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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 Main purpose of the present article is to show the existence of a coarse moduli space of irreducible connections on vector bundles of rank \(2\) on \(\mathbb{P}^1(\mathbb{C})\) having regular singularities in three fixed points, say, \(\{0,1, \infty \}\). Put \(U_0 = \mathbb{P}^1(\mathbb{C})- \{\infty \}= \mathbb{C}\) with coordinate \(z\) and \(U_{\infty}=\mathbb{P}^1(\mathbb{C})-\{0\}=\mathbb{C}\) with coordinate \(x\) such that \(xz=1\). Let \(S=\{s_1,s_2,\ldots,s_n\}\) be a finite collection of points of \(X=P^1(\mathbb{C})\), \(n\geq 2\), \(s_i\ne s_j\), if \(i\ne j\), \(z(s_i)=a_i\). Let \(\mathcal E\) be a holomorphic vector bundle of rank \(2\) on \(X= \mathbb{P}^1(\mathbb{C})\). A connection \(\nabla\) on \(\mathcal E\) having regular singularities on \(S\) is, by definition: (1) \(\nabla\) is a holomorphic connection of \({\mathcal E}_{| X-S}\) (i.e. a \({\mathbb{C}}\)- linear map \(\nabla: \mathcal E_{| X-S}\to \Omega^1(\mathcal E_{| X-S})\) satisfying \(\nabla(fe) = df\otimes e+f\nabla(e)\) for local sections \(f\) and \(e\) of \(\mathcal O_X\) and \(\mathcal E\), respectively); (2) for any \(s\in S\), there exist an open neighbourhood \(U\) of \(s\) in \(X\), a base \(\omega\) of \(\Omega^1_U\) and a base \((e)\) of \(\mathcal E_{| U}\) meromorphic in \(s\) (i.e. there exists an element \(T\in \mathrm{GL}(2,\Gamma(U,\mathcal O_ X(S)))\) and a base \((g)\) of \(\mathcal E_{| U}\) with \((e)=(g)T\) on \(T-\{s\}\) such that if we write \(\nabla(e) = M\cdot(\omega \otimes e)\), then each component of the matrix \(M\) has a pole of order at most one at \(s\) and holomorphic on \(U-\{s\}\).
Put \(_z\nabla(v) = <\nabla(v),d/dz>\), \(_x\nabla(v)=<\nabla(v),d/dx>\), where \(v\) is a local section of \(\mathcal E\). We have \(_z\nabla = -x^2_x\nabla\) on \(U_0\cap U_{\infty}- S\). The author first shows the following theorem, which enable him to reduce the moduli problem of connections to that of pairs of matrices:
Theorem 1.2.1. Let \(\mathcal E\) and \(S\) be the same as above and \(\nabla\) be a connection on \(\mathcal E\) having regular singularities in \(S\). Then there exists a basis of \(\mathcal E\) meromorphic in \(S\) such that with respect to this basis, if \(S\subset U_0\), then
\[ _z\nabla =(d/dz)+(A_1/(z- a_1))+ \ldots +(A_n/(z-a_n)),\]
where \(A_1,\ldots,A_n\) are \(2\times 2\) complex matrices and \(A_1+ \ldots +A_n=0\).
The author shows that the same theorem holds for algebraic vector bundles \(\mathcal E\) on \(P^1(K)\) and an algebraic connection with regular singularities on \(S\), if \(K\) is an algebraically closed field of characteristic \(0\). He also shows that if \(K\) is not algebraically closed with \(\mathrm{char}(K)=0\), then the theorem does not necessarily hold. (He gives the necessary and sufficient conditions that the theorem holds in this case: Theorem 1.3.6.)
Let \(P_m\) be the set consisting of pairs \((A_1,A_2)\) of \(K\)-valued \(2\times 2\) matrices. The author defines two equivalence relations \(\sim,\approx\) on \(P_m\) by:
\((B_1,B_2)\sim(A_1,A_2)\) if \((B_1,B_2)=(T^{-1}A_1T,T^{-1}A_ 2T)\) for some \(T\in \mathrm{GL}(2,K)\);
\((B_1,B_2)\approx(A_1,A_2)\) if \((B_1/z)+(B_2/(z-1)) = T^{-1}((A_1/z)+(A_2/(z-1)))+T^{-1}dT/dz\) for some \(T\in \mathrm{GL}(2,K[z,1/z,1/(z- 1)])\).
Classifying elements of \(P_m\) relative to the first equivalence relation \(\sim\) agrees with classifying Fuchsian systems and classifying elements of \(P_m\) relative to the second equivalence relation \(\approx\) agrees with classifying connections (Theorem 1.2.1 and Lemma 3.1.5).
In {\S} 2, the author classifies certain subsets of Fuchsian systems. Let \(V\) be the category of analytic spaces (or algebraic varieties). For any \(S\in V\), a family of pairs of matrices over \(S\) is a triple (\(\mathcal E,A_1,A_2)\) consisting of a vector bundle \(\mathcal E\) of rank \(2\) on \(S\) and endomorphisms \(A_1,A_2\) of \(\mathcal E\). Two families (\(\mathcal E,A_1,A_2)\) and (\(\mathcal E',A'_1,A'_2)\) on \(S\) are called equivalent, if there exist an open covering \(\{U_j\}_{j\in J}\) of \(S\) and isomorphisms \(\phi_j: \mathcal E_{| U_ j}\to \mathcal E'_{| U_ j}\) such that \(A'_{k| U_ j}=\phi_ j(A_{k| U_ j})\phi_ j^{-1}, k=1,2\), for all j. We denote the equivalence class of (\(\mathcal E,A_1,A_2)\) by \(c(\mathcal E,A_1,A_2)\). By \(F_m\) we mean a contravariant functor from \(V\) to \((Set)\) defined by \(F_m(S)=\{c(\mathcal E,A_1,A_2)\}\). Similarly one defines a subfunctor \(F\) of \(F_m\) by \(F(S)=\{c(\mathcal E,A_1,A_2)| (A_1(s),A_2(s))\) is irreducible, that is, \(A_1(s)\) and \(A_2(s)\) have no common eigenvectors in \(\mathcal E(s)\) for any \(s\in S\}\). The author shows that \(F_m\) has no coarse moduli space (proposition 2.2) but \(F\) has a coarse moduli space (Theorem 2.3.2). Here, a coarse moduli space for a functor \(G\) from \(V\) to \((Set)\) is a pair \((N,\Phi)\) with \(N\in V\), \(\Phi: G\to h_N = \Hom(N,\cdot)\) such that \(\Phi(p): G(p)\to h_N(p)\) is bijective where \(p\) is a point and for each \(L\in V\) and each morphism \(\Psi: N\to h_L,\) there is a unique morphism \(f: N\to L\) such that the diagram \(G\to^{\Phi}h_N\to^{h_f}h_L; G\to^{\Psi}h_L\) is commutative.
In {\S} 3 the author classifies the irreducible connections on holomorphic vector bundles of rank \(2\) on \(\mathbb{P}^1(\mathbb{C})\) having regular singularities in three fixed points \(0,1,\infty\). A family of irreducible connections on an analytic space \(S\) is, by definition, a pair (\(\mathcal E,\nabla)\) where \(\mathcal E\) is a vector bundle of rank 2 on \(S\times \mathbb{P}^1(\mathbb{C})\) and \(\nabla\) is an irreducible relative connection on \(\mathcal E\) having regular singularities in \(\{0,1,\infty \}\). (''Irreducible'' means that the family of pairs \((A_1,A_2)\) on \(S\) associated with \(\nabla\) by Theorem 1.2.1 is irreducible.) Two pairs (\(\mathcal E,\nabla)\) and (\(\mathcal E',\nabla')\) are called equivalent if for each \(x\in X=S\times \mathbb{P}^1(\mathbb{C})\) there exist a neighbourhood \(U\) of \(x\) in \(X\) and an isomorphism \(\phi: E_{| U}\to \mathcal E'_{| U}\) meromorphic along \(Y=S\times \{0,1,\infty \}\) such that on \(U-Y\) we have \(_z\nabla =_z\nabla'\). Then the author shows that the functor \(G\) from \(V\) to \((Set)\) defined by \(G(S)= \) the set of equivalence classes of (\(\mathcal E,\nabla)\) has a coarse moduli space. Fuchsian system; coarse moduli space of irreducible connections on vector bundles of rank 2; regular singularities Algebraic moduli problems, moduli of vector bundles, Connections (general theory), Complex-analytic moduli problems, Structure of families (Picard-Lefschetz, monodromy, etc.), Sheaves in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Other connections Moduli spaces for pairs of \(2\times 2\)-matrices and for certain connections on \(\mathbb{P}^1(\mathbb{C})\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper we study graded Betti numbers of any nondegenerate 3-regular algebraic set \(X\) in a projective space \(\mathbb{P}^n\). More concretely, via Generic initial ideals (Gins) method we mainly consider `tailing' Betti numbers, whose homological index is at least \(\mathrm{codim}(X, \mathbb{P}^n)\). For this purpose, we first introduce a key definition `\(\mathrm{ND}(1)\) property', which provides a suitable ground where one can generalize the concepts such as 'being nondegenerate' or `of minimal degree' from the case of varieties to the case of more general closed subschemes and give a clear interpretation on the tailing Betti numbers. Next, we recall basic notions and facts on Gins theory and we analyze the generation structure of the reverse lexicographic (rlex) Gins of 3-regular \(\mathrm{ND}(1)\) subschemes. As a result, we present exact formulae for these tailing Betti numbers, which connect them with linear normality of general linear sections of \(X \cap {\Lambda}\) with a linear subspace {\(\Lambda\)} of dimension at least \(\mathrm{codim}(X, \mathbb{P}^n)\). Finally, we consider some applications and related examples. graded Betti numbers; generic initial ideals; linear normality; 3-regular scheme Projective techniques in algebraic geometry, Syzygies, resolutions, complexes and commutative rings Linear normality of general linear sections and some graded Betti numbers of 3-regular projective schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review studies the properties of the set of dynamical degrees of a fixed projective surface. If \(X\) is a projective surface and \(f\) is a birational transformation of \(X\), its dynamical degree is defined as
\[
\lambda(f)=\lim_{n\to\infty}||(f^n)_*||^{1/n}
\]
where \(||\cdot||\) is any norm on \(\mathrm{End}(NS_{\mathbb R}(X))\). The set of all dynamical degrees \(\Lambda(X):= \{ \lambda(f)|f \text{ birational transformations of } X \}\) is called the dynamical spectrum of \(X\). By a theorem of Diller and Favre a dynamical degree different from 1 is either a Pisot or a Salem number. Those are algebraic integers whose Galois conjugates lie in the open, resp. closed, unit disk.
The first part of the paper is devoted to a survey of properties of the dynamical degrees, with proofs and several examples, and to the proof of
Theorem A: Let \(k\) be an algebraically closed field. Let \(f\) be a birational transformation of \(X\) defined over \(k\). If \(\lambda(f)\) is a Salem number, there exists a surface \(Y\) and a birational mapping \(\varphi: Y\dashrightarrow X\) such that \(\varphi^{-1}\circ f\circ \varphi \) is an automorphism of \(Y\).
As a corollary, they obtain a spectral gap property: there is no dynamical degree between 1 and the Lehmer number \(\lambda_L\), which is conjecturally the infimum of the set of Salem numbers.
Secondly, the authors study the set \(\Lambda(X)\) for a non-rational surface \(X\) and they prove that
Theorem B: (1) \(\Lambda(X)\) is made of quadratic integers and Salem numbers of degree at most 6 (resp. 22, resp. 10) if \(X\) is birationally equivalent to an abelian surface (resp. a \(K3\) surface, resp. an Enriques surface); (2) \(\Lambda(X)= \{1 \}\) otherwise.
The final part of the paper is about rational surfaces. In this case, the situation is more complicated and the results rely on the study of the Picard-Manin space and the hyperbolic space.
The minimal degree \(\mathrm{mcdeg}(f)\) of a birational transformation \(f\) of the projective plane is the minimum of the set of degrees of all its conjugates.
The main results of this part are a bound on \(\mathrm{mcdeg}(f)\) in terms of \(\lambda(f)\) and the fact that \(\Lambda(\mathbb{P}^2)\) is well ordered and it is also closed if the base field is algebraically closed. dynamical spectrum; projective surfaces; dynamical degrees; Salem numbers; Pisot numbers Jérémy Blanc & Serge Cantat, ''Dynamical degrees of birational transformations of projective surfaces'', J. Amer. Math. Soc.29 (2016) no. 2, p. 415-471 Birational automorphisms, Cremona group and generalizations, Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables Dynamical degrees of birational transformations of projective surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this article, the author focuses on the family of strata \(\mathcal H(a,-a)\), where \(a \geq 2\). Such strata, denoted by \(\mathcal H(a,-a)\) for \(a \geq 2\), are moduli spaces of biholomorphism classes of pairs of a genus-one Riemann surface and a meromorphic 1-form with exactly one zero of order \(a\) and a pole of order \(a\). Modular curves \(X_1(N)\) parametrize elliptic curves with a point of order \(N\). For \(a \geq 2\), the modular curves \(X_1(a)\) can be identified with connected components of projectivized strata \(\mathbb{P}\mathcal H(a,-a)\). They admit a canonical walls-and-chambers structure. Here chambers of projectivized strata are topological disks (with, in some cases, a puncture inside) and walls are real-codimension-one submanifolds which are affine in the period coordinates. In other words, walls are straight lines. They meet each other at the punctures of the algebraic complex curve. In this article, the author provides formulas for the number of chambers and an effective means for drawing the incidence graph of the chamber structure of any modular curve \(X_1(N)\). This defines a family of graphs with specific combinatorial properties. This approach provides a geometric-combinatorial computation of the genus and the number of punctures of modular curves \(X_1(N)\). Although the dimension of a stratum of meromorphic differentials depends only on the genus and the numbers of singularities, the topological complexity of the stratum crucially depends on the order of the singularities. translation surface; walls-and-chambers structure; flat structure; modular curves Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Families, moduli of curves (analytic), Formal category theory Chamber structure of modular curves \(X_1(N)\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Hilbert scheme of \(m\) points \(\text{Hilb}_m(S)\) of a quasiprojective surface \(S\) parametrizes \(m\)-point subschemes of \(S\). The main purpose of this paper is to characterize the ring structure of the small equivariant quantum cohomology rings of \(\text{Hilb}_m(\mathcal{A}_n)\) for all \(m\) and \(n\), where \(\mathcal{A}_n\) is the crepant resolution of the \(A_n\) singularity \(\mathbb{C}/\mathbb{Z}_{n+1}\). This generalizes the work of Okounkov-Pandharipande for \(\text{Hilb}_m(\mathbb{C}^2)\).
The classical equivariant cohomology ring \(H^*_T(\mathcal{A}_n,\mathbb{Q})\) is generated by the \(n\) exceptional divisors \(\omega_i\) and the identity class \(1\). Moreover, \(H^*_T(\text{Hilb}_m(\mathcal{A}_n),\mathbb{Q})\) has a basis, the Nakajima basis, labeled by cohomology weighted partitions \(\overrightarrow{\mu}=\{((\mu^{(1)},\gamma_{i_1}),\dots, (\mu^{(l)},\gamma_{i_l}))\}\) of \(m\), where \(\mu^{(1)}+\dots+\mu^{(l)}=m\) and \(\gamma_{i}\in H^*_T(\mathcal{A}_n,\mathbb{Q})\). Two divisors \(D:=-\{(2,1),(1,1),\dots,(1,1)\}\), a multiple of the boundary divisor of two point collisions, and \((1,\omega_i):=\{(1,\omega_i),(1,1),\dots,(1,1)\}\) play a central role in the paper.
The Fock space \(\mathcal{F}_{\mathcal{A}_n}\) modeled on \(\mathcal{A}_n\) is graded isomorphic to
\[
\mathcal{F}_{\mathcal{A}_n}=\bigoplus\limits_{m\geq0}H^*_T(\text{Hilb}_m(\mathcal{A}_n),\mathbb{Q})\,,
\]
and the results of this paper are naturally stated in terms of this Fock space after extension of coefficients. The underlying reason is that \(\mathcal{F}_{\mathcal{A}_n}\otimes\mathbb{Q}(t_1,t_2)\) is isomorphic to a subspace in the basic representation of the affine Lie algebra \(\widehat{\mathfrak{gl}}(n+1)\). Quantum product on
\[
QH^*_T(\text{Hilb}_m(\mathcal{A}_n)):=H^*_T(\text{Hilb}_m(\mathcal{A}_n),\mathbb{Q}) \otimes\mathbb{Q}(t_1,t_2)((q))[[s_1,\dots,s_n]]
\]
is a deformation of the classical cup product defined via three-point genus \(0\) Gromov-Witten invariants of the Hilbert scheme.
Two-point invariants can be encoded into an operator \(\Theta\) on the Fock space. To generate them, the authors define explicit representation-theoretic operators \(\Omega_0\), \(\Omega_+\) and prove that \(\Theta=(t_1+t_2)(\Omega_0+\Omega_+)\). This leads to their main result, explicit formulas for quantum multiplication operators \(M_D\) and \(M_{(1,\omega_i)}\) in terms of their classical counterparts \(M_D^{cl}\) and \(M_{(1,\omega_i)}^{cl}\):
\[
M_D=M_D^{cl}+(t_1+t_2)\,q\frac{\partial}{\partial q}(\Omega_0+\Omega_+)
\]
\[
M_{(1,\omega_i)}=M_{(1,\omega_i)}^{cl}+(t_1+t_2)\,s_i\frac{\partial}{\partial s_i}\Omega_+\,.
\]
One consequence of these formulas is that the multiplication operators are rational in \(q,s_1,\dots,s_n\) in contrast to the case of general surfaces, where only rationality in \(q\) is expected. They also imply a correspondence between multiplication by divisors in \(QH^*_T(\text{Hilb}_m(\mathcal{A}_n),\mathbb{Q})\) and in the equivariant Gromov-Witten theory of \(\mathcal{A}\times\mathbb{P}^1\) relative to the fibers at \(0\), \(1\) and \(\infty\). Under an additional conjecture that all joint eigenspaces of \(M_D\) and \(M_{(1,\omega_i)}\) are one-dimensional, and hence \(D\), \((1,\omega_i)\) generate \(QH^*_T(\text{Hilb}_m(\mathcal{A}_n),\mathbb{Q})\), the authors prove complete Gromov-Witten/Hilbert correspondence for \(\mathcal{A}_n\). If true, this is another special feature of these surfaces, which fails already for \(\mathbb{P}^2\), at least if no change of curve class variables is allowed. At the end of the paper the authors briefly discuss the quantum differential system \(q\frac{\partial}{\partial q}\psi=M_D\psi\), \(\,s_i\frac{\partial}{\partial s_i}\psi=M_{(1,\omega_i)}\psi\) and its monodromy, and indicate how their formulas can be extended to crepant resolutions of \(D\) and \(E\) singularities. Hilbert scheme of points; \(A_n\) singularity; small quantum cohomology; Nakajima basis; Gromov-Witten/Hilbert correspondence Maulik, D; Oblomkov, A, Quantum cohomology of the Hilbert scheme of points on \({\mathcal{A}}_n\)-resolutions, J. Am. Math. Soc., 22, 1055-1091, (2009) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum cohomology of the Hilbert scheme of points on \(\mathcal {A}_n\)-resolutions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 Artin formalism for \(L\)-series of number fields is known to be satisfied in topology, in the context of the characteristic polynomial in linear algebra, when one has an endomorphism acting functorially on some representation spaces associated with a finite covering.
In the present paper, it is shown that Artin's formalism is satisfied in the infinite-dimensional analogue of a one-parameter group \(\exp (-tA)\), where \(A\) is a positive self-adjoint operator. The kernel representing the operator \(\exp (-tA)\) is called the associated heat kernel and could be called the characteristic kernel, since it plays the role of the characteristic polynomial in the finite-dimensional case. It follows that any homomorphic image of the heat kernel (in a sense precisely defined) also satisfies the Artin formalism, for instance the trace of the heat kernel, the coefficients of the asymptotic expansion of this trace at the origin, the regularized determinant of the Laplacian operating on a metrized sheaf, the Selberg zeta function, etc. Other applications are given to derive systematically relations among theta functions. These include the classical Kronecker, Jacobi and Riemann relations. \(L\)-series of number fields; infinite-dimensional analogue of one- parameter group; Artin formalism; self-adjoint operator; associated heat kernel; characteristic kernel; trace; asymptotic expansion; regularized determinant of the Laplacian; Selberg zeta function; theta functions J. Jorgenson, S. Lang, Artin formalism and heat kernels. Jour. Reine. Angew. Math. 447 (1994), 165-280. Zbl0789.11055 MR1263173 Other Dirichlet series and zeta functions, Spectral theory; trace formulas (e.g., that of Selberg), Heat and other parabolic equation methods for PDEs on manifolds, Theta functions and curves; Schottky problem, Langlands \(L\)-functions; one variable Dirichlet series and functional equations, Zeta functions and \(L\)-functions of number fields, Theta functions and abelian varieties Artin formalism and heat kernels | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \({\mathcal F}\) be a differential field of characteristic zero with derivation \(\delta\), and let \({\mathcal C}\) be its field of constants. Assume that both fields are algebraically closed. In this paper and its sequels, the author studies differential polynomial functions on schemes \(X\) over \({\mathcal F}\) and their applications to the theory of algebraic groups and in diophantine problems. The present paper defines the notion of differential polynomial function and considers applications to algebraic groups.
The definition of differential polynomial function depends on the author's theory of ``infinite prolongations'': this is a functor \(X \mapsto X^ \infty\) from \({\mathcal F}\)-schemes to differential \({\mathcal F}\)- schemes, or \(D\)-schemes (schemes whose structure sheaf has a derivation extending \(\delta)\) which is adjoint to the forgetful functor. Then a differential polynomial morphism \(f:X \to Y\) of \({\mathcal F}\)-schemes is a set theoretic map induced from a morphism \(f^ \infty:X^ \infty \to Y^ \infty\) of \(D\)-schemes. The non-zeros of differential polynomial functions \(X \to \text{Spec} ({\mathcal F})\) determine the open set of the differential, or \(\delta\), topology on \(X\).
Using his theory, the author establishes the following: Let \(G\) be a simple linear algebraic \({\mathcal F}\)-group, then a \(\delta\)-closed Zariski dense subgroup must be either \(G({\mathcal F})\) or conjugate to \(G({\mathcal C})\) (Cassidy's theorem); there is a differential polynomial linear representation \(G \to GL (n,{\mathcal F})\) associated to any irreducible algebraic \({\mathcal F}\)-group \(G\) whose kernel \(G^ \#\) is contained in the kernel of any differential polynomial linear representation of \(G\) and which is Zariski dense if \(G\) has only \({\mathcal F}\) as global functions; when \(A\) is an abelian variety then \(A^ \#\) is shown to be the \(\delta\)-closure of the torsion subgroup of \(A\) and it has the same dimension as \(A\) only if \(A\) is defined over \({\mathcal C}\); and this latter is used to show that there exists a \(\delta\)-open subset of the moduli space of curves of genus \(g\) such that for the Jacobian \(J(X)\) of any curve \(X\) in this subset the dimension of \(J(X)^ \#\) is \(2g\), a result the author plans to use in the investigation of differential polynomial functions on curves. differential schemes; differential field; differential polynomial functions; algebraic groups; abelian variety A. Buium, Geometry of differential polynomial functions, I: algebraic groups , to appear in Amer. J. Math. JSTOR: Group schemes, Differential algebra, Algebraic theory of abelian varieties Geometry of differential polynomial functions. I: Algebraic 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 [For the entire collection see Zbl 0744.00034.]
The article under review is devoted to a theory of linear systems over singular K3-surfaces (i.e. of normal projective algebraic surfaces, such that the minimal resolution is K3 in the usual sense). This is motivated by relations with the classification theory for Fano threefolds. -- The author studies trees of curves arising in the following way: First of all, let \(H\) be an effective divisor on the smooth K3-surface \(X\), \(| H |=| C |+\Delta\) \((\Delta\) the fixed part of the linear system), and consider the condition:
\((*)\) One of the following assertions is satisfied:
(i) \(C^ 2>0\), \(| C |\) contains an irreducible curve and has no fixed points,
(ii) \(C^ 2=0\), \(| C |=m \cdot | E |\), \(| E |\) an elliptic pencil,
(iii) \(C=0\).
Then, for \(C\) and \(\Delta\) satisfying \((*)\), the condition \(| C+\Delta |=| C |+\Delta\) is shown to be equivalent with the property: \(G(C,\Delta)\) (:=dual graph of the intersections of the irreducible components) is a tree, and it has no subtrees \(D^ \sim_ m\), \(D^ \sim_ m(C)\) \((m \geq 4)\), \(E^ \sim_ m\), \(E^ \sim_ m(C)\) \((m=6,7,8)\), \(B^ \sim_ m(C)\) \((m \geq 2)\), \(G^ \sim_ 2(C)\), where \(C\) denotes the terminal element in one ``branch'' of maximal length in the corresponding graph.
This theorem reduces the description of all possible graphs \(G(C,\Delta)\) to the description of trees containing at most one curve \(C\) with \(C^ 2 \geq 0\) and with all other irreducible components nonsingular rational curves. -- Now let \({\mathcal T}\) be such a tree, \(G(C)\) its intersection graph, \(G\) the intersection graph of the smooth rational components of \({\mathcal T}\). The Hodge index theorem allows at most one positive square for the corresponding intersection matrix. Thus a classification is obtained into hyperbolic, parabolic and elliptic type for \(G\). What follows is a closer investigation of all possible trees \(G(C)\) (the rank of the Picard lattice is \(\leq 22\), resp. \(\leq 20\) for the case of characteristic 0), reducing the problem to questions of combinatorics and linear algebra.
The general case is treated in this way: Let \(\sigma:X \to Y\) be the minimal resolution of a singular K3-surface \(Y\), \(\Delta_ s:=\sum b_ jF_ j\) \((b_ j \geq 0)\), and \(F_ j\) the components of the exceptional divisor on \(X\). Further, let \(D\) be an effective divisor on \(X\). A complete Weil linear system is the image \(\overline D=\sigma_ *(| D |+\Delta_ s)\), which is stabilizing for large \((b_ j)\). As before, a graph \(G(C,\Delta)\) can be defined, and there is an analog of the condition \((*)\) with \(\Delta=\Delta_ r+\Delta_ s\), \(\Delta_ r=\sum a_ i \Gamma_ i\) \((a_ i \in \mathbb{N})\) fixed, and \(\Delta_ s=\sum b_ jF_ j\), \((b_ j>>0)\), giving rise to a similar characterization of the equality \(| C+\Delta |=| C |+\Delta\). -- For the case of \(\text{rk(Pic} Y)=1\), a more detailed discussion is included. linear systems over singular K3-surfaces; Fano threefolds; effective divisor; Hodge index theorem; complete Weil linear system Nikulin, V.V.: Weil linear systems on singular \(K3\) surfaces. In: Algebraic Geometry and Analytic Geometry (Tokyo, 1990). ICM-90 Satellite Conference Proceedings, pp. 138-164. Springer, Tokyo (1991) \(K3\) surfaces and Enriques surfaces, Singularities of surfaces or higher-dimensional varieties, Divisors, linear systems, invertible sheaves Weil linear systems on singular K3 surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathcal E}\) be a disjoint decomposition of \(\mathbb R^n\) and let \(X\) be a vector field on \(\mathbb R^n\), defined to be linear on each cell of the decomposition \({\mathcal E}\). Under some natural assumptions, we show how to associate a semiflow to \(X\) and prove that such semiflow belongs to the o-minimal structure \(\mathbb R_{\text{an},\exp}\). In particular, when \(X\) is a continuous vector field and \(\Gamma\) is an invariant subset of \(X\), our result implies that if \(\Gamma\) is non-spiralling then the Poincaré first return map associated \(\Gamma\) is also in \(\mathbb R_{\text{an} ,\exp}\). piecewise linear vector field; o-minimal structure; semiflow Dynamics induced by flows and semiflows, Periodic solutions to ordinary differential equations, Model theory of ordered structures; o-minimality, Semialgebraic sets and related spaces Tame semiflows for piecewise linear vector 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 extend to higher dimensions some of the valuative analysis of singularities of plurisubharmonic (psh) functions developed by the first two authors. Following Kontsevich and Soibelman, we describe the geometry of the space \({\mathcal V}\) of all normalized valuations on \(\mathbb C[x_1,\dots,x_n]\) centered at the origin. It is a union of simplices naturally endowed with an affine structure. Using relative positivity properties of divisors living on modifications of \(\mathbb C^n\) above the origin, we define formal psh functions on \({\mathcal V}\), designed to be analogues of the usual psh functions. For bounded formal psh functions on \({\mathcal V}\), we define a mixed Monge-Ampère operator which reflects the intersection theory of divisors above the origin of \(\mathbb C^n\). This operator associates to any \((n-1)\)-tuple of formal psh functions a positive measure of finite mass on \({\mathcal V}\). Next, we show that the collection of Lelong numbers of a given germ \(u\) of a psh function at all infinitely near points induces a formal psh function \(\widehat{u}\) on \({\mathcal V}\). When \(\varphi\) is a psh Holder weight in the sense of Demailly, the generalized Lelong number \(\nu_\varphi(u)\) equals the integral of \(u\) against the Monge-Ampère measure of \(\widehat{\varphi}\). In particular, any generalized Lelong number is an average of valuations. We also show how to compute the multiplier ideal of \(u\) and the relative type of \(u\) with respect to \(\varphi\) in the sense of Rashkovskii, in terms of \(\widehat {u}\) and \(\widehat{\varphi}\). Monge-Ampère operator; Riemann-Zariski space; Weil divisor; nef Weil divisors; intersection theory Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias, Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci., 44, 2, 449-494, (2008) Lelong numbers, Valuations and their generalizations for commutative rings, Singularities in algebraic geometry, Non-Archimedean analysis, Modifications; resolution of singularities (complex-analytic aspects) Valuations and plurisubharmonic singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is divided into two distinct parts. The first part is about the positivity of a refinable function associated with a mask of positive elements. The authors obtain several characterizations for the positivity of such functions. The second part presents a conjecture about cardinal interpolation by refinable functions. More precisely, the authors conjecture that: \textit{for} or \textit{any} \(n>1\), \textit{the polynomial}
\[
V_n(z)=\sum_j\phi_n(j+1)z^j, \quad z\in C
\]
\textit{never vanishes on the unit circle}, where \(\phi_n\) is the orthonormal Daubechies scaling function with supp\((\phi_n)=[0, 2n-1]\). The authors prove the following fact which strongly supports this conjecture: Let \(\delta_n\) be the unique number in \([0, \pi]\) satisfying \((1-\sin ^2\delta_n)^n=\sin \delta_n\). There exists a positive integer \(m\) such that for \(n\geq m\), there holds for all \(0\leq | \xi| \leq \pi-2\delta_n\) the bound \(| V_n(e^{i\xi})| \geq {3 \over {10}}\). refinable functions; subdivision scheme; cardinal interpolation; orthogonal wavelets; irreducible matrix C.A. Micchelli, D.-X. Zhou, Refinable functions: positivity and interpolation, Anal. Appl., 2003, to appear. Nontrigonometric harmonic analysis involving wavelets and other special systems, Interpolation in approximation theory, General harmonic expansions, frames Refinable functions: positivity and interpolation. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 develops a necessary condition for the regularity of a multivariate refinable function in terms of a factorization property of the associated subdivision mask. The extension to arbitrary isotropic dilation matrices necessitates the introduction of the concepts of restricted and renormalized convergence of a subdivision scheme as well as the notion of subconvergence, i.e., the convergence of only a subsequence of the iterations of the subdivision scheme. Since, in addition, factorization methods pass even from scalar to matrix valued refinable functions, those results have to be formulated in terms of matrix refinable functions or vector subdivision schemes, respectively, in order to be suitable for iterated application. Moreover, it is shown for a particular case that the condition is not only a necessary but also a sufficient one. subdivision; subconvergence; wavelet; factorization methods Sauer, T.: Differentiability of multivariate refinable functions and factorization. Adv. Comput. Math. 26(1--3), 211--235 (2006) Numerical methods for wavelets Differentiability of multivariate refinable functions and factorization | 1 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors present a fast multiscale collocation method for solving Fredholm integral equations of the second kind with weakly singular kernels on polyhedral domains in \(\mathbb R^d\). The polyhedral domain is subdivided into a finite number of simplices, and multiscale bases and corresponding collocation functionals are constructed on a uniform self-similar partition of any simplex to achieve a compression of the matrix representation of the integral operator. The authors present concrete multiscale bases and collocation functionals on simplices in \(\mathbb R^d\), \(d=1,2,3\). They develop a cubature rule for computing the weakly singular integrals based on an error control strategy which is designed to preserve the nearly optimal order of convergence and computational complexity of the method.
Unfortunately, the section devoted to a numerical experiment is incomplete, since Table 3 reporting on the results of the solution of a weakly singular integral equation in \(\mathbb R^3\) is missing. Fredholm integral equations of the second kind; high dimension; fast collocation methods; multi-scale methods Chen, Z., Wu, B., Xu, Y.: Fast collocation methods for high dimensional weakly singular integral equations. J. Integral Equations Appl. 20, 49--92 (2008) Numerical methods for integral equations, Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) Fast collocation methods for high-dimensional weakly singular integral 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 Fractal interpolation provides an efficient way to describe data that have smooth and non-smooth structures. Based on the theory of fractal interpolation functions (FIFs), the Hermite rational cubic spline FIFs (fractal boundary curves) are constructed to approximate an original function along the grid lines of interpolation domain. Then the blending Hermite rational cubic spline fractal interpolation surface (FIS) is generated by using the blending functions with these fractal boundary curves. The convergence of the Hermite rational cubic spline FIS towards an original function is studied. The scaling factors and shape parameters involved in fractal boundary curves are constrained suitably such that these fractal boundary curves are positive whenever the given interpolation data along the grid lines are positive. Our Hermite blending rational cubic spline FIS is positive whenever the corresponding fractal boundary curves are positive. Various collections of fractal boundary curves can be adapted with suitable modifications in the associated scaling parameters or/and shape parameters, and consequently our construction allows interactive alteration in the shape of rational FIS. fractals; iterated function systems; fractal interpolation functions; blending functions; fractal interpolation surfaces; positivity Chand, AKB; Vijender, N., Positive blending Hermite rational cubic spline fractal interpolation surfaces, Calcolo, 52, 1-24, (2015) Fractals, Numerical interpolation, Computational aspects of algebraic surfaces, Attractors of solutions to ordinary differential equations, Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics Positive blending Hermite rational cubic spline fractal interpolation surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Fractal interpolation is a modern technique for fitting of smooth/non-smooth data. In the present article, we develop the \(\mathcal{C}^{1}\)-rational cubic fractal interpolation surface (FIS) as a fixed point of the Read-Bajraktarević (RB) operator defined on a suitable function space. Our \(\mathcal{C}^{1}\)-rational cubic FIS is effective tool to stich surface data arranged on a rectangular grid. Our construction needs only the functional values at the grids being interpolated, therefore implementation is an easy task. We first construct the \(x\)-direction rational cubic FIFs (\(x\)-direction fractal boundary curves) to approximate the data generating function along the grid lines parallel to \(x\)-axis. Then we form a rational cubic FIS as a blending of these fractal boundary curves. An upper bound of the uniform distance between the rational cubic FIS and an original function is estimated for the convergence results. A numerical illustration is provided to explain the visual quality of our rational cubic FIS. An~extra feature of this fractal surface scheme is that it allows subsequent interactive alteration of the shape of the surface by changing the scaling factors and shape parameters. fractals; iterated function systems; fractal interpolation functions; fractal interpolation surfaces; blending function; numerical examples; fitting; Read-Bajraktarević operator Numerical interpolation, Numerical smoothing, curve fitting, Fractals, Computational aspects of algebraic surfaces, Computer-aided design (modeling of curves and surfaces) \(\mathcal{C}^{1}\)-rational cubic fractal interpolation surface using functional values | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems First let us quote from the preface: ``This book began about 20 years ago in the form of supplementary notes for my algebra classes. I wanted to discuss some concrete topics such as symmetry, linear groups, and quadratic number fields in more detail than the text provided, and to shift the emphasis in group theory from permutation groups to matrix groups. Lattices, another recurring theme, appeared spontaneously. My hope was that the concrete material would interest the students and that it would make the abstractions more understandable, in short, that they could get farther by learning both at the same time. This worked pretty well\dots The main novel feature of the book is its increased emphasis on special topics\dots
In writing the book I tried to follow these principles:
1. The main examples should precede the abstract definitions.
2. The book is not intended for a `service course', so technical points should be presented only if they are needed in the book.
3. All topics discussed should be important for the average mathematician.''
And in ``A note for the teacher'' the author says: ``There are few prerequisites for this book. Students should be familiar with calculus, the basic properties of complex numbers, and mathematical induction. Some acquaintance with proofs is obviously useful, though less essential. The concepts from topology, which are used in chapter 8, should not be regarded as prerequisites. An appendix is provided as a reference for some of these concepts; it is too brief to be suitable as a text.'' In some sense the first chapter sets the tone for the book by giving a very down to earth introduction to matrix operations.
Chapters 2, 3, 4 give very concrete introductions to groups and linear algebra, including infinite dimensional spaces, systems of linear differential equations and the matrix exponential.
Chapter 5 treats symmetry, starting with the symmetry of plane figures; it contains also a classification of the finite subgroups of the rotation group in three dimensions.
Chapter 6 continues with group theory, including free groups, generators and relations and the Todd-Coxeter algorithm.
Chapter 7 treats bilinear forms, including spectral theorems. The next two chapters are devoted to linear groups and group representations, including a brief discussion of continuous representations of compact groups. As an example
the representations of the group \(SU_2\) are found.
Chapter 10 is about
rings and presents some rudiments of algebraic geometry.
In chapter 11 the problem of factorization with applications to certain diophantine equations is discussed. Chapter 12 on modules contains also a section about free modules over polynomial rings. The next chapter on fields gives also a very interesting introduction to function fields and their relations to Riemann surfaces. The final chapter on Galois theory contains many classical examples, including the cubic, the quartic and the quintic equations.
Each chapter is followed by many exercises, simple ones which help the reader to understand the basic concepts introduced in the text, but also very interesting ones. -- The book contains also some misprints.
Everyone who uses this book, the student, the teacher or the researcher will certainly agree with me: This is a wonderful and useful book! finite subgroups of rotation group; groups; linear algebra; infinite dimensional spaces; systems of linear differential equations; symmetry; free groups; generators; relations; Todd-Coxeter algorithm; bilinear forms; spectral theorems; linear groups; group representations; rings; algebraic geometry; factorization; modules; function fields and their relations to Riemann surfaces; Galois theory Artin, M.: Algebra. Prentice-Hall, Englewood Cliffs (1991) Mathematics in general, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics in general, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory 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 See the preceding review of the English original. groups; linear algebra; infinite dimensional spaces; systems of linear differential equations; symmetry; finite subgroups of rotation group; free groups; generators; relations; Todd-Coxeter algorithm; bilinear forms; spectral theorems; linear groups; group representations; rings; algebraic geometry; factorization; modules; function fields and their relations to Riemann surfaces; Galois theory Mathematics in general, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics in general, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory 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 Interpolation is a central tool in numerical analysis and in computer-aided geometric design (CAGD) and scientific applications. Especially in imaging in CAGD and in physics applications for instance, not only uni- but also multivariate (sometimes also called multi-variable) interpolation is very frequently used. In this paper this is described in the context of (bivariate) surfaces (therefore most closely related to CAGD) of a method based on contraction mappings where the sought interpolants (here in particular: surfaces) are limits, so-called interpolating fractal surfaces (IFS). Convergence results and existence theorems are established especially for rational fractal interpolating surfaces. fractals; iterated function systems; fractal interpolation functions; self-referential; fractal interpolation surfaces; blending function; convergence; computer-aided geometric design Chand, AKB; Vijender, N, A new class of fractal interpolation surfaces based on functional values, Fractals, 24, 1-17, (2016) Numerical interpolation, Fractals, Computational aspects of algebraic surfaces, Interpolation in approximation theory, Multidimensional problems A new class of fractal interpolation surfaces based on functional values | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Topological recursion associates a double family of differential forms \(\omega_{g,n}\) to a `spectral curve' \(\mathcal{C}\). This paper uses as initial data only a \textit{local} spectral curve, whose archetypal example is an open Riemann surface doubled across cuts. This allows topological recursion to be applied to generalized matrix models which have an eigenvalue representation, also known as `repulsive particle systems'. Previous applications to one-Hermitian and to two-Hermitian matrix models are recovered as special cases, as their Schwinger-Dyson equations arise as special cases of the `abstract loop equations' of this paper.
Four new examples are studied. First, the \(1/N\) expansions of systems of repulsive particles, when it exists. Second, enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies. Third, \(\mathrm{SU}(N)\) Chern--Simons invariants of torus knots. And finally, Liouville theory on surfaces of positive genus, at a formal level and without addressing issues such as convergence.
Aside from the individual significances of these examples and of others, this paper is a further step towards establishing a general universal framework for the topological recursion. The level of generality considered in this paper must both be decreased in order to recover symplectic invariance (by how much is unclear), and increased in order to cover a larger class of generalized matrix models (the authors speculate that topological recursion may be applicable to all quiver matrix models).
In recent years, topological recursion has established itself as an exciting and powerful approach with applications to problems in \(2\)--dimensional enumerative geometry, to the two-Hermitian matrix model, to the chain of Hermitian matrices, to topological string theory and Gromov-Witten invariants, to integrable systems, to intersection numbers on the moduli space of curves, and to quantum invariants of knots and links. The paper under review serves to increase the scope of application of topological recursion while at the same time elucidating its foundations. topological recursion; partition function; spectral curve; matrix models; 1/N expansion; Schwinger-Dyson equation; repulsive particle systems; AGT conjecture; BKMP conjecture; Chern-Simons theory Borot, G., Eynard, B., Orantin, N.: Abstract loop equations, topological recursion and applications. Commun. Numbers Theory Phys. \textbf{9}(1) (2015). arXiv:1303.5808 [math-ph] Relationships between algebraic curves and physics, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Structure of families (Picard-Lefschetz, monodromy, etc.), Noncommutative geometry in quantum theory, Groups and algebras in quantum theory and relations with integrable systems, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, KdV equations (Korteweg-de Vries equations), Quantization in field theory; cohomological methods, Eta-invariants, Chern-Simons invariants Abstract loop equations, topological recursion and new 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 propose a class of affine fractal interpolation surfaces (FISs) that stitch a given set of surface data arranged on a rectangular grid. The proposed FISs are blending of the affine fractal interpolation functions (FIFs) constructed along the grid lines of given interpolation domain. We investigate the stability results of the developed affine FIS with respect to its independent and dependent variables at the grids. These affine FISs preserve the inherited shape of given surface data (like monotonicity, positivity, and convexity), whenever the associated affine FIFs mimic the shape of the univariate data sets along the grid lines of interpolation domain. By using suitable conditions on the scaling factors, we study the monotonicity preserving interpolation via \(\mathcal{C}^0\)-continuous affine FIFs. Under these conditions, apart from one scaling factor, the rest depend only on the functional values but not on both the horizontal contractive factors and slopes at the grids.This weak restriction provides a large flexibility in the selection of the scaling factors for monotonicity preserving \(\mathcal{C}^0\)-continuous affine FIFs/FISs. The positivity criterion for \(\mathcal{C}^0\)-continuous affine FIF is also deduced. fractals; iterated function systems; fractal interpolation functions; fractal interpolation surfaces; blending function; monotonicity; positivity; convexity Vijender, N.; Chand, AKB, Shape preserving affine fractal interpolation surfaces, Nonlinear Stud, 21, 179-194, (2014) Fractals, Numerical interpolation, Computational aspects of algebraic surfaces, Attractors of solutions to ordinary differential equations, Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics Shape preserving affine fractal interpolation surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Stable Khovanov-Rozansky polynomials of algebraic knots are expected to coincide with certain generating functions, superpolynomials, of nested Hilbert schemes and flagged Jacobian factors of the corresponding plane curve singularities. Also, these 3 families conjecturally match the DAHA superpolynomials. These superpolynomials can be considered as singular counterparts and generalizations of the Hasse-Weil zeta-functions. We conjecture that all \(a\)-coefficients of the DAHA superpolynomials upon the substitution \(q \mapsto qt\) satisfy the Riemann Hypothesis for sufficiently small \(q\) for uncolored algebraic knots, presumably for \(q \leq 1/2\) as \(a = 0\). This can be partially extended to algebraic links at least for \(a = 0\). Colored links are also considered, though mostly for rectangle Young diagrams. Connections with Kapranov's motivic zeta and the Galkin-Stöhr zeta-functions are discussed. double affine Hecke algebras; Jones polynomials; HOMFLY-PT polynomials; plane curve singularities; compactified Jacobians; Hilbert scheme; Khovanov-Rozansky homology; iterated torus links; Macdonald polynomial; Hasse-Weil zeta-function; Riemann hypothesis Knots and links in the 3-sphere, Plane and space curves, Hecke algebras and their representations, Braid groups; Artin groups, Root systems, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Singular homology and cohomology theory Riemann hypothesis for DAHA superpolynomials and plane curve singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f_1(x)\) and \(f_2(x)\) be two monic and coprime polynomials in \(\mathbb{F}_q[x]\) and suppose that \(m=\deg f_1>\deg f_2\). The author considers the function field extension \(L\mid\mathbb{F}_q(y)\), where \(L\) is a splitting field over \(\mathbb{F}_q(y)\) for the polynomial \(F(x,y)\):
\[
F(x,y)=f_1(x)-yf_2(x).
\]
The following bound on the genus \(g\) of \(L\) is obtained:
\[
2g\leq(m-3)\cdot\bigl[L:\mathbb{F}_q(y)\bigr]+2.
\]
Quite related to the number of rational places of \(L\) is the number \(N\) of elements \(a\in\mathbb{F}_q\) such that the polynomial \(f_1(x)-af_2(x)\) is a product of distinct linear factors. A bound on \(N\) is then deduced from the Hasse-Weil bound; clearly, one assumes that \(\mathbb{F}_q\) is the full constant field of \(L\) and two interesting criteria for this assumption to hold are given.
Applications to graph theory and coding theory are given (Steiner systems, covering and packing radius of codes,\dots). finite fields; upper bound; Steiner systems; covering radius; codes; function field extension; genus; rational places; packing radius Cohen, S. D., Some function field estimates with applications, (Number Theory and Its Applications, Ankara, 1996, Lect. Notes Pure Appl. Math., vol. 204, (1999), Dekker New York), 23-45 Curves over finite and local fields, Algebraic functions and function fields in algebraic geometry, Algebraic coding theory; cryptography (number-theoretic aspects), Directed graphs (digraphs), tournaments, Applications of the theory of convex sets and geometry of numbers (covering radius, etc.) to coding theory Some function field estimates 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 Let \(C\) be a smooth projective curve of genus \(g\geq 3\), and let \(J(C)\) be its Jacobian variety. Choosing a base point, one can embed \(C\) into \(J(C)\), and define \(W_k\) to be the natural image of the \(k\)-th symmetric product of \(C\). \textit{G. Ceresa} [Ann. Math. (2) 117, 285--291 (1983; Zbl 0538.14024)] showed that \(W_k-W_k^-\) is not algebraically trivial when \(C\) is generic and \(1\leq k\leq g-2\). In this article the author focuses on the Fermat curve of degree \(N\geq 4\) and gives a criterion for the non-vanishing of \(W_k-W_k^-\) modulo algebraic equivalence in terms of special values of generalized hypergeometric functions. algebraic cycle; iterated integral; hypergeometric function Algebraic cycles, Generalized hypergeometric series, \({}_pF_q\), Appell, Horn and Lauricella functions On the Abel-Jacobi maps of Fermat Jacobians | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems With the notation of the previous review (Zbl 0932.13010) let \(G\) be a line-\(S_4.\) Then it is shown that the nondegenerate \(G\)-quartics are parametrized by \(F.\) For an appropriate \(G\) each nondegenerate \(G\)-quartic is a constant multiple of \(h_a = az^4 + (x^2 + yz)(y^2 + xz)\), \(a \in F.\) For each \(a\) the author attaches an integer \(l = l(a)\) that determines completely \(e_n(h_a).\) In particular, it turns out that \(c(h_a) = 3 + 4^{-2l}.\) The surprise is the definition of \(l = l(a).\) The author constructs a \(1\)-parameter family of dynamical systems parametrized by \(F.\) Then \(l(a)\) is defined to be an `escape time' for the system corresponding to \(a.\) Hilbert-Kunz function; Hilbert-Kunz multiplicity; plane quartic; dynamical systems Monsky, P.: Hilbert--kunz functions in a family: line-S4 quartics. J. algebra 208, 359-371 (1998) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Families, moduli of curves (algebraic), Polynomial rings and ideals; rings of integer-valued polynomials Hilbert-Kunz functions in a family: Line-\(S_4\) quartics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Studies of Riemann surfaces of infinite genus have been started in mathematics some years ago [see e.g. \textit{P. Myrberg}, Acta Math. 76, 185-224 (1945; Zbl 0060.21602)]. According to McKean and Trubowitz (1976) there exists a one-to-one correspondence between divisors of Riemann surfaces of infinite genus and smooth almost-periodic solutions of the integrable evolution equations in terms of theta functions similar to what had been proved before for the compact Riemann surfaces by Its and Matveev (1975). This approach is summarized in the books [\textit{B. M. Levitan}, ``The Sturm-Liouville problems'', Moscow: Nauka (1984; Zbl 0575.34001) and \textit{J. Pöschel} and \textit{E. Trubowitz}, ``Inverse spectral theory'', Boston Academic Press (1987; Zbl 0623.34001)].
In his memoir, the author proposes new and more geometrical methods to describe this correspondence based on an appropriate generalization of the Picard group (i.e., the set of all equivalence classes of holomorphic line bundles with their tensor product) for the Riemann surfaces of infinite genus. With this aim he adds in a special way to the spectral curve of the Lax operator (which is in this case the ordinary differential matrix operator with periodic coefficients) the points corresponding to infinite value of the spectral parameter. The resulting object is no longer a Riemann surface in the usual sense but is quite similar to the compact one. It allows to generalize in natural way all basic tools of the theory of compact Riemann surfaces to this spectral curve and thus describe explicitly for the infinite genus the structure of complete integrability: 1) the eigenbundles define holomorphic line bundles on the spectral curve which completely determine potentials; 2) the line bundles are described by divisors of the same degree as the genus; 3) these divisors give the Darboux coordinates, and the Serre duality leads to the symplectic form; 4) the isospectral sets may be identified with open dense subsets of the Jacobian varieties in accordance with the Riemann-Roch theorem; 5) the real parts of the isopectral sets are infinite-dimensional tori etc. integrable systems; Picard group; Riemann surfaces of infinite genus; complete integrability; eigenbundles; holomorphic line bundles; spectral curve; divisors; Darboux coordinates; Serre duality Schmidt M U 1996 \textit{Integrable Systems and Riemann Surfaces of Infinite Genus}\textit{(Memoirs of the American Mathematical Society vol 122)} (Providence, RI: American Mathematical Society) pp 1--111 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.), Singularities of curves, local rings, Theta functions and curves; Schottky problem, NLS equations (nonlinear Schrödinger equations) Integrable systems and Riemann surfaces of infinite 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 Let \(\mathbb {P}^1=\mathbb {P}^1_{k}\) with \(k\) an algebraically closed field and let \(\mathcal{Q}=\mathbb {P}^1\times \mathbb {P}^1\) be a smooth quadric. Let \(S=k[u,u',v,v']\) be the bigraded ring, \(X\) be a zero-dimensional scheme and \(I=I(X)\) its saturated bigraded ideal in \(S\). A zero-dimensional scheme \(X\subseteq \mathcal{Q}\) is said \textit{scheme-theoretically generated} by \(r\) forms \(f_1,\dots,f_r\in S\) with \(\deg f_i=(a_i,b_i)\) if there exists a sheaves surjection \(\bigoplus_{i=1}^r \mathcal{O}_{\mathcal{Q}}(a_i,b_i)\overset{\phi}\rightarrow \mathcal{I}_X\rightarrow 0\) with \(\phi=(f_1,\dots,f_r)\). If \(r=2\), \(X\) is called \textit{scheme-theoretically complete intersection}.
Since in general in \(\mathbb {P}^1\times \mathbb {P}^1\) a bigraded ideal \(I(X)\) generated by a regular sequence is not saturated, it is interesting to study zero-dimensional schemes \(X\) defined by ideals that are the saturation of bigraded ideals generated by regular sequence.
In this paper, the authors study the case \(r=2.\)
In particular, Section 3 is devoted to the minimal case, i.e., when \(X\), arising by the ideal \((f,g),\) with \(\deg f=(a,b)\) and \(\deg g=(c,d)\), is contained in two curves of type \((a+c,0)\) and \((0,b+d).\) They found that \(X\) is the union of two \(0\)-grids and describe a minimal free bigraded resolution, showing that \(X\) has only four minimal generators (in the minimal case).
Section 4 is devoted to the general case, i.e., the curves \(C\) and \(D\) are two general curves of bidegree \((a,b)\geq (1,1)\) and \((c,d)\geq (1,1)\), respectively. They compute the Hilbert function of \(X\) and prove that the saturated ideal of \(X\) can be obtained by saturating the row ideals or the column ideals. bigraded rings; bigraded modules; complete intersection; Hilbert function; minimal free resolution; scheme-theoretic complete intersection Divisors, linear systems, invertible sheaves, Syzygies, resolutions, complexes and commutative rings, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Scheme-theoretic complete intersections in \(\mathbb P^1 \times\mathbb P^1\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper centres on the construction of a right half-plane spectral sequence \(H^ P_{Zar}(X;\underset \tilde{} K_{-q})\Rightarrow K_{-p- q}(X)\), conjectured to be valid for any quasiprojective scheme X (this has subsequently been proved by Thomason and Trobaugh). For a scheme U, take the category \(\underset \tilde{} P(U)\) of coherent vector bundles over U; apply Quillen's construction to get the category \(Q\underset \tilde{} P(U)\); and feed this into an infinite loop space machine to get a fibrant spectrum \(\underset \tilde{} K^ Q(U)\). A key idea of this paper is to extend this to a non-connective spectrum \(\underset \tilde{} K(U)\) using the negative K-groups of Bass. Consider this as a functor on Z-open subspaces U of X: then if this has a homotopy pullback property for pairs (U,V,U\(\cap V,U\cup V)\) a general theorem of \textit{K. S. Brown} and \textit{S. M. Gersten} [Lect. Notes Math. 341, 266-292 (1973; Zbl 0291.18017)] yields the desired spectral sequence. In this note the property is derived for the case that X has isolated singularities, using formal properties and the fact that it was already known when X had a unique singular point [\textit{A. Collino}, Ill. J. Math. 25, 654-666 (1981; Zbl 0496.14005)]. right half-plane spectral sequence; quasiprojective scheme; coherent vector bundles; infinite loop space; fibrant spectrum; non-connective spectrum; isolated singularities Charles A. Weibel, A Brown-Gersten spectral sequence for the \?-theory of varieties with isolated singularities, Adv. in Math. 73 (1989), no. 2, 192 -- 203. Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Varieties and morphisms, Infinite loop spaces A Brown-Gersten spectral sequence for the K-theory of varieties with isolated singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider the usual model of hypermaps or, equivalently, bipartite maps, represented by pairs of permutations that act transitively on a set of edges \(E\). The specific feature of our construction is the fact that the elements of \(E\) are themselves (or are labelled by) rather complicated combinatorial objects, namely, the 4-constellations, while the permutations defining the hypermap originate from an action of the Hurwitz braid group on these 4-constellations. The motivation for the whole construction is the combinatorial representation of the parameter space of the ramified coverings of the Riemann sphere having four ramification points. Riemann surface; ramified covering; dessins d'enfants; Belyi function; braid group; Hurwitz scheme; hypermaps; bipartite maps A. Zvonkin, ''Megamaps: Construction and Examples,'' in: \textit{Discr. Math. Theor. Comput. Sci. Proc., AA} (2001), pp. 329-339. Algebraic combinatorics, Topology of Euclidean 3-space and 3-sphere, Topology of Euclidean 2-space, 2-manifolds, Riemann surfaces; Weierstrass points; gap sequences Megamaps: Construction and examples | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We propose a method for diagonalizing matrices with entries in commutative rings. The point of departure is to split the characteristic polynomial of the matrix over a (universal) splitting algebra, and to use the resulting universal roots to construct eigenvectors of the matrix. A crucial point is to determine when the determinant of the eigenvector matrix, that is the matrix whose columns are the eigenvectors, is regular in the splitting algebra. We show that this holds when the matrix is generic, that is, the entries are algebraically independent over the base ring. It would have been desirable to have an explicit formula for the determinant in the generic case. However, we have to settle for such a formula in a special case that is general enough for proving regularity in the general case. We illustrate the uses of our results by proving the Spectral Mapping Theorem, and by generalizing a fundamental result from classical invariant theory. matrices; determinant; diagonalization; eigenvector; eigenvalue; symmetric function; splitting algebra; universal root; regularity; spectral mapping Polynomials over commutative rings, Galois theory and commutative ring extensions, Rings of fractions and localization for commutative rings, Theory of matrix inversion and generalized inverses, Determinants, permanents, traces, other special matrix functions, Matrices over function rings in one or more variables, Polynomials and finite commutative rings, Determinantal varieties Diagonalization of matrices over rings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let H be the Hilbert scheme of projective space \({\mathbb{P}}^ n\). It is well known that H is a disjoint union of subschemes \(H_ Q\) of finite type, each parametrizing coherent sheaves of ideals with a given Hilbert polynomial Q. In fact, this is the way H is constructed by \textit{A. Grothendieck} [``Techniques de construction et théorèmes d'existence en géométrie algébrique. IV'', Sém. Bourbaki 13 (1960/61), Exposé 221 (1969; Zbl 0236.14003). \textit{R. Hartshorne} proved that the \(H_ Q\) are connected [Publ. Math., Inst. Hautes Étud. Sci. 29, 5-48 (1966; Zbl 0171.415)]. The paper under review generalizes this connectedness result in the following way:
For a given ideal \(I\subseteq {\mathcal O}_{P^ n}\), let h(I) be the Hilbert function of I. Introduce a partial order on the set of all functions \({\mathbb{N}}\to {\mathbb{N}}\) by putting \(f\leq g\) if and only if f(n)\(\leq g(n)\) for all \(n\in {\mathbb{N}}\). Then by standard semicontinuity results, h(I) is a semicontinuous function of I, i.e., if \(f:\quad {\mathbb{N}}\to {\mathbb{N}}\) is given, then the set \(H_{\geq f}=\{I\in H_ Q| \quad h(I)_{\geq f}\}\) is a closed subset of \(H_ Q\). The main theorem of the present paper is that the subschemes \(H_{\geq f}\) are connected in characteristic zero. - The proof makes use of the action of the upper triangular subgroup of \(GL(n+1)\) (and its subgroups) on \({\mathbb{P}}^ n\) and the induced action on \(H_ Q\). Several kinds of specializations are considered (arising from the group actions) and a careful study of the behaviour of the Hilbert function under such specialization is undertaken. Hilbert scheme; Hilbert function Gotzmann, G.: Durch Hilbertfunktionen definierte Unterschemata des Hilbertschemas. Comment. Math. Helv.63, 114--149 (1988) Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients) Durch Hilbertfunktionen definierte Unterschemata des Hilbertschemas. (Subschemes of the Hilbert scheme defined by Hilbert functions) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author addresses the problem of classifying point systems (or \(N\)- uples of points) in the projective plane. More precisely, if \(Z\subset\mathbb{P}^ 2\) is a finite subscheme of degree \(N\) and \(d\) is a positive integer, consider the linear system \(I_ Z(d)\) of plane curves of degree \(d\) containing \(Z\). The subscheme \(Z\) is called superabundant (for the linear system of plane curves of degree \(d)\) if the dimension of \(I_ Z(d)\) is strictly greater than the expected dimension \({d+2\choose 2}-1-N\). The set of superabundant subschemes \(Z\) is a subscheme \(W_ N[d]\) of the Hilbert scheme \(\text{Hilb}^ N\mathbb{P}^ 2\). The author's main result is to give the dimension of \(W_ N[d]\) and determine when it is irreducible, in terms of \(d\) and \(N\). classifying point systems; superabundant subschemes; Hilbert scheme Coppo, M. -A.: Familles maximales de systèmes de points surabondants dans P2. Math. ann. 291, 725-735 (1991) Projective techniques in algebraic geometry, Parametrization (Chow and Hilbert schemes), Divisors, linear systems, invertible sheaves Maximal families of superabundant point systems in \(\mathbb{P}^ 2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems generalized spectral radius theorem; generalized analytic spaces Algebraic geometry Generalized spectral radius theorem and generalized analytic spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is identical with the author's article [Theor. Math. Phys. 159, No. 2, 598--617 (2009; Zbl 1174.81009); translation from Teor. Mat. Fiz. 159, No. 2, 220--242 (2009)]; only the second half of Section~4 and Section~5 from [loc.\ cit.] are not copied here. Korteweg-de Vries systems; Toda systems; Gromov-Witten potential; Abelian integrals; Nekrasov partition function Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Yang-Mills and other gauge theories in quantum field theory, KdV equations (Korteweg-de Vries equations), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) On nonabelian theories and abelian 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 We handle the postulation of a general disjoint union \(X\subset\mathbb P^n\) of an \(m\)-dimensional linear space and a prescribed number of lines and degree 3 planar connected zero-dimensional subschemes. Hilbert function; unions of linear spaces; zero-dimensional scheme Projective techniques in algebraic geometry Postulation of general unions of lines, a linear space and planar length 3 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 This article pursues the series, initiated by [Rend. Semin. Mat. Univ. Padova 133, 125--158 (2015; Zbl 1326.12004)], dedicated to Pulita's \(\pi\)-exponentials and \(p\)-adic differential equations of rank one with coefficient a polynomial in a ultrametric extension of the field of \(p\)-adic numbers. We complement Part I (loc. cit.) with a closed formula for the index. In particular this answers one problem studied in [\textit{Yuri Morofushi}, \(p\)-adic theory of exponential sums on the affine line. Thesis (Ph.D.) University of Florida. 2010]. We also answer a question ({\S}2.4 of [Mém. Soc. Math. Fr., Nouv. Sér. 23, 61--105 (1986; Zbl 0623.14005)]) of \textit{P. Robba} on the comparison from rational cohomology toward Dwork cohomology (i.e. rigid cohomology on a disk with coefficient). We also indicate a procedure to palliate the lack of isomorphy of this comparison. We establish by the way a characterization of soluble equations up to equivalence on the dagger algebra. An appendix determine the polynomial complexity of the derived algorithm. \(\pi\)-exponentials; \(p\)-adic differential equations: Kernel of Frobenius endomorphism of Witt vectors over a \(p\)-adic ring; radius of convergence function; algorithm; index formula; Dwork cohomology; Rationnal cohomology; Boyarsky principle; \(p\)-adic irregularity; Swan conductor \(p\)-adic differential equations, Witt vectors and related rings, Local ground fields in algebraic geometry On \(\pi\)-exponentials. II: Closed formula for the index | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider graded Gorenstein quotients of \(R=k[X_1,X_2, \dots,X_s]\) of codimension \(r\). If \(s=r\), we let \(PGor(H)\) be the space parametrizing all graded Gorenstein \(R\)-algebras \(R/I\) with Hilbert function \(H\) \((H(i)=\dim(R/I)_i)\), with a scheme structure induced by the vanishing of the relevant catalecticant minors [as explained by \textit{S. J. Diesel}, Pac. J. Math. 172, No. 2, 365-397 (1996; Zbl 0882.13021)]. If \(s>r\) we define \(PGor(H)\) similarly and endow it with an apparently different scheme structure (remark 1.6). The main theorems of this note are concerned with \(PGor(H)\) for \(s=r=3\) in which case we prove that \(PGor (H)\) is a smooth irreducible scheme and we compute its dimension. For \(s>r=3\) a corresponding theorem is proved by \textit{J. O. Kleppe} and \textit{R. M. Miró-Roig} [J. Pure Appl. Algebra 127, No. 1, 73-82 (1998)]. As a corollary we prove a conjecture of Geramita, Pucci, and Shin on the Hilbert function of \(R/I^2\). If \(s=r>3\), we find conditions for \(PGor(H)\) to be smooth and we prove a useful linkage result for computing its dimension. We also prove some results on the codimension of \(PGor(H)\) embedded in some natural strata of the punctual Hilbert scheme. In particular if \(s=r=3\), we compute the dimension of \(ZGor(H)\) of all (not necessarily graded) Gorenstein quotients of \(k[[X_1,X_2, \dots, X_s]]\) with symmetric Hilbert function \(H\) at a graded quotient \(R\to A\), leading to a criterion for \(A\) to be non-alignable. Our method of proof applies the theorem of \textit{A. Iarrobino} and \textit{V. Kanev} [``The length of a homogeneous form, determinantal loci of catalecticants and Gorenstein algebras'' (preprint)] where they determine the tangent space of \(PGor(H)\). punctual Hilbert scheme; graded Gorenstein quotients; Hilbert function; dimension Kleppe, JO, The smoothness and the dimension of \({\mathrm PGor}(H)\) and of other strata of the punctual Hilbert scheme, J. Algebra, 200, 606-628, (1998) Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The smoothness and the dimension of \(PGor(H)\) and of other strata 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 This paper studies symbolic powers of homogeneous radical ideals \(I\) of \(R = k[\mathbb P^n] = k[x_1, \dots, x_{n+1}]\) where \(k\) is an algebraically closed field of characteristic zero. Recall that the \(m\)th symbolic power of \(I\) is defined to be
\[
I^{(m)} = R \cap \bigcap _{Q \in \text{Ass}(I)}(I^m)_Q
\]
where the localizations are embedded in a field of fractions of \(R\). It is well-known (using a theorem of Zariski and Nagata) that
\[
I^{(m)} = \bigcap_{p \in V(I)}M_p^m
\]
where \(M_p\) is the maximal ideal of a point \(P\) and \(V(I)\) is the set of zeroes of \(I\). In addition, as the authors point out, the \(m\)th symbolic power is the set of polynomials which vanish to order \(m\) along \(V(I)\) and so provides geometric information about \(I\). Unfortunately, calculating \(I^{(m)}\) is very challenging. Thus, it is natural to try to compute asymptotic versions (by letting \(m\) grow) of usual invariants of interest, such as the Castelnuovo-Mumford regularity and the initial degree of an ideal. In this direction, \textit{M. Mustaţǎ} [J. Algebra 256, No. 1, 229--249 (2002; Zbl 1076.13500)] and \textit{S. Mayes} [Commun. Algebra 42, No. 5, 2299--2310 (2014; Zbl 1298.13021)] and [J. Pure Appl. Algebra 218, No. 3, 381--390 (2014; Zbl 1283.13025)] connect volumes of complements of limiting shapes and the asymptotic multiplicity for ideals of points. More precisely, using the degree reverse lexicographic order, we let gin\((I^{(m)})\) be the generic initial ideal of \(I^{(m)}\) and define the \textit{limiting shape} of \(I\) to be
\[
\Delta(I) = \bigcup_{m=1}^{\infty} \frac{P(\text{gin}(I^{(m)}))}{m}
\]
where \(P(J)\) denotes the Newton polytope for a monomial ideal \(J\). If \(\Gamma(I)\) is the closure of the complement of \(\Delta(I)\) in \(\mathbb R_{\geq 0}^n\) and \(I\) is the zero-dimensional radical ideal of \(r\) points in \(\mathbb P^n\), then
\[
\text{vol}(\Gamma(I)) = \frac{r}{n!}.
\]
The authors of this paper generalize this relationship for higher dimensional objects. In particular, the authors introduce the \textit{asymptotic Hilbert function}
\[
\text{aHF}_I(t) : = \lim_{m \rightarrow \infty} \frac{\text{HF}_{I^{(m)}}(mt)}{m^n}
\]
and the \textit{asymptotic Hilbert polynomial} of \(I\)
\[
\text{aHP}_I(t) := \lim_{m \rightarrow \infty} \frac{\text{HP}_{I^{(m)}}(mt)}{m^n}.
\]
Here HF\(_J(t) = \dim_k(R_t/J_t)\) denotes the usual Hilbert function of a homogeneous ideal \(J \subseteq R\) and \(HP_J\) is the usual Hilbert polynomial of \(J\).
The main theorem is: If \(I\) is a homogeneous radical ideal, then for each integer \(t \geq 0\), we have
\[
\text{vol}(\Gamma(I) \cap \{(x_1, \dots, x_n) \mid x_1 + \cdots x_n \leq t\}) = \text{aHF}_I(t).
\]
Moreover, for an ideal of \(r\) points in \(\mathbb P^n\) and \(t >> 0\) we have
\[
\text{aHF}_I(t) = \text{aHP}_I(t) = \frac{r}{n!}.
\]
symbolic generic initial systems; limiting shapes; asymptotic Hilbert function and polynomial M. Dumnicki, Asymptotic Hilbert polynomials and limiting shapes, J. Pure Appl. Algebra, 219, 4446, (2015) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Configurations and arrangements of linear subspaces, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Asymptotic Hilbert polynomials and limiting shapes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 coordinate ring of \(\mathbb P^1\times\mathbb P^1\) is the bigraded ring \(R = k[x_0,x_1,y_0,y_1]\) where \(\deg(x_i) = (1,0)\), \(\deg(y_j)=(0,1)\) and \(k\) is an algebraically closed field. A fat point scheme in \(\mathbb P^1\times\mathbb P^1\) is a subscheme \(Z\) defined by an ideal \(I_Z ={\mathfrak p}^{m_1}_1\cap \cdots \cap {\mathfrak p}^{m_s}_s\) where \({\mathfrak p}_i\) is the ideal defining a reduced point and \(m_i\geq 1\). The authors study the Hilbert function \(H_Z(i,j)=\dim_kR_{i,j}/(I_Z)_{i,j}\) of such schemes \(Z\).
Since the coordinate ring of \(Z\) contains homogeneous non-zero divisors of degree \((1,0)\) and \((0,1)\), they can easily show that \(H_Z\) is non-decreasing and eventually constant in each row and colum. A much less obvious result is that the eventual values of \(H_Z\) in each row and column can be calculated directly from the multiplicities of the points and the relative positions of the points in the support of \(Z\) with respect to the lines of the two rulings.
Next, the authors use the eventual behaviour of \(H_Z\) to derive further information about the scheme \(Z\). In particular, they prove that the arithmetically Cohen-Macaulay (ACM) property (i.e. the property that \(R/I_Z\) is a Cohen-Macaulay ring) can be checked using these eventual values. They also derive a number of further informations about the multiplicities and relative position of the points. The paper concludes with some special configurations of ACM fat point schemes. Hilbert function; fat point; zero-dimensional scheme; quadric surface Guardo, Elena; Van Tuyl, Adam, Fat points in \(\mathbb{P}^1\times\mathbb{P}^1\) and their Hilbert functions, Canad. J. Math., 56, 4, 716-741, (2004) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Syzygies, resolutions, complexes and commutative rings Fat points in \(\mathbb{P}^1\times\mathbb{P}^1\) and their Hilbert functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A new class of infinite-dimensional Lie algebras, called Lax operator algebras, is presented, along with a related unifying approach to finite-dimensional integrable systems with a spectral parameter on a Riemann surface such as the Calogero-Moser and Hitchin systems. In particular, the approach includes (non-twisted) Kac-Moody algebras and integrable systems with a rational spectral parameter. The presentation is based on quite simple ideas about the use of gradings of semisimple Lie algebras and their interaction with the Riemann-Roch theorem. The basic properties of Lax operator algebras and the basic facts about the theory of the integrable systems in question are treated (and proved) from this general point of view. In particular, the existence of commutative hierarchies and their Hamiltonian properties are considered. The paper concludes with an application of Lax operator algebras to prequantization of finite-dimensional integrable systems. gradings of semisimple Lie algebras; Lax operator algebras; integrable systems; spectral parameter on a Riemann surface; Tyurin parameters; Hamiltonian theory; prequantization Sh_UMN_2015 Sheinman, O.K. \emph Lax operator algebras and integrable systems. Russian Math. Surveys, 71:1 (2016), 109--156. Infinite-dimensional Lie (super)algebras, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Riemann surfaces; Weierstrass points; gap sequences, Applications of Lie algebras and superalgebras to integrable systems, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Lax operator algebras 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 For a nonnegative symmetric weakly irreducible tensor, its spectral radius is an eigenvalue of the tensor corresponding to a unique positive eigenvector called the Perron vector. But including the Perron vector, it may have more than one eigenvector corresponding to the spectral radius. The projective eigenvariety of the tensor associated with the spectral radius is the set of the eigenvectors of the tensor corresponding to the spectral radius considered in the complex projective space.
In this paper we proved that such projective eigenvariety admits a module structure, which is determined by the support of the tensor and can be characterized explicitly by the Smith normal form of the incidence matrix of the tensor. We introduced two parameters: the stabilizing index and the stabilizing dimension of the tensor, where the former is exactly the cardinality of the projective eigenvariety and the latter is the composition length of the projective eigenvariety as a module. We give some upper bounds for the two parameters, and characterize the case that there is only one eigenvector of the tensor corresponding to the spectral radius, i.e. the Perron vector. By applying the above results to the adjacency tensor of a connected uniform hypergraph, we give some upper bounds for the two parameters in terms of the structural parameters of the hypergraph such as path cover number, matching number and the maximum length of paths. tensor; spectral radius; projective variety; module; hypergraph Multilinear algebra, tensor calculus, Eigenvalues, singular values, and eigenvectors, Hypergraphs, Solving polynomial systems; resultants, Projective techniques in algebraic geometry Eigenvariety of nonnegative symmetric weakly irreducible tensors associated with spectral radius and its application to hypergraphs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems As confirmed by the title, this is a book about G-functions. Moreover, as far as I know, it is the first book on this subject. So it is worth while to give an elaborate discussion. Before we consider its contents further, let us give a brief description of G-functions.
Consider a Taylor series of the form \(f(z)=\sum^{\infty}_{n=0}a_ nz^ n\), where the numbers \(a_ n\) belong to the same algebraic number field K \(([K:{\mathbb{Q}}]<\infty)\). Suppose it satisfies the following conditions, (i) f satisfies a linear differential equation with polynomial coefficients. (ii) \(| a_ n| =O(c^ n_ 1)\) for all \(n>0\) and a fixed \(c_ 1>0\). (iii) (common denominator of \(a_ 0,...,a_ n)=O(c^ n_ 2)\) for all \(n>0\) and a fixed \(c_ 2>0.\)
Roughly speaking, they can be considered as (very interesting) variations on the geometric series. They are not the same as the Meyer G-functions. The most common examples are -log(1-z), arctg(z) and the ordinary hypergeometric functions with rational parameters. They were defined by C. L. Siegel in 1929, along with their relatives, the E-functions, which can be considered as variations on \(e^ z\). Although Siegel states some irrationality results for values of G-functions at algebraic points, he never published the details of his computations. This turned out to be understandable when subsequent work of Galochkin and others showed that there are many more obstacles to get arithmetic results for values of G- functions then for E-functions. Significant progress was achieved in the 1980's, notably by E. Bombieri and G. V. Chudnovsky. Much of this progress was related to the properties of G-functions themselves and to the question of what G-functions really are.
If one takes any linear differential equation with coefficients in \({\mathbb{Q}}(z)\), its power series solutions will usually not be G-functions. Having a G-function solution poses very large constraints on the arithmetic of a linear differential equation. There exist several conjectures in this direction, the most important being the Bombieri- Dwork conjecture that the differential equation should come from algebraic geometry in a suitable sense. All known G-functions actually arise in this way. The converse statement is known to be true. So statements on the arithmetic nature of values of G-functions can also have consequences for problems in algebraic geometry. This explains the `Geometry' part of the title of the book. Despite all progress the amount of arithmetic results on values of G-functions at algebraic points is still very meager indeed.
The book under review is in the first place an account of recent developments in connection with G-function, which are otherwise scattered over the literature. Secondly, it is a very interesting attempt to point out directions in which it might be possible to have some `mature' applications of G-function theory, notably to algebraic geometry. In addition there are a number of new results by the author. Let us try to give a summary of the contents. The first part deals with definitions and the introduction of two heights, \(\rho\) (f) and \(\sigma\) (f), of a G- function f. Then an important example (conjecturally the only), namely the case of geometric differential equations is presented. The second part deals with Fuchsian differential systems \(\Lambda\), their formal and arithmetic aspects. Again two heights are introduced, \(\rho\) (\(\Lambda)\) and \(\sigma\) (\(\Lambda)\). A corrected proof of Chudnosky's remarkable theorem: `y cyclic and \(\sigma (y)<\infty\) implies \(\sigma (\Lambda)'\) is presented and finally the main results of this part are assembled on page 125. Part three deals with the arithmetic of values of G-functions. Here the author gives an unusual but original approach to produce linear independence results which is inspired by Gel'fond's method. We also find Bombieri's important idea of global relations in this part. It is on this principle that the author's hope for future applications is based, although up till now I have not seen this hope vindicated by any example. Finally, in part four we find two applications, found by the author, of the previous results to algebraic geometry. One concerns Grothendieck's conjecture on algebraic relations between periods of algebraic varieties. The other gives a bound for the heights of certain abelian varieties with a large endomorphism ring. Although both results apply to very limited situations it is very much worth while to keep these potential application areas for G-functions in mind.
In an appendix we find a new proof of the transcendence of \(\pi\) as a bonus. Unfortunately it contains an error. On line 6 on page 128 it is stated that \(\tau_ v\) belongs to the fundamental domain of SL(2,\({\mathbb{Z}})\) and on line 12 we find that \(\tau_ v\in \{ni,\frac{i+m}{n}| -[\frac{n}{2}]\leq m\leq [\frac{n-1}{2}]\}\). This is a contradiction. Moreover, I am afraid that this error is beyond repair.
To conclude, the book is written on a high level and not easy to read. Sometimes the author has a tendency to impress the reader unnecessarily. The proofs are written in concise, but usually intelligible way. They are not always reliable as is shown by the incorrect transcendence proof of \(\pi\). So the book should be handled with care in this respect. In particular someone ought to check the main theorem of chapter X very carefully. Despite these misgivings I enjoyed studying the book. Because of its originality and the stimulus it gives to provoke new research in the field of G-functions. Gauss-Manin connection; Bombieri-Dwork conjecture; arithmetic results; values of G-functions at algebraic points; applications of G-function theory; geometric differential equations; Fuchsian differential systems; heights; linear independence; global relations; Grothendieck's conjecture; algebraic relations between periods of algebraic varieties; bound for the heights of certain abelian varieties with a large endomorphism ring; transcendence André, Y.: G-functions and Geometry, Aspects of Mathematics, vol. E13. Friedr. Vieweg & Sohn, Braunschweig (1989) Transcendence (general theory), Research exposition (monographs, survey articles) pertaining to number theory, \(p\)-adic differential equations, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Structure of families (Picard-Lefschetz, monodromy, etc.), Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions), Ordinary differential equations in the complex domain G-functions and geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper deals with the following well known conjectures in Algebraic Geometry whose version is due to B. Segre (1961):
``A set of general multiple points in \({\mathbb P}^2 _{\mathbb C}\) imposes independent conditions to plane curves of a given degree if and only if there are no multiple fixed rational curves in the linear system of such curves''.
The conjecture has been reformulated later independently by several authors and with different equivalent statements, and many partial results have been proved.
Let \({\mathcal L}_d(m_1,\dots,m_k)\) be the linear system of plane curves of degree \(d\) having multiplicity at least \(m_i\) at \(P_i\), for \(k\) general points \(P_1,\dots,P_k\); in this paper the above conjecture is proved for \(m_i\leq 7\), \(i=1,\dots,k\).
The first step consists in proving a theorem which is very interesting per se; namely that
\(\forall M\in {\mathbb N}\), \(\exists D(M)\in {\mathbb N}\) such that if the conjecture is true for all \({\mathcal L}_d(m_1,\dots,m_k)\) with \(d < D(M)\) and \(m_i < M\), then it is also true for all \({\mathcal L}_d(m_1,\dots,m_k)\) with \(m_i < M\).
So, if we bound the multiplicities, only a finite number of linear systems need to be checked for the conjecture, and the theorem gives an explicit formula for \(D(M)\).
The theorem is proved via a degeneration technique introduced by C. Ciliberto and R. Miranda, which has already produced many results on this problem, expecially when \(m_1=\dots=m_k\).
With only a finite number of cases to check, the author can use a clever combinatorial method (a ``game of checkers on a triangular board'') which is based on specializations (collisions) of the base points. This idea turns out to be surprisingly powerful: it allows to treat ``almost all'' of about \(10^8\) cases which must be checked for all \(m_i\leq 7\), leaving only 42 cases uncovered (the author wrote a computer program to enumerate all the cases and play the ``checkers game'' on them).
The remaining 42 cases are treated with ``ad hoc'' methods which mainly use particular adaptations of the degeneration argument or of the checkers game. linear systems; fat points; Hilbert function Yang, S., Linear systems in \(\mathbb P^2\) with base points of bounded multiplicity, J. Algebraic Geom., 16, 1, 19-38, (2007) Divisors, linear systems, invertible sheaves Linear systems in \(\mathbb P^2\) with base points of bounded multiplicity | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(A={\mathbb{C}}\{X_ 1,...,X_ r\}/I\), where I is an \({\mathfrak m}\)-primary ideal, be a finite local analytic \({\mathbb{C}}\)-algebra. The main results of this thesis are lower bounds for the \({\mathbb{C}}\)-dimension of \(Der_{{\mathbb{C}}}(A)\), the module of derivations of A, (theorems 4.3 and 4.9) and of \(D_{{\mathbb{C}}}(A)\), the universally finite module of differentials (theorem 5.7 and 5.8). E.g., theorem 4.9 states: Let \(n=\dim_{{\mathbb{C}}}(A)\), \(\nu =ord(I)\) and \(\alpha =\min \{k| \quad {\mathfrak m}^ k\subset I\}\) then \(\dim_{{\mathbb{C}}}Der_{{\mathbb{C}}}(A)=r(n- 1)-\dim_{{\mathbb{C}}}Hom(I/{\mathfrak m}^{\alpha},{\mathfrak m}^{\quad \nu}/I).\) The author arrives at these results by considering those algebras as fibers of finite morphisms of complex analytic spaces. Very useful is an explicit construction of the Hilbert scheme \(Hilb^ n{\mathbb{P}}^ r\), given in section 3, and the concept of an embedded deformation (section 2). This interesting paper concludes with explicit examples and a computer program for computing the Hilbert-Samuel function of an ideal generated by homogeneous polynomials in four variables having the same total degree. finite analytic algebras; modules of differentials; module of; derivations; Hilbert scheme; embedded deformation; computing the Hilbert- Samuel function Modules of differentials, Analytic algebras and generalizations, preparation theorems, Deformations and infinitesimal methods in commutative ring theory, Parametrization (Chow and Hilbert schemes), Software, source code, etc. for problems pertaining to commutative algebra, Morphisms of commutative rings, Formal methods and deformations in algebraic geometry Über Deformationen und Derivationen endlicher \({\mathbb{C}}\)-Algebren. (On deformations and derivations of finite \({\mathbb{C}}\)-algebras) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems If X is a 0-dimensional subscheme of a smooth quadric \(Q\cong {\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) we investigate the behaviour of X with respect to the linear systems of divisors of any degree (a,b). This leads to the construction of a matrix of integers which plays the role of a Hilbert function of X; we study numerical properties of this matrix and their connection with the geometry of X. Further we put into relation the graded Betti numbers of a minimal free resolution of X on Q with that matrix, and give a complete description of the arithmetically Cohen- Macaulay 0-dimensional subschemes of Q. postulation; arithmetically Cohen-Macaulay 0-dimensional subschemes of a smooth quadric; linear systems of divisors; Hilbert function Giuffrida, S; Maggioni, R; Ragusa, A, On the postulation of \(0\)-dimensional subschemes on a smooth quadric, Pac. J. Math., 155, 251-282, (1992) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On the postulation of 0-dimensional subschemes on a smooth quadric | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0542.00006. A preliminary version was published in Groupe Étude Anal. Ultramétrique 10e Année 1982/83, No.2, Exp. No.16, 15 p. (1984).]
The function in the title is the classical hypergeometric one. The purpose of this paper is to study this function \(_ 3F_ 2\) and the linear differential equation of order 3, \(_ 3L_ 2\), whose solution is \({}_ 3F_ 2\). The method is the one that \textit{B. Dwork} used in his book ''Lectures on p-adic differential equations'' (1982; Zbl 0502.12021) for studying the hypergeometric function \({}_ 2F_ 1\). The starting point is an integral formula. In fact, two formulae were available: one gives \({}_ 3F_ 2\) by a double integration of an algebraic function and the other gives \({}_ 3F_ 2\) by a simple integration of a function involving \({}_ 2F_ 1\). Here the author uses the second one.
The main results are concerned with: an explicit construction of the differential module associated with \({}_ 3L_ 2\) (an explicit base is given), the symplectic structure (in connection with the dual theory), the p-adic radius of convergence of solutions of \({}_ 3L_ 2\) (but the growth of these solutions is not studied) and strong Frobenius structure for \({}_ 3L_ 2\). Moreover, the results of Dwork's book that are needed are briefly recalled. Therefore this paper gives a new application of Dwork's ideas. It was not at all obvious that calculations can be achieved for a ''second level'' integral formula. In fact, the calculations are rather intricate. p-adic cohomology; linear differential equation of order 3; p-adic differential equations; hypergeometric function; differential module; symplectic structure; p-adic radius of convergence; Frobenius structure; integral formula F. BALDASSARRI - Cohomologie p-adique pour la fonction 3F2(a,b1,b2c1,c2 ; \lambda ) , Astérisque, 119-120 ( 1984 ), 51-110. Zbl 0545.12013 \(p\)-adic differential equations, Linear differential equations in abstract spaces, \(p\)-adic cohomology, crystalline cohomology, Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) Cohomologie \(p\)-adique pour la fonction \(_3F_2\left(\begin{matrix} a,b_1,b_2 \\ c_1,c_2\end{matrix} ;\lambda\right)\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study \(K\)-theoretic Donaldson-Thomas theory of the affine space \(\mathbb A^3\) regarded as a local Calabi-Yau \(3\)-fold. For a positive integer \(r\), let \(\mathrm{Quot}_{\mathbb A^3}(\mathcal O^{\oplus r}, n)\) be the Quot scheme parametrizing length-\(n\) quotients of the free sheaf \(\mathcal O^{\oplus r}\). This Quot scheme admits the action of the algebraic group \(\mathbb T = (\mathbb C^*)^3 \times (\mathbb C^*)^r\) whose equivariant parameters are denoted by \(t = (t_1, t_2, t_3)\) and \(w = (w_1, \ldots, w_r)\). The rank-\(r\) \(K\)-theoretic Donaldson-Thomas invariant of \(\mathbb A^3\), denoted by \(\mathrm{DT}_r^{\mathrm{K}}(\mathbb A^3, q, t, w) \in \mathbb Z((t, (t_1 t_2 t_3)^{1/2}, w))[[q]],\) is defined via the critical structure on \(\mathrm{Quot}_{\mathbb A^3}(\mathcal O^{\oplus r}, n)\). The rank \(r = 1\) case has previously been defined and studied by \textit{A. Okounkov} [IAS/Park City Math. Ser. 24, 251--380 (2017; Zbl 1402.19001)].
The main result of the present paper states that \(\mathrm{DT}_r^{\mathrm K}(\mathbb A^3, (-1)^r q, t, w) = \mathrm{Exp}(\mathrm{F}_r(q, t_1, t_2, t_3))\) where \(\mathrm{Exp}(\cdot)\) is the plethystic exponential and
\[
\mathrm{F}_r(q, t_1, t_2, t_3) = \frac{[\mathfrak t^r]}{[\mathfrak t] [\mathfrak t^{\frac{r}{2}}q] [\mathfrak t^{\frac{r}{2}}q^{-1}]} \cdot \frac{[t_1 t_2] [t_1 t_3] [t_2 t_3]}{[t_1] [t_2] [t_3]}
\]
with \(\mathfrak t = t_1 t_2 t_3\) and \([x] = x^{1/2} - x^{-1/2}\). This confirms a conjecture proposed in string theory by \textit{H. Awata} and \textit{H. Kanno} [``Quiver matrix model and topological partition function in six dimensions'', J. High Energy Phys. 2009, No. 7, Paper No. 76, 23 p. (2009; \url{doi:10.1088/1126-6708/2009/07/076})].
The key ingredient in the proof of the main result is the observation that \(\mathrm{DT}_r^{\mathrm K}(\mathbb A^3, q, t, w)\) is independent of \(w\) and hence can be simply denoted by \(\mathrm{DT}_r^{\mathrm K}(\mathbb A^3, q, t)\). Reducing from \(K\)-theoretic to cohomological invariants, the authors prove a conjecture of \textit{R. J. Szabo} [J. Geom. Phys. 109, 83--121 (2016; Zbl 1348.81328)] by showing that the rank-\(r\) cohomological Donaldson-Thomas partition function of \(\mathbb A^3\) is given by \(\mathrm{DT}_r^{\mathrm{coh}}(\mathbb A^3, q, s) = M((-1)^rq)^{-r \frac{(s_1+s_2)(s_1+s_3)(s_2+s_3)}{s_1 s_2 s_3}}\) where \(s=(s_1,s_2,s_3)\) with \(s_i = c_1^{\mathbb T}(t_i)\), and \(M(t) = \prod_{m \ge 1} (1-t^m)^{-m}\) is the MacMahon function counting plane partitions. Moreover, the higher rank Donaldson-Thomas theory of a pair \((X, F)\) is obtained, where \(F\) is an equivariant exceptional locally free sheaf on a projective toric \(3\)-fold \(X\).
Section 2 recalls background materials such as perfect obstruction theories, virtual classes, virtual structure sheaves, and the \(K\)-theoretic virtual localisation theorem. Section 3 is devoted to \(\mathrm{Quot}_{\mathbb A^3}(\mathcal O^{\oplus r}, n)\) such as its critical structure, its \(\mathbb T\)-action, the \(\mathbb T\)-fixed locus in terms of \(r\)-colored plane partitions, and its equivariant critical obstruction theory. Section 4 introduces the cohomological and \(K\)-theoretic Donaldson-Thomas invariants of \(\mathbb A^3\) via \(\mathrm{Quot}_{\mathbb A^3}(\mathcal O^{\oplus r}, n)\). Section 5 develops a higher rank topological vertex formalism based on the combinatorics of \(r\)-colored plane partitions. The main result is proved in Section 6, while the above formula of \(\mathrm{DT}_r^{\mathrm{coh}}(\mathbb A^3, q, s)\) is verified in Section 7. In Section 8, the authors present a mathematically rigorous definition of a chiral version of the virtual elliptic genus and use it to define the elliptic Donaldson-Thomas invariants. Section 9 studies the higher rank Donaldson-Thomas invariants of a compact toric \(3\)-fold by gluing vertex contributions from its toric charts. Donaldson-Thomas theory; local Calabi-Yau \(3\)-fold; Quot scheme; partition; MacMahon function; string theory; obstruction theory; virtual fundamental class Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) Higher rank \(K\)-theoretic Donaldson-Thomas theory of points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The motivic integration was introduced by M. Kontsevich (Lecture at Orsay, 1995) in an attempt to prove some results about the topology of Calabi-Yau varieties. Originally formulated over smooth varieties, it has been developed further for some class of varieties with mild singularities in the works of [\textit{J. Denef} and \textit{F. Loeser}, Invent. Math. 135, No. 1, 201--232 (1999; Zbl 0928.14004); Prog. Math. 201, 327--348 (2001; Zbl 1079.14003); Duke Math. J. 99, No. 2, 285--309 (1999; Zbl 0966.14015); J. Algebr. Geom. 7, No. 3, 505--537 (1998; Zbl 0943.14010)]. After that more generalizations have been proposed, one among them being the motivic integration over formal schemes and rigid spaces [\textit{F. Loeser} and \textit{J. Sebag}, Duke Math. J. 119, No. 2, 315--344 (2003; Zbl 1078.14029); \textit{J. Nicaise}, Math. Ann. 343, No. 2, 285--349 (2009; Zbl 1177.14050)]. The article under review gives an introduction to this theory, with some applications to complex hypersurface singularities and rational points on varieties over a field with a non-archimedean valuation. We are working in the category of special formal schemes. For \(R\) a Noetherian adic ring, a special formal scheme over \(R\) is a separated Noetherian adic formal scheme \(\mathcal{X} \rightarrow Spf R\), such that the set \(V(\mathcal{I})\) is of finite type over \(R\) for any ideal of definition \(\mathcal{I}\). Such a scheme \(\mathcal{X}\) is separated of topologically finite type (stft) if \(\mathcal{X} \rightarrow Spf R\) is an adic morphism.
The Greenberg schemes [\textit{M. J. Greenberg}, Ann. Math. (2) 73, 624--648 (1961; Zbl 0115.39004); Ann. Math. (2) 78, 256--266 (1963; Zbl 0126.16704)] play the role of the arc spaces in the classic motivic integration.
For \((R, m)\) a DVR with residue field \(k\) and quotient field \(K\), let \(R_n = R/m^{n+1}\), and let \(X\) be a separated \(R_n\)-scheme of finite type. Using the \(R_n\)-algebra \(\mathcal{R}_n(A)\) (defined as \(R \otimes_k A\), if \(R\) has equal characteristic), there is a functor \(\mathcal{A}lg_k \rightarrow \mathcal{S}ets, A \mapsto X(\mathcal{R}_n(A))\). It could be shown that this functor is represented by a separated scheme of finite type \(Gr_n(X)\), the Greenberg scheme of \(X\). If \(\mathcal{X}\) is stft formal scheme, the Greenberg scheme of \(\mathcal{X}\) at level \(n\) is \(Gr_n(\mathcal{X} \times_R R_n)\), and there are truncation morphisms \(GR_n(X) \rightarrow GR_m(X), n \geq m\), as in the case of jet schemes. The fact these are affine morphisms permits to define \(\mathrm{Gr}(\mathcal{X}) = \lim_{\leftarrow n} GR_n(\mathcal{X})\), the Greenberg scheme of \(\mathcal{X}\), which is not Noetherian in general. It could be shown that it parametrizes the étale sections on \(\mathcal{X}\). Also, there are natural morphisms \(\theta_n \colon \mathrm{Gr}(\mathcal{X}) \rightarrow GR_n(\mathcal{X})\), and any morphism of stft formal schemes \(g \colon \mathcal{Y} \rightarrow \mathcal {X}\) induces a morphism of the corresponding Greenberg schemes \(\mathrm{Gr}(g)\). The schemes \(GR_n(\mathcal{X}), \mathrm{Gr}(\mathcal{X})\) over \(\mathcal{X}\) share the main properties of the jet schemes and arc spaces over \(X\).
The ring in which the motivic integral takes its value, when \(R\) has equal characteristic, is the localization of \(K_0(Var_X)\), the Grothendieck ring of varieties over \(X\), by the class of the line over \(X\) (in mixed characteristic is used the modified Grothendieck ring of varieties over \(X\)). For \(\mathcal{X}\) a smooth stft formal \(R\)-scheme, a cylinder set of level \(n\) is \(C = \theta^{-1}_n(C_m)\) for some constructible \(C_m \subset Gr_m(\mathcal{X})\). Then the motivic measure \(\mu_{\mathcal{X}}(C)\) is defined (for \(\mathcal{X}\) of pure relative dimension), which does not depend on \(n\). An integrable function \(\beta \colon \mathrm{Gr}(\mathcal{X}) \rightarrow \mathbb{Z} \cup \{\infty\}\) is one which takes finitely many values, and all its fibers are cylinders. For such \(\beta\) the motivic integral \(\int_{\mathrm{Gr}(\mathcal{X})} {\mathbb{L}}^{\beta}\) is defined, belonging to the localization of \(K_0(Var_{\mathcal{X}_0})\), where \(\mathcal{X}_0\) is the reduction of \(\mathcal{X}\). A typical example is \(\int_{\mathrm{Gr}(\mathcal{X})} {\mathbb{L}}^{-\mathrm{ord}(\omega)}\), where \(\omega\) is nowhere vanishing differential form of maximal degree, defined on each connected component of \(\mathcal{X}_{\eta}\), and called a gauge form.
The main tool in any theory of motivic integration is the change of variables. Let \(h \colon \mathcal{Y} \rightarrow \mathcal{X}\) be morphism of smooth stft formal \(R\)-schemes, inducing \(\mathrm{Gr}(h) \colon \mathrm{Gr}(\mathcal{Y}) \rightarrow \mathrm{Gr}(\mathcal{X})\). After defining the notion of the Jacobian of \(h\) is given a corrected form of the original change of variables [\textit{J. Sebag}, Bull. Soc. Math. Fr. 132, No. 1, 1--54 (2004; Zbl 1084.14012)].
The dilatation of a flat special formal \(R\)-scheme is a tool, permitting to reduce some constructions from special formal schemes to stft formal schemes [\textit{J. Nicaise}, Math. Ann. 343, No. 2, 285--349 (2009; Zbl 1177.14050)]. It is related with the notions of Néron smoothening of a special formal scheme, and that of weak Néron model for rigid variety. After their definitions given with some basic properties, for generically smooth special formal \(R\)-scheme \(\mathcal{X}\), with \(\mathcal{X}_{\eta}\) admitting a gauge form \(\omega\), is defined the motivic integral \({\int}_{\mathcal{X}} |\omega|\). For given \(X\) a separated smooth rigid \(K\)-variety, admitting a weak Néron model, and a gauge form \(\phi\), is defined the motivic integral \(\int_X |\omega|\), showing that it is independent of the choice of the weak Néron model.
The applications to rational points on rigid variety are using the motivic Serre invariant [\textit{F. Loeser} and \textit{J. Sebag}, Duke Math. J. 119, No. 2, 315--344 (2003; Zbl 1078.14029)]. It gives a motivic measure of the set of rational points on a separated rigid \(K\)-variety, admitting a weak Néron model, and could be modified for algebraic varieties as well. Then is formulated a trace formula J. Nicaise [Zbl 1177.14050], which provides a cohomological interpretation of it. In the case of analytic manifolds and \(p\)-adic integration, the \(p\)-adic Serre invariant is also defined. By a result of Serre, it classifies the \(K\)-analytic manifolds, and the motivic Serre invariant specializes to it.
For the applications of this theory of motivic integration to singularities, a central notion is that of the analytic Milnor fiber. If \(Z\) is a complex manifold, \(g\) is analytic function on \(Z\), and \(g(x) = 0\), take \(B\) to be an open ball centered at \(x\), and \(D^*\) to be a puncured open disc at \(0 \in \mathbb{C}\), both small enough. Then at \(x\) is defined the Milnor fibration \(g_x \colon g^{-1}(D^*) \cap B \rightarrow D^*\). There is a monodromy action on the singular cohomology \(\sum_n H^n_{\mathrm{sing}}(F_x, \mathbb{Z})\) of the universal fiber of the fibration \(F_x\). An eigenvalue of the monodromy is any \(\alpha \in \mathbb{C}\), which is eigenvalue of the monodromy on \(H^n_{\mathrm{sing}}(F_x, \mathbb{Z})\) for some \(n\). The monodromy theorem states that all eigenvalues are roots of unity.
Let \(X\) be smooth irreducible variety over \(k\), and \(f \colon X \rightarrow {\mathbb{A}}^1_k\) a non-constant morphism. Then is defined the motivic zeta function \(Z_f(T)\) associated to \(f\), and by a result of Denef and Loeser, when \(\mathrm{char}k = 0\), \(Z_f(T)\) is a rational function [Zbl 1079.14003]. They proposed also the motivic monodromy conjecture, relating the poles of \(Z_f(T)\) with the monodromy eigenvalues of \(f\), generalizing the Igusa's conjecture about the \(p\)-adic zeta function of \(f\) over a number field.
The analytic Milnor fiber is a rigid \(k((t))\)-variety, which is the non-archimedean model of the topological Milnor fibration. It contains information about the invariants of the singularity of an \(f\) as above. The points on the analytic Milnor fiber can be described in terms of some subsets of the arc space \(X_{\infty}\). In this way it connects motivic zeta functions, arc spaces and the monodromy. special formal scheme; rigid variety; Greenberg scheme; motivic zeta function; analytic Milnor fiber Johannes Nicaise and Julien Sebag, Motivic invariants of rigid varieties, and applications to complex singularities, Motivic integration and its interactions with model theory and non-Archimedean geometry. Volume I, London Math. Soc. Lecture Note Ser., vol. 383, Cambridge Univ. Press, Cambridge, 2011, pp. 244 -- 304. Rigid analytic geometry, Arcs and motivic integration, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Rational points Motivic invariants of rigid varieties, and applications to complex singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The necessity to process data which live in nonlinear geometries (e.g. capture data, unit vectors, subspace, positive definite matrices) has led to some recent development in nonlinear multiscale representation and subdivision algorithms. The present paper analyzes convergence and \(C^1\)- and \(C^2\)-smoothness of subdivision schemes which operate in the matrix groups or general Lie groups, and which are defined by the so-called log-exponential analogy. It is shown that a large class of such schemes has essentially the same smoothness as the linear schemes derived from them.
The bibliography contains 17 sources. Lie group; matrix group; subdivision scheme; smoothness properties; nonlinear multiscale representation; convergence; log-exponential analogy P. Grohs and J. Wallner, \textit{Log-exponential analogues of univariate subdivision schemes in Lie groups and their smoothness properties}, in Approximation Theory XII, M. Neamtu and L. L. Schumaker, eds., Nashboro Press, Nashville, TN, 2008, pp. 181--190. Numerical aspects of computer graphics, image analysis, and computational geometry, Representations of Lie and linear algebraic groups over real fields: analytic methods, Group actions on varieties or schemes (quotients), Group actions on affine varieties, Associated Lie structures for groups, Analysis on real and complex Lie groups Log-exponential analogues of univariate subdivisional schemes in Lie groups and their smoothness properties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A fat point ideal is an ideal of the form \(\mathfrak{p}_1^{s_1} \cap \mathfrak{p}_2^{s_2} \cap \cdots \cap \mathfrak{p}_m^{s_m}\), where \(\mathfrak{p}_i\) is the vanishing ideal of a point \(p_i \in \mathbb{P}^n.\) The Hilbert function in degree \(d\) of \(K[x_0,\ldots,x_{n}]/ \mathfrak{p}_1^{s_1} \cap \mathfrak{p}_2^{s_2} \cap \cdots \cap \mathfrak{p}_m^{s_m}\) encodes the number of independent conditions that are imposed by the vanishing to the order \(s_1, s_2, \ldots, s_m\) of forms of degrees \(d\) on the points \(p_1,p_2,\ldots, p_m\). The Alexander-Hirschowitz theorem provides a classification of the Hilbert function in the case \(K=\mathbb{C}\), \(s_1 = s_2 = \cdots = s_m=2\), but in general, not much is at present known about the Hilbert series of this natural class of ideals.
In the paper the corresponding problem for \(\mathbb{P}^1 \times \mathbb{P}^1\) is considered. Specifically the Hilbert function of the module of Kähler differentials associated to the coordinate ring is determined in many special cases, including when the support is a complete or almost complete intersection. Also, arithmetically Cohen-Macaulay reduced schemes having the Cayley-Bacharach property are characterized. A large number of examples are provided, which serves as a good help for a reader which is new to the area. ACM fat point scheme; complete intersection; fat point scheme; Hilbert function; Kähler different; Kähler differentials; separators Modules of differentials, Linkage, complete intersections and determinantal ideals, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Projective techniques in algebraic geometry Kähler differentials for fat point schemes in \(\mathbb{P}^1\times\mathbb{P}^1\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For any \(P\in \mathbb {P}^n\) and any integer \(m>0\) let \(mP\) be the closed subscheme of \(\mathbb {P}^n\) with \((\mathcal {I}_P)^m\) as its ideal sheaf. A fat scheme \(X\subset \mathbb {P}^n\) is a finite disjoint union of schemes \(m_iP_i\). Trung and, independently, Fatabbi and Lorenzini, conjectured a sharp upper bound for the regularity index of a fat scheme (it is called Segre's bound). In this paper the authors give an affermative answer to the equimultiple case (all \(m_i\) are the same) for a low number of points. Later, Segre's conjecture was solved by \textit{U. Nagel} and \textit{B. Trok} [``Segre's regularity bound for fat points schemes'', Preprint, \url{arXiv:1611.06279}]. fat point scheme; regularity index; Segre conjecture; Castelnuovo-Mumford regularity; Hilbert function Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Divisors, linear systems, invertible sheaves The regularity index of up to \(2n-1\) equimultiple fat points of \(\mathbb{P}^n\) | 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 It is predicted that the principal specialization of the partition function of a B-model topological string theory, that is mirror dual to an A-model enumerative geometry problem, satisfies a Schrödinger equation, and that the characteristic variety of the Schrödinger operator gives the spectral curve of the B-model theory, when an algebraic K-theory obstruction vanishes. In this paper the authors present two concrete mathematical A-model examples whose mirror dual partners exhibit these predicted features on the B-model side. The A-model examples they discuss are the generalized Catalan numbers of an arbitrary genus and the single Hurwitz numbers. In each case, they show that the Laplace transform of the counting functions satisfies the Eynard-Orantin topological recursion, that the B-model partition function satisfies the KP equations, and that the principal specialization of the partition function satisfies a Schrödinger equation whose total symbol is exactly the Lagrangian immersion of the spectral curve of the Eynard-Orantin theory. The paper is organized as follows. In Section 2, the authors give the definition of the Eynard-Orantin topological recursion. They emphasize the aspect of Lagrangian immersion in their presentation. In Section 3, they review the generalized Catalan numbers of [\textit{O. Dumitrescu} et al., Contemp. Math. 593, 263--315 (2013; Zbl 1293.14007)]. Then in Section 4, they derive the Schrödinger equation for the Catalan partition function. The equation for the Hurwitz partition function is given in Section 5. In Section 6, concerning the Schur function expansion of the Hurwitz partition function, the authors give the proof of a differential-difference equation (or a delay differential equation). They use the Schur function expansion of the Hurwitz generating function and its principal specialization. spectral curves; partition function; Hurwitz numbers; Schur function; Schrödinger equation; Eynard-Orantin recursion Mulase, M. and Sulkowski, P.: Spectral curves and the Schr''odinger equations for the Eynard--Orantin recursion. Adv. Theor. Math. Phys.19 (2015), no. 5, 955--1015. String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Families, moduli of curves (analytic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumeration in graph theory, Lattice points in specified regions, Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics, Special algebraic curves and curves of low genus, Mirror symmetry (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Calabi-Yau manifolds (algebro-geometric aspects) Spectral curves and the Schrödinger equations for the Eynard-Orantin recursion | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors elucidate the cohomology meaning of the Kadomtsev-Petviashvili (KP) hierarchy of integrable equations which was considered for the first time by Mulase (1984). The KP flows are known to describe a point movement in the universal Grassmannian. The authors show that the Faà di Bruno polynomials, which are defined recursively, form a basis in a subspace of the universal Grassmannian associated to the KP hierarchy. For the algebraic geometrical solutions of the KP equations a point in the universal Grassmannian is associated via the Krichever map to the datum of a smooth algebraic spectral curve, a point on it with an appropriate local coordinate, a line bundle and a local trivialization in a neighborhood of the point. In this situation the Faà di Bruno recursion relations appear to be the cocycle condition for the Welters hypercohomology group which describes the deformations of the line bundle over the spectral curve. The authors illustrate the general theory by the example of an elliptic spectral curve. integrable systems; Kadomtsev-Petviashvili hierarchy; elliptic spectral curve; hypercohomology groups Falqui, G.; Reins, C.; Zampa, A.: Krichever maps, fa à di bruno polynomials, and cohomology in KP theories. Lett. math. Phys. 42, No. 4, 349-361 (1997) KdV equations (Korteweg-de Vries equations), Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Other completely integrable equations [See also 58F07], Curves in algebraic geometry Krichever maps, Faà di Bruno polynomials, and cohomology in KP theory | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We use Hamiltonian reduction to simplify Falqui and Mencattini's recent proof of Sklyanin's expression providing spectral Darboux coordinates of the rational Calogero-Moser system. This viewpoint enables us to verify a conjecture of \textit{G. Falqui} and \textit{I. Mencattini} [``Bi-Hamiltonian geometry and canonical spectral coordinates for the rational Calogero-Moser system'', Preprint, \url{arXiv:1511.06339}] and to obtain Sklyanin's formula as a corollary. integrable systems; Calogero-Moser type systems; spectral coordinates; Hamiltonian reduction; action-angle duality , Momentum maps; symplectic reduction, Relationships between algebraic curves and integrable systems A simple proof of Sklyanin's formula for canonical spectral coordinates of the rational 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 For a reduced complete intersection set of points \(X= \text{CI}(a, b)\) in \(\mathbb{P}^2\), the Hilbert function of \(X\) minus a point depends only on the type \((a, b)\) of the complete intersection. The authors study the Hilbert functions of fat point schemes \(Y\) defined by ideals \(I_Y= I^m_{P_1}\cap\cdots\cap I^m_{P_{ab-1}}\) and supported on \(X\setminus\{P_{ab}\}\). Examples show that the Hilbert function of \(Y\) depends on a number of subtle factors, e.g. whether or not the forms defining \(X\) are irreducible or what point is removed from \(X\).
For \(a< b\), they prove that the initial degree \(\alpha(Y)= \min\{i\mid(I_Y)_i\neq 0\}\) depends only on \(a\), \(b\) and \(m\), and they give a conjecture on the behavior of \(\alpha(I_Y)\) when \(a= b\). Moreover, in the case of double points, i.e. when \(m= 2\), they show that the schemes \(Y_{\text{gen}}\) supported on a generic CI\((a, b)\) and the schemes \(Y_{\text{grid}}\) supported on a grid (i.e. on a complete intersection defined by products of linear forms) have the same graded Betti numbers, except for some special cases. Hilbert function; Cayley-Bacharach property; fat point scheme; graded Betti number Guardo, E.; Tuyl, A., Some results on fat points whose support is a complete intersection minus a point, 257-266, (2005), Berlin Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Syzygies, resolutions, complexes and commutative rings, Divisors, linear systems, invertible sheaves, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Schemes and morphisms Some results on fat points whose support is a complete intersection minus a point | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(A\) be an abelian variety defined over \(\mathbb{Q}\) and with a principal polarization. Take an even invertible sheaf \(L\) on \(A\). The author proves that there exists a uniquely defined height function \(h_L : Z_k (A) \to \mathbb{R}\) with the following properties:
(i) \(Z_k (A)\) is a free group generated by the cycles of dimension \(k\),
(ii) \(h\) is homogeneous of degree \(2k + 2\),
(iii) \(h\) equals to \(\deg (L)\) modulo a bounded function.
This is a generalization of the Néron-Tate function for \(k = 0\). The definition uses the Arakelov intersection theory developed by Gillet and Soulet and the arithmetical compactification of the moduli scheme of abelian varieties constructed by \textit{G. Faltings} and \textit{C.-L. Chai}.
Independently similar results were obtained by \textit{P. Philippon} [Math. Ann. 289, No. 2, 255-283 (1991; Zbl 0704.14017) and \textit{W. Gubler} [ibid. 298, No. 3, 427-455 (1994; Zbl 0792.14012)]. principal polarization; height function; Arakelov intersection theory; moduli scheme of abelian varieties Arithmetic varieties and schemes; Arakelov theory; heights, Algebraic theory of abelian varieties On a generalization of the Néron-Tate height on abelian 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 In this paper the well-known transformation properties of theta-functions with (polynomial) harmonic coefficients as well as the associated Epstein zeta function are extended to theta functions with certain rational harmonic coefficients of low degree and a small number of variables. These theta functions are no longer modular forms, but the classical functional equation for the associated Epstein zeta function still holds. The proof of the various transformation laws are based on Chen's theory of iterated integrals.
As an application this new Epstein zeta function divided by a product of two L-functions is used to generate for the Fermat quartic \(F_ 4:\) \(X_ 4+Y_ 4=1\) the Abel Jacobi image of the 1-cycle in \(Jac(F_ 4)\) given by \([F_ 4]-[\iota (F_ 4)]\). Jacobians; theta-functions; iterated integrals; Epstein zeta function; product of two L-functions; Fermat quartic Theta functions and abelian varieties, Other Dirichlet series and zeta functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Iterated integrals and Epstein zeta functions with harmonic rational function coefficients | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider a discrete classical integrable model on a three-dimensional cubic lattice. The solutions of this model can be used to parameterize the Boltzmann weights of various three-dimensional spin models. We find the general solution of this model constructed in terms of the theta functions defined on an arbitrary compact algebraic curve. Imposing periodic boundary conditions fixes the algebraic curve. We show that the curve then coincides with the spectral curve of the auxiliary linear problem. For a rational curve, we construct the soliton solution of the model. three-dimensional integrable systems; Bäcklund transformations; spectral curves Exactly solvable models; Bethe ansatz, Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems, Relationships between algebraic curves and integrable systems, Theta functions and curves; Schottky problem, Relationships between algebraic curves and physics Spectral curves and parameterization of a discrete integrable three-dimensional model | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The content of this article, which has very important applications in analytical number theory and other fields, covers the following topics: the logarithm function, multiple zeta values, polylogarithms, partial orders, iterated integrals, the differentials operator $(dt)/t)$ and $(dt)/(1-t)$, the Beta function, the gamma function. The author provides iterated integrals on products of one variable multiple polylogarithms in the algebra of multiple zeta values with a convergent condition. In the divergent case, the author also defines the regularized iterated. Using the same method, the author gives also regularized iterated integrals in the algebra of multiple zeta values. The author provides some applications involving series representations for multiple zeta values. He also gives some remarks and examples involving iterated integrals and multiple zeta values. logarithm function; multiple zeta values; polylogarithms; partial orders; iterated integrals; differentials operator Multiple Dirichlet series and zeta functions and multizeta values, Polylogarithms and relations with \(K\)-theory, Other Dirichlet series and zeta functions, Motivic cohomology; motivic homotopy theory Iterated integrals on products of one variable multiple polylogarithms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let k be an algebraically closed field of characteristic 0, \(d\in {\mathbb{N}}\), and H d the Hilbert scheme parametrizing ideals \({\mathcal I}\subset {\mathcal O}_{{\mathbb{P}}^ 2_ k}\) of \(colength\quad d.\) For \(p\in H\) d, let h(p) denote the Hilbert function of the ideal of \({\mathbb{P}}\) \(2\otimes k(p)\), which corresponds to the point p. If \(\phi: {\mathbb{N}}\to {\mathbb{N}}\) is any function, the subset \(H_{\phi}=\{p\in H\quad d| \quad h(p)=\phi \}\) is locally closed in H d (and possibly empty).
The following result is proved: If \(H_{\phi}\) is provided with the reduced induced scheme structure, then \(H_{\phi}\) is connected and smooth over k.
Moreover, in the appendix a formula is described, by means of which the dimension of \(H_{\phi}\) can be computed in terms of \(\phi\). Hilbert function; stratification of Hilbert scheme of the projective plane Gotzmann, G.: A stratification of the Hilbert scheme of points in the projective plane. Math. Z.199, 539--547 (1988) Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, Projective techniques in algebraic geometry A stratification of the Hilbert scheme of points in the projective plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Bei diesem Buch handelt es sich um einen unveränderten Nachdruck der ersten Auflage aus dem Jahre 1993; insbesondere wird das Fermatproblem auf Seite 501 immer noch als ungelöst eingestuft. Das ist aber schon alles, was ich an diesem Buch auszusetzen habe: schließlich handelt es sich hierbei immer noch um eine extrem anfängerfreundliche und kompetente Einführung in die Algebra: welcher Mathematiker von Rang läßt sich schon dazu herab, bei der Einführung des Körpers mit \(4\) Elementen darauf hinzuweisen, daß es sich hierbei nicht um \(\mathbb Z/4\mathbb Z\) handelt? Hilfestellungen wie diese ziehen sich durch das ganze Buch, und die vielen Beispiele aus weiten Bereichen der Mathematik (für den Inhalt siehe die Besprechung des englischen Originals Zbl 0788.00001) bieten auch dem Kenner der Materie noch etwas. Die freundliche Preisgestaltung läßt auf eine weite Verbreitung hoffen, die dieses Buch uneingeschränkt verdient hat. groups; linear algebra; infinite dimensional spaces; systems of linear differential equations; symmetry; finite subgroups of rotation group; free groups; generators; relations; Todd-Coxeter algorithm; bilinear forms; spectral theorems; linear groups Artin E.: Geometric Algebra. Interscience Publishers, New York (1998) Mathematics in general, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory Algebra. Aus dem Engl. übersetzt von Annette A'Campo. (Algebra). Unveränd. Nachdr | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a field. Let \(m\in\mathbf{N}_{>0}\) be a positive integer. Let \(f\in k[x_1,\ldots,x_m]\) be a polynomial with degree \(d\geq 1\) and associated hypersurface \(H:=H(f):=\mathrm{Spec}(k[x_1, \ldots, x_m]/\langle f\rangle)\). In this article, we firstly provide a structure property of the weighted-homogeneity of \(f\) in terms of the jet schemes \(\mathscr{L}_H\) of \(H\). As a by-product, we deduce from this property a new and very effective method for the computation of the motivic Poincaré power series \(P_H (T):=\sum_{n\geq 0} [\mathscr{L}_n (H)]T^n \in K_0 (\mathrm{Var}_k)[[T]]\) associated with a homogeneous hypersurface \(H\) with a single isolated singularity at the origin \(\mathfrak{o}\) (and more generally with a specific class of isolated quasi-homogeneous hypersurface singularities). With this point of view we obtain various consequences. For the considered class of varieties, our method provides a characteristic-free proof of the rationality of \(P_H (T)\) which does not use motivic integration nor the existence of resolutions of singularities; we obtain a precise description of the numerator and the possible poles in the rational expression of \(P_H (T)\); when the field is assumed to be of characteristic zero, this allows us to prove the validity of the motivic monodromy conjecture. jet scheme; motivic monodromy conjecture; motivic zeta function; quasi-homogeneous hypersurface singularities Arcs and motivic integration, Singularities in algebraic geometry, Hypersurfaces and algebraic geometry, Commutative rings of differential operators and their modules Jet schemes of quasi-homogeneous hypersurfaces and motivic monodromy conjecture for isolated quasi-homogeneous hypersurface singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0516.00012.]
Let F be a totally real number field of finite degree g over \({\mathbb{Q}}\), and let B be a division quaternion algebra over F which is unramified at one infinite prime of F and ramified at all the other infinite primes of F. In [Jap. J. Math. 9, 1-25 (1983; Zbl 0527.10023)], the author constructed for odd g an abelian scheme A over the Shimura curve V attached to B and expressed the Hasse-Weil zeta function of a k-fold fiber product of A over V as a product of Dedekind zeta-functions and automorphic L-functions associated with \(B^*\). As an application, the Ramanujan-Petersson conjecture was proved for certain automorphic forms on \(B^*\) for almost all finite primes of F.
In the present article, the author extends these results to the case of even g and proves the Ramanujan-Petersson conjecture for all ''good primes'' of F. \textit{Morita} [Congruence relations and the Ramanujan conjecture (to appear)] has recently proved the Ramanujan-Petersson conjecture under weaker assumptions than those made here. Eichler-Shimura cohomology; quaternion algebras; abelian scheme; Shimura curve; Hasse-Weil zeta function; Ramanujan-Petersson conjecture M. Ohta, On the zeta function of an abelian scheme over the Shimura curve II , Galois Groups and Their Representations (Nagoya, 1981), Advanced Studies in Pure Mathematics, vol. 2, North-Holland, Amsterdam-New York, 1983, pp. 37-54. Automorphic forms, one variable, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Arithmetic ground fields for abelian varieties, Group schemes, Local ground fields in algebraic geometry On the zeta function of an Abelian scheme over the Shimura curve. 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 Let \(n\) and \(d\) be positive integers, with \(n > d\) and let \({\mathcal A} \subset \mathbb{N}^{d}\setminus \{(0, \ldots, 0)\}\) be a set of \(n\) elements. Let \(S = k[X_{1}, \ldots , X_{n}]\) and let \(I_{{\mathcal A}}\) be the toric ideal defined by \({\mathcal A}\). A homogeneous ideal \(M \subset S\) is called \({\mathcal A}\)-graded if \(S/M\) has the same multigraded Hilbert function as \(S/I_{{\mathcal A}}\). A main result in the theory of \({\mathcal A}\)-graded ideals states that if \(d = 1\) and \(n = 3\) then every \({\mathcal A}\)-graded ideal is coherent. In 1994 Sturmfels conjectured that if \(n - d = 2\) then every \({\mathcal A}\)-graded ideal is coherent. The main result of this paper is the proof of this conjecture, which is achieved after a detailed study of the syzygies of \(I_{\mathcal A}\). toric ideal; toric Hilbert scheme; codimension 2 toric varieties; multigraded Hilbert function; syzygies; coherence of graded ideal; polynomial ring V. Gasharov and I. Peeva, Deformations of codimension \(2\) toric varieties , Compositio Math. 123 (2000), 225--241. Toric varieties, Newton polyhedra, Okounkov bodies, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Polynomial rings and ideals; rings of integer-valued polynomials Deformations of codimension 2 toric varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems One can try to study the fundamental group of a smooth, complex projective variety \(S\) by looking at spaces of representations of \(\pi_ 1 (S)\). If the space of representations is a union of isolated points, then this structure reduces to the discrete structure of the set of representations. So a natural problem is to try to find examples of varieties \(S\) where there exist nontrivial continuous families of representations (or, equivalently, local systems on \(S) \). It is not too hard to see that if \(S\) is a projective algebraic curve of genus \(g \geq 2\), then the moduli space of representations of rank \(r \geq 1\) has dimension \((2g - 2) r^ 2 + 2\); or, for example, if \(S\) is a variety with \(\dim H^ 1(S, \mathbb{C}) = a\), then the space of representations of rank 1 has dimension \(a\). Taking tensor products of pullbacks of families of local systems arising in these ways, we obtain some more families. To pursue the problem we ask: Do families other than these exist?
The idea in this article is to use the next natural construction, taking higher direct images of local systems. Suppose that \(f:X \to S\) is a smooth projective morphism and \(\{W_ t\}\) is a family of local systems on \(X\), chosen in a simple way. Let \(V_ t = R^ i f_ * (W_ t)\). This is a collection of local systems, and if the ranks are constant, then it is a continuous family. We can hope that \(\{V_ t\}\) will be an interesting family of local systems on \(S\). The principal question that needs to be addressed is whether, if the family \(\{W_ t\}\) varies nontrivially, the family of direct images \(\{V_ t\}\) varies nontrivially. We are also interested in constructing examples where we can show that the family \(\{V_ t\}\) does not, by some miracle, arise from a simpler construction such as the one described before.
This article consists essentially of two parts. The first (sections 1-5) is devoted to answering our principal question in a fairly general situation. For this we develop the technique of taking the direct image of a harmonic bundle and its associated Higgs bundle. We give a way to calculate the spectral varieties of the Higgs bundles associated to \(V_ t\), as a way of verifying that the \(V_ t\) vary nontrivially. -- The second part (sections 6-8) is concerned with the construction of a particular class of examples and the verification of some additional properties about monodromy groups and possible factorization through morphisms to algebraic curves; these properties serve to show that our examples do not come from tensor products of pullbacks. The methods used in this part are all fairly well known. representations of fundamental group; local systems on complex projective variety; moduli space of algebraic curve; families of representations; direct image of a harmonic bundle; Higgs bundle; spectral varieties of the Higgs bundles Simpson C.: Some families of local systems over smooth projective varieties. Ann. Math. 138, 337--425 (1993) Homotopy theory and fundamental groups in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, moduli of curves (algebraic), Variation of Hodge structures (algebro-geometric aspects) Some families of local systems over smooth projective varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a reductive group scheme over the algebraic curve \(X\) over the field \(k\). Let \(K\) be the function field of \(X\). We show how \(G\) induces an additional structure on the root system \(\Phi = \Phi(G_ K,T)\) for any split maximal torus \(T\) of \(G_ K\), the pullback of \(G\) to \(\text{Spec }K\). We call this structure a complementary polyhedron for \(\Phi\). The study of these polyhedra leads to a proof of the existence and uniqueness of a canonical parabolic subgroup \(P\) of \(G\). If \(G =\text{GL}(V)\) for a vector bundle \(V\) over \(X\), the parabolic \(P\) is the stabilizer of the Harder-Narasimhan filtration of \(V\). reductive group scheme; algebraic curve; function field; root system; split maximal torus; complementary polyhedron; parabolic subgroup; vector bundle; Harder-Narasimhan filtration Behrend K, Semi-stability of reductive group schemes over curves, Math. Ann. 301 (1995) 281--305 Linear algebraic groups over adèles and other rings and schemes, Group schemes, Simple, semisimple, reductive (super)algebras, Classical groups (algebro-geometric aspects) Semi-stability of reductive group schemes over curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We take a new look at the curvilinear Hilbert scheme of points on a smooth projective variety \(X\) as a projective completion of the nonreductive quotient of holomorphic map germs from the complex line into \(X\) by polynomial reparametrisations. Using an algebraic model of this quotient coming from global singularity theory we develop an iterated residue formula for tautological integrals over curvilinear Hilbert schemes. Hilbert scheme of points; curve counting; Göttsche formula; tautological integrals; nonreductive quotients; equivariant localisation; iterated residue Bérczi, G., Tautological integrals on curvilinear Hilbert schemes, Geom. Topol., 2897-2944, (2017) Parametrization (Chow and Hilbert schemes), Enumerative problems (combinatorial problems) in algebraic geometry, Equivariant homology and cohomology in algebraic topology Tautological integrals on curvilinear 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 We study an integrable vertex model with a periodic boundary condition associated with \(Uq(An^{(1)})\) at the crystallizing point \(q=0\). It is an \((n+1)\)-state cellular automaton describing the factorized scattering of solitons. The dynamics originates in the commuting family of fusion transfer matrices and generalizes the ultradiscrete Toda/KP flow corresponding to the periodic box-ball system. Combining Bethe ansatz and crystal theory in quantum group, we develop an inverse scattering/spectral formalism and solve the initial value problem based on several conjectures. The action-angle variables are constructed representing the amplitudes and phases of solitons. By the direct and inverse scattering maps, separation of variables into solitons is achieved and nonlinear dynamics is transformed into a straight motion on a tropical analogue of the Jacobi variety. We decompose the level set into connected components under the commuting family of time evolutions and identify each of them with the set of integer points on a torus. The weight multiplicity formula derived from the \(q=0\) Bethe equation acquires an elegant interpretation as the volume of the phase space expressed by the size and multiplicity of these tori. The dynamical period is determined as an explicit arithmetical function of the \(n\)-tuple of Young diagrams specifying the level set. The inverse map, i.e., tropical Jacobi inversion is expressed in terms of a tropical Riemann theta function associated with the Bethe ansatz data. As an application, time average of some local variable is calculated. soliton cellular automaton; crystal basis; combinatorial Bethe ansatz; inverse scattering/spectral method; tropical Riemann theta function Kuniba, A.; Takagi, T., Bethe ansatz, inverse scattering transform and tropical Riemann theta function in a periodic soliton cellular automaton for \(A_n^{(1)}\), SIGMA, 6, (2010), (52pp) Exactly solvable models; Bethe ansatz, Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems, Dynamical aspects of cellular automata, Combinatorics on words, , Zeta and \(L\)-functions: analytic theory Bethe ansatz, inverse scattering transform and tropical Riemann theta function in a periodic soliton cellular automaton for \(An^{(1)}\) | 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 prove model completeness for the expansion of the real field by the Weierstrass \(\wp\) function as a function of the variable \(z\) and the parameter (or period) \({\tau}\). We need to existentially define the partial derivatives of the \(\wp\) function with respect to the variable \(z\) and the parameter \({\tau}\). To obtain this result, it is necessary to include in the structure function symbols for the unrestricted exponential function and restricted sine function, the Weierstrass \({\zeta}\) function and the quasi-modular form \(E_{2}\) (we conjecture that these functions are not existentially definable from the functions \(\wp\) alone or even if we use the exponential and restricted sine functions). We prove some auxiliary model-completeness results with the same functions composed with appropriate change of variables. In the conclusion, we make some remarks about the non-effectiveness of our proof and the difficulties to be overcome to obtain an effective model-completeness result, and how to extend these results to appropriate expansion of the real field by automorphic forms. model completeness; Weierstrass systems; elliptic functions; \(\wp\) function; Weierstrass \({\zeta}\) function; modular forms; o-minimality; definable Quantifier elimination, model completeness, and related topics, Model theory of ordered structures; o-minimality, Applications of model theory, Elliptic curves, Elliptic functions and integrals Model completeness for the Real Field with the Weierstrass \(\wp\) function | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(D\) be a quasi-homogeneous Saito free divisor in \(S={\mathbb C}^m,\) \(m\geq 2.\) Then the \({\mathcal O}_S\)-module \(\Omega^1_S(\log D)\) of logarithmic differential \(1\)-forms on \(S\) with poles along \(D\) is free of rank \(m\) and one can define a logarithmic connection \(\nabla : \Omega^1_S(\log D) \longrightarrow \Omega^1_S(\log D) \otimes_{{\mathcal O}_S} \Omega^1_S(\log D)\) [\textit{K. Saito}, ``On the uniformization of complements of discriminant loci'', in: Hyperfunctions and linear partial differential equations, RIMS Kōkyūroku 287, 117--137 (1977)]. The set of all the torsion free integrable connections \(\nabla\) form a finite dimensional affine algebraic variety; it is denoted by \({\mathcal U}(S,D).\) Curiously the structure of this variety is not clear even in the simplest cases. For example, if \(D=D_{A_3}\) is the discriminant of the versal deformation of an \(A_3\)-singularity then \({\mathcal U}(S,D)\) is a 1-dimensional variety with two irreducible components [loc. cit.], while for its ``twin'' given by Sato's polynomial this set is empty [\textit{A. G. Aleksandrov}, ``Moduli of logarithmic connections along a free divisor'', Contemp. Math. 314, 1--23 (2002; Zbl 1021.32011)].
The paper under review is devoted to the study of \({\mathcal U}(S,D)\) by explicit computations of the corresponding systems of uniformization equations for a number of interesting examples, mainly in the 3-dimensional case. It should be remarked that the divisors in question are Koszul free [\textit{F. Calderón-Moreno} and \textit{L. Narváez-Macarro}, ``Locally quasi-homogeneous free divisors are Koszul free'', Proc. Steklov Inst. Math. 238, No. 3, 72--76 (2002) and Tr. Mat. Inst. Steklova 238, 81--85 (2002; Zbl 1031.32006)]. Therefore, any such system is, in fact, a maximally overdetermined system of linear partial differential equations, that is, a holonomic system [Aleksandrov, loc. cit.]. Moreover, in all known cases the fundamental solutions of such systems of uniformization equations are expressed via elementary functions or hypergeometric functions and their generalizations. For example, the fundamental solution of the system associated with one irreducible component of \({\mathcal U}(S,D_{A_3})\) consists of three \(2\)-dimensional hypergeometric series of Horn's type [\textit{A. G. Aleksandrov} and \textit{A. N. Kuznetsov}, ``Regular singular holonomic systems of differential equations with given integrals'', J. Appl. Funct. Anal. 2, No. 1, 21--56 (2007; Zbl 1118.35016)], while the fundamental solution corresponding to another component is expressed via elementary functions [\textit{M. Kato} and \textit{J. Sekiguchi}, ``Systems of uniformization equations with respect to the discriminant sets of complex reflections groups of rank three'' (to appear)]. Conjecturelly, similar assertions are true for discriminants associated with versal deformations of all isolated singularities, finite reflection groups, etc.
The most part of computations in the paper under review relates to these problems for divisors which occur as discriminant hypersurfaces given by anti-invariants associated with irreducible real and complex reflection groups of rank 3 from the list of [\textit{G. C. Shephard} and \textit{J. A. Todd}, ``Finite unitary reflection groups'', Can. J. Math. 6, 274--304 (1954; Zbl 0055.14305)] involving \(A_2, A_3, B_3, H_3\) and \(G_{648}, G_{1296}, G_{336}, G_{2160},\) respectively. To be more precise, the author constructs explicitly systems of uniformization equations for 17 Saito free divisors in 3-dimensional space obtained in [\textit{J. Sekiguchi}, ``A classification of weighted homogeneous Saito free divisors'', J. Math. Soc. Japan 61, No. 4, 1071--1095 (2009; Zbl 1189.32017)] and then finds their particular solutions. In conclusion, he discusses some relations with results from [\textit{K. Saito} and \textit{T. Ishibe}, ``Monoids in the fundamental groups of the complement of logarithmic free divisors in \(\mathbb C^3\)'', J. Algebra 344, No. 1, 137--160 (2011; Zbl 1266.20067)] concerning an explicit computation of the fundamental groups of complements of these 17 divisors, and analyzes four Saito free divisors associated with an exceptional \(W_{12}\)-singularity. torsion free integrable connections; uniformization equations; holonomic systems; Appell's hypergeometric function; monodromy groups; reflection groups; invariant polynomials; hyperelliptic integrals; free divisors Sekiguchi, J., Systems of uniformization equations along Saito free divisors and related topics, the Japanese-Australian workshop on real and complex singularities -- JARCS III, Proc. Centre Math. Appl. Austral. Nat. Univ., 43, 83-126, (2010) Overdetermined systems of PDEs with variable coefficients, Reflection and Coxeter groups (group-theoretic aspects), Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Hypersurfaces and algebraic geometry, Linear first-order PDEs, Generalized hypergeometric series, \({}_pF_q\) Systems of uniformization equations along Saito free divisors and related topics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0553.00001.]
On présente les conjectures de Beilinson, Bass et l'A. concernant le lieu entre le zéro au point entier j de la fonction zêta d'un schéma quasi-projectif sur \({\mathbb{Z}}\) et la K-théorie des faisceaux cohérents sur ce schéma:
(i) \(K'_ m(X)_{(j)}\) est nul pour presque tout entier m; (ii) \(K'_ m(X)_{(j)}\) est de dimension finie sur \({\mathbb{Q}}\); (iii) l'ordre du zéro de \(\zeta_ X(s)\) au point \(s=j\) est égal à \(\sum_{m\geq 0}(-1)^{m+1}\dim_ QK'_ m(X)_{(j)}\). \((K'_ m(X)_{(j)}\) est la partie de poids j de \(K'_ m(X)\otimes {\mathbb{Q}}\) pour les opérations d'Adams, \(K'_ m(X)\) étant les groupes de Quillen). Le lieu entre ceci et les questions connexes (par exemple, les espaces de cycles et les conjectures de Tate, schémas sur un corps fini ou sur \({\mathbb{Q}},...)\) est également présenté. algebraic K-theory; algebraic cycles; zeta-function of scheme; p-adic cohomology; Tate conjecture on cycles; Beilinson conjecture Soulé, C.: K-théorie et zéros aux points entiers de fonctions zêta. Proc. ICM (1983) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) K-théorie et zéros aux points entiers de fonctions zêta | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(C \subset \mathbb{P}^2\) be an integral degree \(d\) curve and let \(Y\) be the normalization of \(C\). Here the author studies the Castelnuovo-Hilbert function of \(\text{Sing} (C)\) to obtain informations on the linear systems \(g^r_{d - \varepsilon}\), \(\varepsilon \geq 0\), on \(Y\). In particular she considers the case in which \(C\) has only a singular ordinary multiple point or a small number of nodes and cusps (here if \(r \geq 2\), then \(r = 2\), \(\varepsilon = 0\) and the \(g^r_d\) is induced by the morphism \(Y \to C \subset \mathbb{P}^2)\). For more on the second example, see \textit{M. Coppens} and \textit{T. Kato}, Manuscr. Math. 70, No. 1, 5-25 (1990); correction: ibid. 71, No. 3, 337-338 (1991; Zbl 0725.14005).
The paper under review contains nice historical remarks. gonality; singular plane curve; Castelnuovo function; line bundle; Castelnuovo-Hilbert function; linear systems Singularities of curves, local rings, Divisors, linear systems, invertible sheaves, Special algebraic curves and curves of low genus, History of algebraic geometry Linear series of low degree on plane curves and the Castelnuovo function | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let f(x,y,z) be a germ of an analytic function defined in a neighbourhood of the origin and assume that the Newton boundary \(\Gamma\) (f) is non- degenerate and \(V=f^{-1}(0)\) has an isolated singularity at the origin. Let \(\Gamma^*(f)\) be the dual Newton diagram. Let \(\Sigma^*\) be a simplicial subdivision of \(\Gamma^*(f)\). It is well known that there is an associated resolution \(\pi:\tilde V\to V.\) However \(\Sigma^*\) is not unique. The author proves that there is a canonical way to get a simplicial subdivision \(\Sigma^*\) so that the graph of the resolution is obtained by a canonical surgery from \(\Gamma^*(f)\) which is considered as a graph. He also proves that a compact two-face \(\Delta\) of \(\Gamma\) (f) corresponds to an exceptional divisor of genus \(g(\Delta)\) which is equal to the number of the integral points on the interior of \(\Delta\). dual Newton diagram; canonical primitive sequence; germ of an analytic function; simplicial subdivision M. OKA, On the resolution of two-dimensional singularities, Proc. Japan Acad., 60 (1984), 174-177 Modifications; resolution of singularities (complex-analytic aspects), Complex singularities, Germs of analytic sets, local parametrization, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry On the resolution of two-dimensional singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper, we study real forms of the complex generic Neumann system. We prove that the real forms are completely integrable Hamiltonian systems. The complex Neumann system is an example of the more general Mumford system. The Mumford system is characterized by the Lax pair \((L^{\mathbb{C}}(\lambda),M^{\mathbb{C}}(\lambda))\) of \(2 \times 2\) matrices, where
\(L^{\mathbb{C}}(\lambda)=\begin{bmatrix} V^{\mathbb{C}}(\lambda) & W^{\mathbb{C}}(\lambda)\\ U^{\mathbb{C}}(\lambda)& -L^{\mathbb{C}}(\lambda)\end{bmatrix}\) and \(U^{\mathbb{C}}(\lambda)\), \(V^{\mathbb{C}}(\lambda)\), \(W^{\mathbb{C}}(\lambda)\) are suitable polynomials. The topology of a regular level set of the moment map of a real form is determined by the positions of the roots of the suitable real form of \(U^{\mathbb{C}}(\lambda)\), with respect to the position of the values of suitable parameters of the system. For two families of the real forms of the complex Neumann system, we describe the topology of the regular level set of the moment map. For one of these two families the level sets are noncompact.
In the paper, we also give the formula which provides the relation between two systems of the first integrals in involution of the Neumann system. One of these systems is obtained from the Lax pair of the Mumford type, while the second is obtained from the Lax pair whose matrices are of dimension \((n+1) \times(n+1)\). integrable systems; Neumann system; Arnold-Liouville level sets; spectral curves; real structures; real forms Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Relationships between algebraic curves and integrable systems, Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics Geometry of real forms of the complex 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 Let G be a group scheme over a suitable base scheme S, acting on a noetherian locally separated algebraic space X. \(G\to S\) is supposed to be flat, separated and of finite type. Then one can associate to the exact category of coherent sheaves of \({\mathcal O}_ X\)-modules with compatible G-action and to the exact subcategory of algebraic G-vector bundles, the K-theory spectra G(G,X) and K(G,X), respectively. These (topological) spectra admit multiplication by a prime power \(l^{\nu}\), such that one has a homotopy fibre sequence \(G\to^{l^{\nu}}G\to G/l^{\nu}\) (similarly for K). \(G/l^{\nu}=G/l^{\nu}(G,X)\) is called the mod \(l^{\nu} reduction\) of G(G,X). It is known that this algebraic \(G/l^{\nu}\) bears some relation to the topological \(G/l^{\nu,Top}\). Analogous facts hold for K. Topological K-theory goes equipped with Bott periodicity. Algebraic K-theory has no such property. To overcome this, one imposes Bott periodicity by inverting the so-called Bott element \(\beta\) to obtain spectra \(G/l^{\nu}(G,X)[\beta^{-1}]\) and \(K/l^{\nu}(G,X)[\beta^{-1}]\). These are the objects of the present paper.
First, the basic properties of G(G,X) and K(G,X) are recalled: functorial behaviour, localization exact sequences, projective space bundle theorem, homotopy property and Poincaré duality. For \(H\subset G\) a closed subgroup acting on \(G\times X\) by \(h(g,x):=(gh^{-1},hx)\), one denotes \(G\times X\) modulo this action by \(G\times^ HX\) and one obtains Morita equivalence \(G(G,G\times^ HX)\simeq G(H,H\times^ HX)\simeq G(H,X)\) and in case G is reductive with a maximal torus T of a Borel subgroup of G, there is a naturally split monomorphism \(G(H,X)\to G(T,G\times^ HX)\). Next, for a group scheme G acting on two algebraic spaces X and Y, one calls a map \(f: X\to Y\) isovariant if f is equivariant and if, moreover, it induces an isomorphism of isotopy groups \(G_ X\simeq G_ Y\simeq_ YX\). Then the orbit topos ``X/G'' is defined as the Grothendieck topos of the site whose objects are locally separated algebraic spaces U with \(U\to X\) finitely presented, étale and G- isovariant. A morphism is a G-equivariant map \(U\to V\) compatible with the maps to X. Such a morphism is étale and isovariant. The topos ``X/G'' is well adapted to geometric questions. On the other hand one can define another topos ``X//G'', better adapted to topos-theoretic questions, like finiteness of cohomological dimension and pasting. ``X/G'' and ``X//G'' are shown to be equivalent topoi.
Part of a fundamental result is the following descent theorem, which says that under certain conditions on S and the prime l, there is a homotopy equivalence
\[
G/l^{\nu}(G,X)[\beta^{-1}]\simeq {\mathbb{H}}^.(``X/G'';\quad G/l^{\nu}(G,p^{-1}(\quad))[\beta^{-1}]),
\]
where \(G(G,p^{-1}(\quad))\) is the presheaf of spectra on the site of ``X/G'', sending an isovariant étale \(U\to X\) to \(G(G,p^{- 1}(U/G))=G(G,U)\). \({\mathbb{H}}^.\) denotes the hypercohomology spectrum. For X of finite Krull dimension a Zariski version also exists without the need of \(\beta\)-inversion and with ``X/G'' replaced by \(``X/G_{Zar}''\), the topos of the site of G-invariant open subspaces of X. Translation in terms of homotopy groups (with \(F_ n\) short for \(\pi_ nF\) for a spectrum F...) gives a strongly converging Fary spectral sequence
\[
E_ 2^{p,q}=H^ p(``X/G'';\quad \tilde G/l_ q^{\nu}(G,p^{- 1}(\quad))[\beta^{-1}])\Rightarrow G/l^{\nu}_{q-p}(G,X)[\beta^{- 1}].
\]
For X a scheme of finite type over an algebraically closed field k of characteristic 0, acted on by a smooth linear algebraic group G over k, one obtains a proof of the fact that \(G/l^{\nu}(G,X)[\beta^{-1}]\) satisfies the Kazhdan-Lusztig axioms for an equivariant K-homology theory suitable for p-adic representation theory. Next a proof is given of Segal's Concentration Theorem, which says that localization at a prime \(\rho\) of the representation ring R(G) of a linear algebraic group G over an algebraically closed field k of characteristic 0 induces a homotopy equivalence
\[
G/l^{\nu}(G,X^{(\rho)})[\beta^{-1}]_{(\rho)}\simeq G/l^{\nu}(G,X)[\beta^{-1}]_{(\rho)},
\]
where \(X^{(\rho)}\to X\) is a well defined equivariant immersion. Such a prime \(\rho\) corresponds to a conjugacy class of a diagonalizable subgroup of G. The last paragraph contains comparison results for a linear algebraic group G over \({\mathbb{C}}\), with a maximal compact subgroup M of G(\({\mathbb{C}})\). Under suitable conditions there are homotopy equivalences with topological K- homology
\[
G/l^{\nu}(G,X)[\beta^{-1}]\simeq G/l^{\nu,Atiyah- Segal}(M,X({\mathbb{C}}))
\]
and with topological K-cohomology
\[
K/l^{\nu}(G,X)[\beta^{-1}]\simeq K/l^{\nu,Atiyah- Segal}(M,X({\mathbb{C}})),
\]
where \(K/l^{\nu,Atiyah-Segal}(M,X({\mathbb{C}}))\) denotes the mod \(l^{\nu}\) reduction of the spectrum of M-equivariant complex topological vector bundles on X(\({\mathbb{C}})\). Similarly for G. Actually one has to be careful with the notion of equivariant homotopy equivalence. In the above the equivalences are with the ``Bredon-type'' equivariant topological K-(co)homologies. group scheme over a base scheme acting on a noetherian locally separated algebraic space; algebraic G-vector bundles; K-theory spectra; Topological K-theory; Bott periodicity; Algebraic K-theory; Bott element; descent theorem; Fary spectral sequence; Kazhdan-Lusztig axioms for an equivariant K-homology theory; Segal's Concentration Theorem; localization R. W. Thomason, Equivariant algebraic vs. topological \?-homology Atiyah-Segal-style, Duke Math. J. 56 (1988), no. 3, 589 -- 636. Topological \(K\)-theory, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Equivariant homology and cohomology in algebraic topology, Cohomology theory for linear algebraic groups Equivariant algebraic vs. topological K-homology Atiyah-Segal-style | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A set of \(s\) points in \(\mathbb{P}^ d\) is called a Cayley-Bacharach scheme (CB-scheme), if every subset of \(s-1\) points has the same Hilbert function. We investigate the consequences of this ``weak uniformity.'' The main result characterizes CB-schemes in terms of the structure of the canonical module of their projective coordinate ring. From this we get that the Hilbert function of a CB-scheme \(X\) has to satisfy growth conditions which are only slightly weaker than the ones gives by Harris and Eisenbud for points with the uniform position property [cf. \textit{J. Harris} (with the collaboration of \textit{D. Eisenbud}), ``Curves in projective space'', Sém. Math. Supér. 85 (1982; Zbl 0511.14014)].
We also characterize CB-schemes in terms of the conductor of the projective coordinate ring in its integral closure and in terms of the forms of minimal degree passing through a linked set of points. Applications include efficient algorithms for checking whether a given set of points is a CB-scheme, results about generic hyperplane sections of arithmetically Cohen-Macaulay curves and inequalities for the Hilbert functions of Cohen-Macaulay domains. CB-scheme; Cayley-Bacharach scheme; Hibert function; hyperplane sections; arithmetically Cohen-Macaulay curves A. Geramita, M. Kreuzer, and L. Robbiano, \textit{Cayley-Bacharach schemes and their canonical modules}, Trans. Amer. Math. Soc., 399 (1993), pp. 163--189, . Schemes and morphisms, Plane and space curves, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Relevant commutative algebra, Projective techniques in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Special varieties, Computational aspects in algebraic geometry 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 [For the entire collection see Zbl 0638.00012.]
The aim of this paper is to investigate the local cohomology of noncommutative rings \(\Lambda\) by using the noncommutative scheme Spec(\(\Lambda)\) endowed with appropriate structure (pre)sheaves. Let \(\Lambda\) be a left and right noetherian ring, and Spec(\(\Lambda)\) the set of all prime ideals of \(\Lambda\), endowed with the Zariski topology. It is shown that \(H^ 0_{X/Z}E_ M=E_{Q_{\sigma}(M)}\), where \(E_ M\) is the structure presheaf associated to the left \(\Lambda\)- module M, Z is any subset of \(X=Spec(\Lambda)\), and \(\sigma\) is the idempotent kernel functor attached to the generic closure of Z in X. In the second part of the paper the author deals with sheaves; the ring \(\Lambda\) is called geometrically realizable if for any \(M\in \Lambda\)- mod, the presheaves \(E_ M\) are actually sheaves. Any compatible ring \(\Lambda\) (this means that for each ideals I, J of \(\Lambda\) there exists a positive integer n such that \(I^ nJ^ n\subseteq JI)\) is geometrically realizable. Examples of compatible rings are: commutative rings, Azumaya algebras over the center, Zariski central rings, left classical rings, almost commutative rings, etc. It is shown that the geometric and the algebraic closure operators coincide, and that there exist spectral sequences relating the geometric and algebraic local cohomology groups. Artin-Rees property; geometrically realizable ring; local cohomology; noncommutative scheme; left and right noetherian ring; prime ideals; structure presheaf; idempotent kernel functor; compatible rings; Azumaya algebras; closure operators; spectral sequences; local cohomology groups A. Verschoren, ''Local cohomology of noncommutative rings: a geometric interpretation,''Lect. Notes Math.,1328, 316--331 (1988). (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), Torsion theories; radicals on module categories (associative algebraic aspects), Homological methods in associative algebras, Localization and associative Noetherian rings, Noetherian rings and modules (associative rings and algebras), Modules, bimodules and ideals in associative algebras, Local cohomology and algebraic geometry, Schemes and morphisms Local cohomology of noncommutative rings: A geometric interpretation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 Givental type decomposition
for partition functions that arise out of topological recursion
applied to spectral curves. Copies of the Konstevich-Witten
Korteweg-de Vries (KdV) tau function arise out of regular spectral
curves and copies of the Brezin-Gross-Witten KdV tau function
arise out of irregular spectral curves. The authors present the
example of this decomposition for the matrix model with two hard
edges and spectral curve \((x^2-4)y^2=1\).
This paper is organized
as follows: The first section is an introduction to the subject.
In the second section the authors recall the definitions of the
two KdV tau functions which form the fundamental pieces of the
decomposition. In the third section, they introduce the
decomposition without the differential operator
\(\mathcal{\widehat{R}}\) via the elementary topological part of the
correlators. The forth section deals with the main result of this
paper which is a generalization of the decomposition theorem to
allow irregular singularities. One application of the main result
applied to the curve above is a Givental type decomposition for
the partition function of the Legendre ensemble. The paper is
supported by an appendix where the authors first consider
quantization in finite dimensions which easily generalizes to
infinite dimensions. Givental decomposition; topological recursion; spectral curve; KdV tau function Enumerative problems (combinatorial problems) in algebraic geometry, Exact enumeration problems, generating functions, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Topological recursion with hard edges | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Binary linear codes with few weights have wide applications in communication, secret sharing schemes, authentication codes, association schemes, strongly regular graphs, etc. Projective binary linear codes are among the most important subclasses of binary linear codes for practical applications. In this paper, motivated by the two excellent recent papers (Li et al. in IEEE Trans Inf Theory 67(7):4263--4275, 2021) and (Wang et al. in IEEE Trans Inf Theory 67(8):5133--5148, 2021), several new families of few-weight projective binary linear codes are constructed from the defining sets, and then their Hamming weight distributions are determined by employing the Walsh transform of the corresponding two-to-one functions over finite fields with the even characteristic. Our constructions can produce binary linear codes with new parameters. Some of the constructed binary linear codes are optimal or almost optimal according to the online Database of Grassl, and the duals of some of them are distance-optimal with respect to the sphere packing bound. This paper also shows once again that the two-to-one functions initially studied in (Mesnager and Qu in IEEE Trans Inf Theory 65(12):7884--7895, 2019) are also promising objects in coding theory. Although our derived codes use objects considered in the very recent literature, the analysis of our designed codes involves functions having different algebraic structures (and, therefore, other Walsh transform distribution) and requires solving new systems of equations over finite fields, which is an essential step in determining the weight distribution of our constructed codes. As applications, some of the codes presented in this paper can be used to construct association schemes and secret sharing schemes with interesting access structures. projective code; Hamming weight distribution; two-to-one function; association scheme; secret sharing scheme Linear codes (general theory), Authentication, digital signatures and secret sharing, Cryptography, Applications to coding theory and cryptography of arithmetic geometry, Algebraic coding theory; cryptography (number-theoretic aspects) Further projective binary linear codes derived from two-to-one functions and their duals | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Fixed an algebraically closed field \(k\), the Hilbert scheme \({\mathbf H}\) parametrizing the zero-dimensional subschemes of the affine plane Spec \(k[x,y]\) is a disjoint union of its components \({\mathbf H}^l\) parametrizing the subschemes of length \(l\). The linear action of the torus \(k^*\times k^*\) on the polynomial ring \(k[x,y]\), defined by \((t_1,t_2) \cdot x^\alpha y^\beta=(t_1\cdot x)^\alpha (t_2\cdot y)^\beta\), induces an action on the Hilbert schemes \({\mathbf H}\) and \({\mathbf H}^l\). Given two relatively prime integers \(a\) and \(b\), one may consider the closed subschemes \({\mathbf H}_{ab}\subset{\mathbf H}\) and \({\mathbf H}_{ab}^l \subset{\mathbf H}^l\), parametrizing the zero-dimensional subschemes invariant under the action of the subtorus \(T_{ab}=\{(t^{-b}, t^a),\;t\in k^*\}\). Sometimes, information on \({\mathbf H}_{ab}\) may be lifted to \({\mathbf H}\). In fact \textit{G. Ellingsrud} and \textit{S. A. Strømme} [Invent. Math. 87, 343--352 (1987; Zbl 0625.14002)] have computed the Chow group of \({\mathbf H}^l\) examining the embedding \({\mathbf H}^l_{ab}\subset {\mathbf H}^l\) for general \((a,b)\), and \textit{M. Brion} [Transform. Groups 2, No. 3, 225--267 (1997; Zbl 0916.14003)] has described the equivariant cohomology of \({\mathcal H}^l\) in terms of the equivariant cohomology of all the \({\mathbf H}^l_{ab}\). This accounts for the interest in studying the schemes \({\mathbf H}_{ab}\). In the paper under review, the author determines the irreducible components of the schemes \({\mathbf H}_{ab}\) (by \textit{R. Hartshorne} [Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.14502)] and \textit{J. Fogarty} [Am. J. Math. 70, 511--521 (1968; Zbl 0176.18401)] one already knows that the smooth subschemes \({\mathbf H}^l\) are the irreducible components of \({\mathbf H})\). The basic and natural idea is that one may separate \({\mathbf H}_{ab}^l\) into disjoint subschemes by fixing a Hilbert function. More precisely, define the degree \(d\) of a monomial by the formula \(d(x^\alpha y^\beta)=-b\alpha +a\beta\). Then a subscheme \(Z\subset\text{Spec}\,k[x,y]\) is in \({\mathbf H}_{ab}\) if and only if its ideal \(I(Z)\) is quasi-homogeneous with respect to \(d\), i.e. \(I(Z)= \bigoplus_{n\in\mathbb{Z}}I(Z)_n\), where \(I(Z)_n=I(Z)\cap k[x,y]_n\) and \(k[x,y]_n\) denotes the vector space generated by the monomials \(m\) of degree \(d(m)=n\). One defines the numerical sequence \(H=(\text{codim} (I(Z)_n\), \(k[x,y]_n))_{n\in \mathbb{Z}}\) as the Hilbert function of \(Z\). Now, fix any sequence of integers \(H=(h_n)_{n\in \mathbb{Z}}\), and denote by \({\mathbf H}_{ab}(H)\) the (possibly empty) closed subscheme of \({\mathbf H}_{ab}\) parametrizing the subschemes \(Z\) whose Hilbert function is \(H\). It turns out that \({\mathbf H}_{ab}\) is the disjoint union of the subschemes \({\mathbf H}_{ab}(H)\), and the main result of the paper consists in proving that, when it is not empty, then \({\mathbf H}_{ab}(H)\) is smooth and irreducible. The methods involved in the proof allow the author to recover some of the results appearing in previous papers of \textit{G. Ellingsrud} and \textit{S. A. Strømme} [loc. cit.; Invent. Math. 91, No. 2, 365--370 (1988; Zbl 1064.14500)]. Bialynicki-Birula stratification; zero-dimensional scheme; affine plane; Hilbert function Evain, L., Irreducible components of the equivariant punctual Hilbert schemes, Adv. Math., 185, 328-346, (2004) Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Group actions on varieties or schemes (quotients) Irreducible components of the equivariant punctual 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 Given a 0-dimensional scheme \(\mathbb{X}\) in a projective space \(\mathbb P_K^n\) over a field \(K\), we characterize the Cayley-Bacharach property (CBP) of \(\mathbb{X}\) in terms of the algebraic structure of the Dedekind different of its homogeneous coordinate ring. Moreover, we characterize Cayley-Bacharach schemes by Dedekind's formula for the conductor and the complementary module, we study schemes with minimal Dedekind different using the trace of the complementary module, and we prove various results about almost Gorenstein and nearly Gorenstein schemes. zero-dimensional scheme; Cayley-Bacharach scheme; almost Gorenstein; Dedekind different; Hilbert function; Dedekind's formula Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Other special types of modules and ideals in commutative rings, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Projective techniques in algebraic geometry The Dedekind different of a Cayley-Bacharach scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathbb{P}^ 2\) be the projective plane over an algebraically closed field \(k\). Given \(s\) distinct points \(P_ 1,\dots,P_ s\) lying on a nonsingular conic in \(\mathbb{P}^ 2\) and \(s\) positive integers \(m_ 1,\dots,m_ s\), let \(I\subset k[X_ 0,X_ 1,X_ 2]\) be the ideal of homogeneous polynomials in three variables vanishing at \(P_ i\) with multiplicity at least \(m_ i\) \((1\leq i\leq s)\). We give algorithms for computing the Hilbert function of the graded ring \(k[X_ 0,X_ 1,X_ 2]/I\), and the shape of the minimal free resolution of \(I\). linear systems; fat points on a conic; computing the Hilbert function Catalisano, M. V., ``Fat'' points on a conic, Commun. Algebra, 19, 2153-2168, (1991) Divisors, linear systems, invertible sheaves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) ``Fat'' points on a conic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 early 90's the works of \textit{B. Dubrovin} [``Integrable systems in topological field theory'', Nuclear Physics B 379 (1992)] and \textit{M. Kontsevich} and \textit{Yu. Manin} [Commun. Math. Phys. 164, No. 3, 525--562 (1994; Zbl 0853.14020)] pointed a striking relationship between quantum cohomology and certain integrable systems, through the so called WDVV-equation. Solutions of an equation of the family of WDVV-equations (which are integrable systems) turned out to be generating functions of Gromov-Witten invariants of certain symplectic manifolds. In later works (quoted in the paper) further similar relations between integrable systems and cohomological invariants were founded.
The question emerged if to any integrable system corresponds a set of cohomological invariants of some variety. A conjecture proposed by \textit{V. Bouchard} et al. [Commun. Math. Phys. 287, No. 1, 117--178 (2009; Zbl 1178.81214)] is that, if the spectral curve of an integrable system is mirror symmetric to a toric Calabi-Yau 3-fold \(\mathfrak{X}\), then the solutions to this system are generating functions of the Gromov-Witten invariants of \(\mathfrak{X}\). \smallskip
The present paper is a step towards the setting of Bouchard-Klemm-Mariño-Pasquetti's conjecture. The main result of the paper is the correspondence between an integrable system and some intersection numbers in a space \(\mathcal{M}_{g,n}^{\mathfrak{b}}\) of Riemann surfaces with `colored' marked points. As explained by the author, such correspondence relates the spectral curve of an integrable system, and the Gromov-Witten invariants of a finite set of points. The author is able to recover with his result several previously known cases of the conjecture. To fully settle BKMP-conjecture it would remain to prove that the Gromov-Witten theory of a toric CY 3-fold reduces to the Gromov-Witten theory of a finite set of points. spectral curve; integrable systems; Gromov-Witten theory B. Eynard, \textit{Intersection numbers of spectral curves}, arXiv:1104.0176 [INSPIRE]. Relationships between algebraic curves and integrable systems, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Invariants of spectral curves and intersection theory of moduli spaces of complex 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 An easy calculation (see, for instance, \textit{K. Joshi} and \textit{C. S. Yogananda} [Acta Arith. 91, No. 4, 325--327 (1999; Zbl 0959.11037)]) shows that
\[
\zeta(s- 1)\zeta(s)^{-1}= \prod^\infty_{n=1}\, \prod_{\chi\in X(n)} L(\xi,s)\quad\text{for Re\,} s> 2,
\]
where \(X(n)\) stands for the group of the Dirichlet characters modulo \(n\). The left-hand side of this identity may be regarded as the Hasse-Weil zeta function of the scheme \(G_m:= \text{Spec\,}\mathbb Z[t, t^{-1}]\). In the work of \textit{Y. Taniyama} [J. Math. Soc. Japan, 9, 330--366 (1957; Zbl 0213.22803)] analogous formulae had been obtained for the Hasse-Weil zeta functions of Abelian schemes. The authors prove such formulae for extensions of Abelian schemes by a torus and discuss some further generalizations. Hasse-Weil zeta function; Abelian scheme; algebraic torus; \(\ell\)-adic representations; Artin \(L\)-series Zeta functions and \(L\)-functions of number fields, Abelian varieties of dimension \(> 1\), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Arithmetic ground fields for abelian varieties, Varieties over global fields Horizontal factorizations of certain Hasse-Weil zeta functions -- a remark on a paper of Taniyama | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 part I of this paper [cf. C. R. Acad. Sci., Paris, Ser. I 325, 183-188 (1999; Zbl 0906.14010)] \textit{Min He} introduced the moduli spaces \(S_\alpha\) of \(\alpha\)-semistable coherent systems \(\Lambda= (\Gamma,F (\ell))\), where \(F\) is an algebraic coherent sheaf of rank 2 on the projective plane \(\mathbb{P}_2\) with Chern classes \((0,n)\), \(\ell\) an integer \(\geq 1\), and \(\Gamma \subset H^0 (F (\ell))\) a vector subspace of rank 1. In this paper for each non-critical value \(\alpha\) on \(S_\alpha\) the author defines the determinant bundles and the corresponding integral. By studying the changes of this integral when \(\alpha\) passes through a critical value, one obtains relations between the Donaldson numbers on \(\mathbb{P}_2\) and certain intersection numbers on the Hilbert scheme \(\text{Hilb}^{n+ \ell^2} (\mathbb{P}_2)\) \((3\leq n\leq 11)\). \(\alpha\)-semistable coherent systems; Donaldson numbers; intersection numbers; Hilbert scheme He M., Espaces de modules de systèmes cohérents. II. Nombres de Donaldson, C. R. Acad. Sci. Paris Sér. I Math., 1997, 325(3), 301--306 Algebraic moduli problems, moduli of vector bundles, Characteristic classes and numbers in differential topology, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Moduli spaces of coherent systems. II: Donaldson 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 the present survey paper, we present several new classes of Hochster's spectral spaces ``occurring in nature,'' actually in multiplicative ideal theory, and not linked to or realized in an explicit way by prime spectra of rings. The general setting is the space of the semistar operations (of finite type), endowed with a Zariski-like topology, which turns out to be a natural topological extension of the space of the overrings of an integral domain, endowed with a topology introduced by Zariski. One of the key tool is a recent characterization of spectral spaces, based on the ultrafilter topology, given in [the first author, Commun. Algebra 42, No. 4, 1496--1508 (2014; Zbl 1310.14005)]. Several applications are also discussed. spectral space and spectral map; star and semistar operations; Zariski, constructible, inverse and ultrafilter topologies; Gabriel-Popescu localizing system; Riemann-Zariski space of valuation domains; Kronecker function ring Finocchiaro, Carmelo A.; Fontana, Marco; Spirito, Dario, New distinguished classes of spectral spaces: a survey, (Multiplicative Ideal Theory and Factorization Theory: Commutative and Non-Commutative Perspectives, (2016), Springer Verlag) Ideals and multiplicative ideal theory in commutative rings, Integral domains, Morphisms of commutative rings, Chain conditions, finiteness conditions in commutative ring theory, Injective and flat modules and ideals in commutative rings, Relevant commutative algebra New distinguished classes of spectral spaces: a survey | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 smooth connected \(k\)-group scheme over a field \(k\). In this article the author investigates
\[
\mathrm{Ext}^1(G,\mathbb{G}_m):=\{ \mathcal{L} \in \mathrm{Pic}(G\ |\ m^*\mathcal{L} \simeq \pi_1^*\mathcal{L} \otimes \pi_2^* \mathcal{L}\} \subseteq \mathrm{Pic}(G),
\]
which is the subgroup of line bundles on \(G\) that are translation-invariant. (Here \(m\colon G \times G \to G \) denotes the multiplication, and \(\pi_1,\ \pi_2\colon G \times G \to G\) denote the natural projections.)
\smallskip
The main results of this article are the following:
\begin{itemize}
\item If \(G\) is affine and \(k\) is a global function field, then \(\mathrm{Ext}^1(G,\mathbb{G}_m)\) is finite (Theorem 1.1).
\item If \(k\) is a global field, then \(\mathrm{Ext}^1(G,\mathbb{G}_m)\) is a finitely generated abelian group (Theorem 1.2).
\item Let \(k_s\) be a separable closure of \(k\). If \(G_{k_s}\) is unirational (this is the case in particular if \(G\) is affine and \(k\) is perfect), then \(\mathrm{Ext}^1(G,\mathbb{G}_m)=\mathrm{Pic}(G)\) is finite (Theorem 1.3, due to Ofer Gabber).
\item Assume that \(G\) is affine and \(\mathrm{Pic}(G)\) is finite. If \(G(k)\) is Zariski dense in \(G\) or if \(k\) is separably closed, then \(\mathrm{Pic}(G)=\mathrm{Ext}^1(G,\mathbb{G}_m)\) (Theorem 1.5).
\end{itemize}
\smallskip
Moreover, the author also proves that if \(\mathrm{Ext}^1(U,\mathbb{G}_m)\) is finite for every \(k\)-form \(U\) of \(\mathbb{G}_a\), then \(\mathrm{Ext}^1(G,\mathbb{G}_m)\) is finite (Proposition 4.1). This result indicates that \(k\)-forms of \(\mathbb{G}_a\) might be a source of various pathologies, and Section 5 is then dedicated to the construction of examples of pathological behavior for the cohomology of commutative linear algebraic groups over local and global function fields. group scheme; Picard group; global function field; cohomology; Tate-Shafarevich set Group schemes, Linear algebraic groups over arbitrary fields, Linear algebraic groups over global fields and their integers, Picard groups Translation-invariant line bundles on linear algebraic 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 paper is devoted to nonsingular intersections of three real quadrics in \({\mathbb P}^6\) (the author of the paper calls these intersections \textit{real three-dimensional triquadrics}).
Recall that a real algebraic variety~\(X\) is called \textit{maximal} (or an \(M\)-\textit{variety}) if the total mod \(2\) Betti number of the real point set \({\mathbb R}X\) of~\(X\) is equal to the total mod \(2\) Betti number of the complex point set~\({\mathbb C}X\) of~\(X\) (according to the Harnack-Smith-Thom inequality the total mod \(2\) Betti number of \({\mathbb R}X\) does not exceed the total mod \(2\) Betti number of~\({\mathbb C}X\)).
The paper under review contains a construction of maximal real three-dimensio\-nal triquadrics with \(k\) connected components of the real point set, where \(k\) is any integer such that \(1 \leq k \leq 14\). (It is proved in [\textit{A. Degtyarev, I. Itenberg} and \textit{V. Kharlamov}, Progr. Math. 296, 81--108 (2012; Zbl 1266.14046)] that the real point set of any real three-dimensional triquadric has at most \(14\) connected components.) intersection of quadrics; maximal varieties; spectral curve; theta-characteristics; index function Topology of real algebraic varieties, Varieties of low degree, \(3\)-folds On the number of components of a three-dimensional maximal intersection of three real quadrics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 abelian varieties with complex multiplication, the dimension of the \({\bar {\mathbb{Q}}}\)-vector space generated by the periods of first and second kind is computed. Applied to the Jacobian of the Fermat curves, this result and the criteria of Shimura-Taniyama and Deligne-Koblitz-Ogus give an optimal theorem on the linear independence of the values \(B(a_ 1,b_ 1),...,B(a_ n,b_ n)\) of the Beta-function at rational arguments \(a_ j, b_ j:\) They are \({\bar {\mathbb{Q}}}\)-linearly dependent only in the obvious case, namely if this dependence already arises from the classical Gauss-Legendre identities for the values of the \(\Gamma\)- function. - This theorem gives in turn a partial answer to a transcendence question in uniformization theory raised by S. Lang: Let X be a smooth projective algebraic curve, defined over \({\bar {\mathbb{Q}}}\) and of genus \(g>1\), and \(\phi:\quad U_ r:=\{\zeta \in {\mathbb{C}}| | \zeta | <r\}\to X\) a normalized holomorphic covering map, i.e. with \(\phi\) (0)\(\in X({\bar {\mathbb{Q}}})\) and tangential map \(\phi\) '(0) defined over \({\bar {\mathbb{Q}}}\); is then the ''covering radius'' r a transcendental number? The answer is ''yes'' if X has many automorphisms - e.g. Fermat curves, Klein's curve - and \(\phi\) (0) is a fixed point, because in this case the covering radius is the quotient of two Beta-values which are \({\bar {\mathbb{Q}}}\)-linearly independent. abelian varieties with complex multiplication; periods of first and second kind; Jacobian of the Fermat curves; linear independence of values of the Beta-function; covering radius Wolfart, J., Der überlagerungsradius gewisser algebraischer kurven und die werte der betafunktion an rationalen stellen, Math. Ann., 273, 1-15, (1985) Coverings of curves, fundamental group, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Transcendence (general theory), Complex multiplication and abelian varieties, Rational points, Jacobians, Prym varieties Der Überlagerungsradius gewisser algebraischer Kurven und die Werte der Betafunktion an rationalen Stellen | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper studies a new definition (studied independently by \textit{M. Kontsevich}) of Hochschild cohomology for schemes. Namely, let \(X\) be a separated scheme of finite type over a field \(k\), and let \({\mathcal F}\) be a sheaf of \({\mathcal O}_X\)-modules. Let \(\delta: X\to X\times X\) be the diagonal morphism, and identify \({\mathcal O}_X\) with \(\delta_*({\mathcal O}_X)\), \({\mathcal F}\) with \(\delta_*({\mathcal F})\). Define the Hochschild cohomology of \(X\) with coefficients in \({\mathcal F}\) to be \(H^n({\mathcal O}_X, {\mathcal F}): =\text{Ext}^n_{{\mathcal O}_{X\times X}} ({ \mathcal O}_X, {\mathcal F})\). This definition is motivated by the definition \(H^n(A,M) =\text{Ext}^n_{A^e} (A,M)\) of Hochschild cohomology for an \(A\)-module \(M\) \((A\) a commutative \(k\)-algebra, \(A^e=A \otimes_kA)\). Two other definitions of the Hochschild cohomology of \(X\) with coefficients in \({\mathcal F}\) are also discussed, namely \(HH^n(X,{\mathcal F}): =\mathbb{E}\text{xt}^n_{{\mathcal O}_X} ({\mathcal C}., {\mathcal F})\), where \({\mathcal C}.\) is the sheafification of the cyclic bar complex and \(\mathbb{E}\)xt denotes hyperext (an approach suggested by Loday for Hochschild homology), and a definition given by Gerstenhaber and Schack [cf. \textit{M. Gerstenhaber} and \textit{S. D. Schack}, ``Algebraic cohomology and deformation theory'', in: Deformation theory of algebras and structures and applications, Nato ASI Ser., Ser. C, 11-264 (1988; Zbl 0676.16022)]. Cyclic cohomology \(HC^n(X, {\mathcal F})\) is defined in a manner similar to \(HH^n(X, {\mathcal F})\). The main result of this paper is that \(H^n({\mathcal O}_X, {\mathcal F})\) is isomorphic to both of the other two definitions of Hochschild cohomology. Each of the three definitions has an associated Hodge spectral sequence. It is shown that if \(X\) is smooth than these spectral sequences are isomorphic. Furthermore it is immediate from the \(H^n({\mathcal O}_X, {\mathcal F})\) definition that if \(X\) is projective and \({\mathcal F}\) is coherent then the Hochschild cohomology and cyclic cohomology are finite-dimensional \(k\)-vector spaces. quasiprojective scheme; Hochschild cohomology for schemes; sheafification; hyperext; deformation theory; Hodge spectral sequence; cyclic cohomology R. G. Swan, ''Hochschild cohomology of quasiprojective schemes,'' J. Pure Appl. Algebra, vol. 110, iss. 1, pp. 57-80, 1996. Other (co)homology theories (cyclic, dihedral, etc.) [See also 19D55, 46L80, 58B30, 58G12], Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Spectral sequences, hypercohomology Hochschild cohomology of quasiprojective 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 classical theory of braids is deeply connected with the theory of reflection groups and there are many relations between Artin groups and Coxeter groups. It turns out that the classifying spaces of Artin groups of finite type are affine varieties, the complement of the singularities associated to Coxeter groups.
In order to study the topology of the Milnor fiber of these non-isolated singularities together with the monodromy action it is useful to compute the cohomology of the Artin groups with coefficients in an Abelian representation.
In this book a description of this cohomology for Artin groups of type \(A\) and \(B\) and for affine Artin groups of the same type is given. affine Artin groups; arrangements of hyperplanes; group cohomology; local systems; Salvetti complexes; Milnor fibrations; spectral sequences; braid groups; Coxeter groups; classifying spaces Cohomology of groups, Research exposition (monographs, survey articles) pertaining to group theory, Braid groups; Artin groups, Classifying spaces of groups and \(H\)-spaces in algebraic topology, Milnor fibration; relations with knot theory, Relations with arrangements of hyperplanes, Reflection and Coxeter groups (group-theoretic aspects), Singularities in algebraic geometry Cohomology of finite and affine type Artin groups over Abelian representations. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Recent work by Grigoriev and Shpilrain [8] suggests looking at the tropical semiring for cryptographic schemes. In this contribution we explore the tropical analogue of the \textit{Hessian pencil} of plane cubic curves as a source of group-based cryptography. Using elementary tropical geometry on the tropical Hessian curves, we derive the addition and doubling formulas induced from their Jacobian and investigate the discrete logarithm problem in this group. We show that the DLP is solvable when restricted to integral points on the tropical Hesse curve, and hence inadequate for cryptographic applications. Consideration of point duplication, however, provides instances of solvable chaotic maps producing random sequences and thus a source of fast keyed hash functions. tropical geometry; discrete dynamical systems; keyed hash function; elliptic curve cryptography Applications to coding theory and cryptography of arithmetic geometry, Jacobians, Prym varieties, Elliptic curves, Theta functions and abelian varieties, Dynamical systems involving maps of the interval, Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets Cryptography from the tropical Hessian pencil | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main purpose of this paper is to give a proof that any Hilbert function stratum, i.e. the set of points of a Hilbert scheme with a fixed Hilbert function, is connected in characteristic zero. Furthermore, we give a criterion when the union of two (or more) Hilbert function strata is itself connected. We also give short proofs of the theorems of \textit{G. Gotzmann} [Comment. Math. Helv. 63, 114-149 (1988; Zbl 0656.14004)] and \textit{R. Hartshorne} [Publ. Math., Inst. Hautes Étud. Sci. 29, 5-48 (1966; Zbl 0171.41502)] in characteristic zero. Furthermore, we prove the connectedness of the subsets of the Hilbert scheme consisting of points with a fixed Castelnuovo-Mumford regularity, as well as the connectedness of the intersections of these sets with Hilbert function strata. connectedness of subsets of the Hilbert scheme; Hilbert function stratum; Castelnuovo-Mumford regularity \textit{AimPL: Components of Hilbert Schemes}. Available at http://aimpl.org/hilbertschemes Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Topological properties in algebraic geometry, Connected and locally connected spaces (general aspects) Connectedness of Hilbert function strata and other connectedness results | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study geometric and algebraic geometric properties of the continuous and discrete Neumann systems on cotangent bundles of Stiefel varieties \(V_{n,r}\). The systems are integrable in the non-commutative sense, and by applying a \(2r\times 2r\)-Lax representation, we show that generic complex invariant manifolds are open subsets of affine Prym varieties on which the complex flow is linear. The characteristics of the varieties and the direction of the flow are calculated explicitly. Next, we construct a family of multi-valued integrable discretizations of the Neumann systems and describe them as translations on the Prym varieties, which are written explicitly in terms of divisors of points on the spectral curve. integrable systems; spectral curves; Prym varieties; invariant tori; multi-valued mappings Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Relationships between algebraic curves and integrable systems, Applications of Lie algebras and superalgebras to integrable systems, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics, Constrained dynamics, Dirac's theory of constraints Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi-Mumford systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\subset \mathbb {P}^n\) be a closed subscheme. Following [\textit{E. Carlini} et al., J. Algebra 324, No. 4, 758--781 (2010; Zbl 1197.13016)] \(X\) is said to have bipolynomial Hilbert function (also known as to have maximal rank) if \(X\) has the expected postulation, i.e. the expected Hilbert function, i.e. \(h^0(\mathcal {I}_X(t)) =\max \{0,\binom{n+t}{n}-h^0(\mathcal {O}_X(t))\}\) for all \(t\in \mathbb {N}\). \(X\) is known to have maximal rank, except in a few completely described exceptional cases, if it is a general union of lines [\textit{R. Hartshorne} and \textit{A. Hirschowitz}, Lect. Notes Math. 961, 169--188 (1982; Zbl 0555.14011)] or a general union of lines and one multiple point or a linear subspace and several general lines (many authors, all quoted in the paper under review). In the paper under review, this is proved when \(n\geq 4\) and \(X\) is a general union of \(s\) lines and one double line, with a unique exception (\(n=4\), \(s=t=2\)). The author conjectures that \(X\) has bipolynomial Hilbert function if it is a general union of \(s\geq 1\) lines and one \(m\)-ple \(c\)-dimensional linear space when \(n\geq c+2\geq 3\), with the only exception \(n=c+3\), \(t=m\) and \(2\leq s \leq t\). good postulation; specialization; degeneration; double line; double point; generic union of lines; sundial; residual scheme; Hartshorne-Hirschowitz theorem; Castelnuovo's inequality; Hilbert function Projective techniques in algebraic geometry, Configurations and arrangements of linear subspaces, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Divisors, linear systems, invertible sheaves Postulation of generic lines and one double line in \(\mathbb {P}^n\) in view of generic lines and one multiple linear 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 The authors describe the graded Betti numbers occurring in the minimal graded free resolution of the homogeneous vanishing ideal of a reduced 0-dimensional scheme \(X \subseteq \mathbb{P}^3\) which is contained in a smooth quadric surface \(Q \subseteq \mathbb{P}^3\). This resolution is constructed from the knowledge of a particular type of locally free resolution of the relative ideal sheaf \(\overline {{\mathcal I}}_X\) of \(X\) in \({\mathcal O}_Q\), namely a resolution built from summands of the form \({\mathcal O}_Q (a,b)\) with bidegrees \((a,b)\) in a certain ``good'' rectangle. They also prove that any 0-dimensional subscheme of \(Q\) is determinantal, and that the Hilbert function of a generic set of points on \(Q\) determines its graded Betti numbers which are exactly given by the fourth difference function of the Hilbert function of \(X\). The paper is part of a programme outlined by the authors previously [see \textit{S. Giuffrida}, \textit{R. Maggioni} and \textit{A. Ragusa} in: zero-dimensional schemes, Proc. Conf., Ravello 1992, 191-204 (1994; Zbl 0826.14029)]. subscheme of quadric surface; determinantal zero-dimensional subscheme; graded Betti numbers; ideal of a reduced 0-dimensional scheme; Hilbert function S. Giuffrida, R. Maggioni, and A. Ragusa, Resolutions of generic points lying on a smooth quadric,Manuscripta Math. 91 (1996), 421--444. Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Determinantal varieties, Global theory and resolution of singularities (algebro-geometric aspects), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Syzygies, resolutions, complexes and commutative rings Resolutions of generic points lying on a smooth quadric | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Der Verf. legt in dieser Abhandlung die Methoden dar, auf denen sich eine arithmetische Theorie der algebraischen Funktionen und der \textit{Abel}schen Integrale in voller Strenge und Allgemeinheit aufbauen läßt, und welche inzwischen in einem in Gemeinschaft mit dem Referenten verfaßten Lehrbuche ihre ausführliche Entwicklung und ihre Weiterführung gefunden haben.
Nachdem die zu einer irreduktiblen algebraischen Gleichung \(f(x, y)=0\) gehörige \textit{Riemann}sche Fläche vermöge der Reihenentwicklungen der Funktion \(y\) konstruiert worden ist, erfolgt zunächst die bekannte Definition der Ordnungszahl, welche einer beliebigen Funktion des Körpers \(K(x,y)\) in einem gegebenen Punkte \({\mathfrak P}\) der \textit{Riemann}schen Fläche zukommt. Jedem Punkte \({\mathfrak P}\) wird ein gleich bezeichneter Primdivisor in der Weise zugeordnet, daß\ eine Funktion des Körpers durch die Potenz \({\mathfrak B}^\lambda\) teilbar genannt wird, wenn ihre Ordnungszahl in \({\mathfrak B}\) mindestens gleich \(\lambda\) ist; hierbei kann \(\lambda\) positiv, Null oder negativ sein. Eine Anzahl derartiger Potenzen von Primdivisoren kann zu einem algebraischen Divisor
\[
{\mathfrak Q = \mathfrak P_1^{\lambda_1} \mathfrak P_2^{\lambda_2} \dots \mathfrak P}_h^{\lambda_h}
\]
der Ordnung \(\lambda_1 + \lambda_2 + \cdots + \lambda_h\) vereinigt werden. Jeder Funktion \(\xi\) des Körpers gehört alsdann ein bestimmter algebraischer Divisor der Ordnung Null zu. Umgekehrt aber braucht einem Divisor der Ordnung Null nicht ohne weiteres eine Funktion des Körpers zu entsprechen, vielmehr bilden also solche Divisoren im allgemeinen nur einen Teilbereich des Gebietes aller Divisoren der Ordnung Null, der als die Hauptklasse der Divisoren bezeichnet wird. Zwei Divisoren heißen äquivalent, wenn ihr Quotient der Hauptklasse angehört: äquivalente Divisoren haben gleiche Ordnung und lassen sich in eine Divisorenklasse vereinigen. Nach dieser Einteilung der Divisoren in Klassen besteht die Grundfrage der Theorie darin, zu entscheiden, ob in einer gegebenen Klasse \textit{ganze} Divisoren vorkommen, und wenn sie vorhanden sind, sie vollständig aufzustellen. Alle die hier eingeführten Begriffsbestimmungen erweisen sich als invariant gegenüber birationaler Transformation; wenn man also die gerade zu Grunde gelegte \textit{Riemann}sche Fläche umkehrbar eindeutig auf eine andere abbildet, so bleiben die Definitionen der Ordnungszahlen und der Äquivalenz unverändert bestehen.
Um nun die oben gestellte Frage zu entscheiden, wird zunächst eine beliebige, aber bestimmte Variable \(x\) des Körpers vom Grade \(n\) und die zugehörige \(n\)-blättrige \textit{Riemann}sche Fläche \(\Re_x\) der Untersuchung zu Grunde gelegt und sodann das folgende Problem behandelt, das zunächst noch nicht invarianter Natur ist, dafür aber stets ein positives Resultat ergibt: Es sollen alle Funktionen des Körpers \(K(x, y)\) gefunden werden, welche in den Punkten \({\mathfrak P_1, \mathfrak P_2,} \dots, {\mathfrak P}_h\) mindestens die Ordnungszahlen \(\lambda_1, \lambda_2, \dots, \lambda_h\) haben und sonst für \textit{alle im Endlichen liegenden Punkte} von \(\Re_x\) regulär sind, während sie für \(x=\infty\) beliebiges Verhalten zeigen dürfen. Die Gesamtheit dieser Funktionen bildet ein zu dem Divisor \({\mathfrak Q} = {\mathfrak P_1^{\lambda_1} \mathfrak P_2^{\lambda_2} \cdots \mathfrak P}_h^{\lambda_h}\) gehöriges Ideal \(J(\mathfrak Q)\). Jedes Ideal besitzt ein Fundamentalsystem \(\eta_1, \eta_2, \dots, \eta_n\), derart, daß\ jede Funktion \(\eta\) des Ideals auf eine einzige Weise in die Form gesetzt werden kann:
\[
\eta = u_1 \eta_1 + u_2 \eta_2 + \cdots + u_n \eta_n,
\]
worin \(u_1, u_2, \dots, u_n\) ganze Funktionen von \(x\) sind. Bezeichnet man die konjugierten Reihenentwicklungen von \(\eta_i\) mit \(\eta_i', \eta_i^{\prime\prime}, \dots, \eta_i^{(n)}\) und bildet die Determinante:
\[
D=| \eta_i^{(k)} | \qquad (i,k = 1,2, \dots, n)
\]
so kann das Fundamentalsystem stets so bestimmt werden, daß\ für eine beliebige endliche Stelle \(x= a\) von \(\Re_x\) die Kolonnenteiler von \(D\) (das sind die größten gemeinsamen Teiler der \(n\) konjugierten Reihenentwicklungen von \(\eta_i\)) mit den Elementarteilern von \(D\) übereinstimmen. Es kann sogar so gewählt werden, daß\ modulo einer beliebig hohen Potenz von \(x-a\) das quadratische System \((\eta_i^{(k)})\) in so viele unzerlegbare Partialsysteme zerfällt, als verschiedene Primdivisoren in \(x-a\) enthalten sind. Sondert man alsdann aus dem Ideale \(J(\mathfrak Q)\) diejenigen Funktionen aus, welche sich auch im Unendlichen regulär verhalten, so erhält man die Gesamtheit der durch \(\mathfrak Q\) teilbaren Funktionen; dieselben bilden eine Funktionenschar:
\[
c_1 \xi_1 + c_2 \xi_2 + \cdots + c_{\mu} \xi_{\mu}.
\]
Die Zahl \(\mu\) der in Schar enthaltenen linearer unabhängigen Funktionen heißt die Dimension der Schar und ist mit der Dimension \(\left\{ \frac 1Q \right\}\) der in der Divisorenklasse \(\frac 1Q\) enthaltenen linear unabhängigen ganzen Divisoren identisch.
Bildet man zu einem Systeme \((\eta_i^{(k)})\) das reziproke, so gehört dasselbe zu einem zweiten Ideale \(J(\overline{\mathfrak Q})\), und die beiden Divisoren \(\mathfrak Q\) und \(\overline{\mathfrak Q}\) stehen in der Beziehung:
\[
{\mathfrak Q} \overline{\mathfrak Q} = \frac{1}{{\mathfrak Z}_x}, \quad {\mathfrak Z}_x = \varPi {\mathfrak P}^{a-1},
\]
wo \({\mathfrak Z}_x\) den Verzweigungsdivisor der \textit{Riemann}schen Fläche \(\Re_x\) bedeutet und das obige Produkt also über alle Punkte \(\mathfrak P\) von \(\Re_x\) in der Weise zu erstrecken ist, daß\ \(\alpha -1\) die Verzweigungsordnung des zugehörigen Punktes \(\mathfrak P\) bedeutet. Ist \(\mathfrak n_x\) der Nenner von \(x\), so ist einem \textit{Abel}schen Differentiale \(d\omega = \zeta dx\) des Körpers ein bestimmter Divisor
\[
{\mathfrak W}_\omega = \zeta \cdot \frac{{\mathfrak Z}_x}{{\mathfrak n}_x^2}
\]
zugeordnet; alle diese sogenannten Differentialteiler gehören einer und derselben Klasse \(W\), der Differentialklasse, an deren Ordnung \(2p-2\) ist, wenn \(p\) das Geschlecht des Körpers ist. Divisorenklassen \(Q\) und \(Q'\), deren Produkt die Differentialklasse \(W\) ist, heißen Ergänzungsklassen; sind ihre Ordnungen \(q\) und \(q'\), so erscheint alsdann der \textit{Riemann-Roch}sche Satz in der Gestalt
\[
\{ Q\} - \frac q2 = \{ Q' \} - \frac{q'}{2} \qquad (q+q' = 2p-2).
\]
Aus ihm lassen sich sodann die Aufstellung der Integrale der drei Gattungen und die zugehörigen Dimensionsbestimmungen mühelos ableiten. Arithmetic theory of algebraic functions; Dedekind-Weber theory; algebraic function fields; linear systems; divisors; Abelian differentials Algebraic functions and function fields in algebraic geometry On the theory of algebraic functions of one variable and \textit{Abel}ian integrals | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 examine the cohomology and representation theory of a family of finite supergroup schemes of the form \((\mathbb{G}_a^- \times \mathbb{G}_a^-) \rtimes (\mathbb{G}_{a (r)} \times (\mathbb{Z} / p)^s)\). In particular, we show that a certain relation holds in the cohomology ring, and deduce that for finite supergroup schemes having this as a quotient, both cohomology mod nilpotents and projectivity of modules is detected on proper sub-supergroup schemes. This special case feeds into the proof of a more general detection theorem for unipotent finite supergroup schemes, in a separate work of the authors joint with Iyengar and Krause.
We also completely determine the cohomology ring in the smallest cases, namely \((\mathbb{G}_a^- \times \mathbb{G}_a^-) \rtimes \mathbb{G}_{a (1)}\) and \((\mathbb{G}_a^- \times \mathbb{G}_a^-) \rtimes \mathbb{Z} / p\). The computation uses the local cohomology spectral sequence for group cohomology, which we describe in the context of finite supergroup schemes. cohomology; finite supergroup scheme; invariant theory; Steenrod operations; local cohomology spectral sequence Group schemes, Superalgebras, Modular representations and characters, Cohomology of groups, Hopf algebras and their applications, ``Super'' (or ``skew'') structure, Homological methods in Lie (super)algebras Representations and cohomology of a family of finite supergroup 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 present an operator-coefficient version of Sato's infinite-dimensional Grassmann manifold, and \(\tau\)-function. In this setting the classical Burchnall-Chaundy ring of commuting differential operators can be shown to determine a \(C^*\)-algebra. For this \(C^*\)-algebra topological invariants of the spectral ring become readily available, and further, the Brown-Douglas-Fillmore theory of extensions can be applied. We construct \(KK\) classes of the spectral curve of the ring and, motivated by the fact that all isospectral Burchnall-Chaundy rings make up the Jacobian of the curve, we compare the (degree-1) \(K\)-homology of the curve with that of its Jacobian. We show how the Burchnall-Chaundy \(C^*\)-algebra extension by the compact operators provides a family of operator \(\tau\)-functions. \(C^*\)-algebras; Hilbert module; Calkin algebra; Baker function; \(K\)-homology; \(\tau\)-function; spectral curve; Jacobian torus; infinite-dimensional Grassmann manifold; Burchnall-Chaundy rings Dupré, M.J.; Glazebrook, J.F.; Previato, E., Differential algebras with Banach-algebra coefficients I: from C^{*}-algebras to the K-theory of the spectral curve, Complex anal. oper. theory, 7, 4, 739, (2013) Ext and \(K\)-homology, Kasparov theory (\(KK\)-theory), Jacobians, Prym varieties, Linear operators in \(C^*\)- or von Neumann algebras Differential algebras with Banach-algebra coefficients. I: From \(C^*\)-algebras to the \(K\)-theory of the spectral curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author proves a natural extension of the polynomial Positivstellensatz to algebras of Borel measurable functions defined on the Euclidean space. A more general statement on a non-commutative \(C^*\)-algebra can be stated in a similar way. positivstellensatz; Borel measurable function; Euclidean space; spectral theorem; selfadjoint operator Linear operator methods in interpolation, moment and extension problems, Semialgebraic sets and related spaces, Nonconvex programming, global optimization A Striktpositivstellensatz for measurable 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 For any arbitrary algebraic curve, we define an infinite sequence of invariants (meromorphic forms \(W^{(g)}_k(p_1,\dots,p_k)\) on the curve where \(k,g \in {\mathbb N}\) and \(p_1,\dots,p_k\) are points of the curve -- \textit{the reviewer}). We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the curve becomes singular. In addition, we find that they can be used to defines a formal series, which satisfies formally a Hirota equation, and we thus obtain a new way of onstructing a \(\tau\)-function attached to an algebraic curve.
These invariants are constructed in order to coincide with the topological expansion of a matrix model integral, when the algebraic curve is chosen as the large \(N\) limit of the matrix model's spectral curve. Surprisingly, we find that the same invariants also give the topological expansion of other models, in paerticular the matrix model with an external field, and the so-called double scaling limit of matrix models, i.e., the \((p,q)\) minimal models of conformal field theory.
As an example to illustrate the efficiency of our method, we apply it to the Kontsevich integral, and give a new and extremely easy proof that the Kontsevich integral depends only on odd times, and that it is a KdV \(\tau\)-function. algebraic curve; spectral curve; matrix model; topological expansion of a matrix integral; tau-function; Hirota equation B. Eynard and N. Orantin, \textit{Invariants of algebraic curves and topological expansion}, \textit{Commun. Num. Theor. Phys.}\textbf{1} (2007) 347 [math-ph/0702045] [INSPIRE]. Relationships between algebraic curves and physics, Groups and algebras in quantum theory and relations with integrable systems, String and superstring theories in gravitational theory, KdV equations (Korteweg-de Vries equations), Quantization in field theory; cohomological methods Invariants of algebraic curves and topological expansion | 0 |
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