rewarded by contemplating this introduction to some very powerful mathematics. finite fields; character sums; Weil conjectures; Riemann-Roch theorem; points on curves over finite fields; zeta-functions; \(L\)-functions; idele class characters; modular forms; automorphic representations; Ramanujan graphs; Alon-Boppana theorem; regular graphs; Riemann hypothesis for zeta functions of curves over finite fields; exponential sums; Cayley graphs; finite upper half plane graphs; valuations of function fields; projective curve; Hecke operators; automorphic representations of quaternion groups; expander; simple random walk; spectral theory of graphs Li, W. -C. Winnie: Number theory with applications. Series of university mathematics 7 (1996) Research exposition (monographs, survey articles) pertaining to number theory, Modular and automorphic functions, Graph theory, Arithmetic theory of algebraic function fields, Curves over finite and local fields, Representation-theoretic methods; automorphic representations over local and global fields, Holomorphic modular forms of integral weight, Estimates on exponential sums, Exponential sums, Adèle rings and groups, Representations of Lie and linear algebraic groups over global fields and adèle rings, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Number theory with applications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present a probabilistic Las Vegas algorithm for computing the local zeta function of a genus-\(g\) hyperelliptic curve defined over \(\mathbb{F}_q\) with explicit real multiplication (RM) by an order \(\mathbb{Z} [\eta]\) in a degree-\(g\) totally real number field.
It is based on the approaches by Schoof and Pila in a more favourable case where we can split the \(\ell \)-torsion into \(g\) kernels of endomorphisms, as introduced by \textit{P. Gaudry} et al. [Lect. Notes Comput. Sci. 7073, 504--519 (2011; Zbl 1227.94045)] in genus 2. To deal with these kernels in any genus, we adapt a technique that the author, \textit{P. Gaudry}, and \textit{P.-J. Spaenlehauer} [Found. Comput. Math. 19, No. 3, 591--621 (2019; Zbl 1470.11170)] introduced to model the \(\ell \)-torsion by structured polynomial systems. Applying this technique to the kernels, the systems we obtain are much smaller and so is the complexity of solving them.
Our main result is that there exists a constant \(c > 0\) such that, for any fixed \(g\), this algorithm has expected time and space complexity \(O ((\log q)^c)\) as \(q\) grows and the characteristic is large enough. We prove that \(c \leq 9\) and we also conjecture that the result still holds for \(c = 7\). hyperelliptic curves; real multiplication; local zeta function; multi-homogeneous polynomial systems; Schoof-Pila's algorithm Curves over finite and local fields, Number-theoretic algorithms; complexity, Rational points, Cryptography Counting points on hyperelliptic curves with explicit real multiplication in arbitrary genus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We give a combinatorial proof for a multivariable formula of the generating series of type D Young walls. Based on this we give a motivic refinement of a formula for the generating series of Euler characteristics of Hilbert schemes of points on the orbifold surface of type D. Hilbert scheme of points; Young walls; generating function Parametrization (Chow and Hilbert schemes), Combinatorial aspects of representation theory, Quantum groups (quantized enveloping algebras) and related deformations, Singularities in algebraic geometry Young walls and equivariant Hilbert schemes of points in type \(D\) | 0 |
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