text stringlengths 571 40.6k | label int64 0 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 We define and study odd analogues of classical geometric and combinatorial objects associated to permutations, namely odd Schubert varieties, odd diagrams, and odd inversion sets. We show that there is a bijection between odd inversion sets of permutations and acyclic orientations of the Turán graph, that the dimension of the odd Schubert variety associated to a permutation is the odd length of the permutation, and give several necessary conditions for a subset of \([ n ] \times [ n ]\) to be the odd diagram of a permutation. We also study the sign-twisted generating function of the odd length over descent classes of the symmetric groups. permutation; generating function; descent class; diagram; odd length; Schubert variety Permutations, words, matrices, Exact enumeration problems, generating functions, Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds, Symmetric groups Odd length: odd diagrams and descent classes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 under review is to generalize the author's early work on rank 2 Drinfeld modular forms to the case of of higher rank. The theory of Drinfeld modular forms of rank 2 was introduced by \textit{D. Goss} in his Harvard PhD Thesis (and subsequent papers [Bull. Am. Math. Soc., New Ser. 2, 391--415 (1980; Zbl 0433.14017); J. Reine Angew. Math. 317, 16--39 (1980; Zbl 0422.10021); Compos. Math. 41, 3--38 (1980; Zbl 0422.10020)]) and further studied by the author of this paper during the 80's. Drinfeld modular forms of higher ranks began to draw experts' attention only recently. In this paper, the author gives a detailed study of the fundamental Drinfeld modular forms of higher rank which are important references for future study. To state the results of this paper, we begin by setting up necessary notations as follows.
Let \(A = {\mathbb F}_q[T]\) be the polynomial ring over the finite field \({\mathbb F}_q\) of \(q\) elements with field of fractions \(K = {\mathbb F}_q(T).\) The completion of \(K\) at infinity is the formal Laurent series field \(K_{\infty} = {\mathbb F}_q((1/T))\) and the completed algebraic closure is denoted by \({\mathbb C}_{\infty}.\) The Drinfeld symmetric space \(\Omega^r \subset {\mathbb P}^{r-1}({\mathbb C}_\infty)\) with \(r\geq 2\) has a natural structure as rigid analytic space and \(\Gamma := \mathrm{GL}(r, A),\) acts on it, and the quotient \(\Gamma\backslash \Omega^r\) parameterizes classes of \(A\)-lattices \(\Lambda\) of rank \(r\) in \({\mathbb C}_\infty.\) As in the case of rank two, there is a one to one correspondence, \(\Lambda \rightsquigarrow \phi^{\Lambda}\), between classes up to scaling of \({\mathbf A}\)-lattices of rank \(r\) and isomorphism classes of Drinfeld \({\mathbf A}\)-modules of rank \(r\). An \({\mathbf A}\)-basis \(\{\omega_{1}, \ldots, \omega_{r}\}\) of \(\Lambda\) determines a point \({\boldsymbol \omega} = (\omega_{1}: \cdots: \omega_{r})\in \Omega^{r}.\) Assuming \(\omega_{r} = 1\) in the homogeneous coordinate for \({\boldsymbol \omega}\), a rank \(r\) Drinfeld module \(\phi^{{\boldsymbol \omega}}\) corresponding to \({\boldsymbol \omega}\) is determined by a polynomial operator
\[
\phi_{T}^{{\boldsymbol \omega}}(X) = T X + g_{1} X^{q} + \cdots + g_{r-1} X^{q^{r-1}} + g_{r} X^{q^{r}},
\]
where the coefficients \(g_{i} = g_{i}({\boldsymbol \omega})\) are Drinfeld modular forms of weight \(q^{i}-1\) and type \(0\). In particular, they are holomorphic on \(\Omega^{r}\) in the rigid analytic sense. Furthermore, the discriminant \(\Delta := g_{r}\) is nowhere zero on \(\Omega^{r}. \)
The main results of this paper is to describe the behavior of \(|g_{i}({\boldsymbol \omega})|\) on \(\Omega^{r}.\) It suffices to study \(|g_{i}({\boldsymbol \omega})|\) on an open admissible subspace \({\mathcal F}\subset \Omega^{r}\) which plays the role of a fundamental domain for \(\Gamma\) on \( \Omega^{r}.\) The key objects over \(g_{i}\) and \(\Delta\) are the division functions \(\mu_{i}\, (1\leq i \leq r)\) which are defined as an \({\mathbb F}_{q}\)-basis of the \(T\)-torsion of the generic Drinfeld module \(\phi^{{\boldsymbol \omega}} ({\boldsymbol \omega}\in \Omega^{r}).\) They are modular forms of negative weight \(-1\) for the congruence subgroup \(\Gamma(T)\) of \(\Gamma.\)
The absolute value \(\log_q |\mu_i ({\boldsymbol \omega})|\) and similarly \(\log_{q} |g_{i}({\boldsymbol \omega})| ,\) for \({\boldsymbol \omega}\in {\mathcal F} ,\) can be regarded as functions on a standard Weyl chamber \(W\) in the realization \(\mathcal{BT}({\mathbb R})\) of the Bruhat-Tits building \(\mathcal{ BT}\) of \(\mathrm{PGL}(r, K_{\infty}).\) The increments of \(\log_q |\mu_i ({\boldsymbol \omega})|\) are given in terms of certain combinatorial numbers \(v_{{\mathbf k}, i}^{\ell}\) when \({\mathbf k}\in W({\mathbb Z})\) is replaced by a neighbor vertex \({\mathbf k}' = {\mathbf k} + {\mathbf k}_{\ell}\) (Proposition~4.10). As a result, the increments of \(\log_{q}|\Delta({\boldsymbol \omega})|\) and of \(\log_{q} |g_i({\boldsymbol \omega})|\) in terms of these numbers \(v_{{\mathbf k}, i}^{\ell}\) can be deduced (Theorem~4.13 and its Corollary~4.16) . From these formulas, one can also conclude that the function \(\Delta\) is strictly monotonically decreasing on \(W({\mathbb Q})\) (Corollary~4.15). An explicit formula for evaluating the increments of \(\log_{q}\,|\Delta({\boldsymbol \omega})|\) is also obtained (Theorem~5.5).
The zero loci \(V(g_i)\cap {\mathcal F}\) are also studied. The author shows that \(V(g_i)\cap {\mathcal F}\) is contained in \({\mathcal F}_{r-i}\subset {\mathcal F},\) called the \((r-i)\)-th wall of \({\mathcal F}\) (Proposition~4.4). Furthermore, the author considers the set \({\mathcal F}_o = \{{\boldsymbol \omega} = ( \omega_1, \ldots, \omega_{r-1}, 1)\in \Omega^r \mid |\omega_1| = \cdots = |\omega_{r-1}| = 1\} \subset {\mathcal F} \) and studies the behavior of \(g_1, \ldots, g_{r-1}, g_r = \Delta\) on \({\mathcal F}_o .\) In particular, the reduction of the vanishing locus of \(g_i\) on \({\mathcal F}_o\) under the canonical reduction map \(\mathrm{red}: {\mathcal F}_o \to \Omega^r(\overline{{\mathbb F}_q})\) is shown to be equal to the vanishing locus of the coefficient \(\bar{\alpha}_i\) of the reduction of the Drinfeld exponential function \(e_{\Lambda_{{\boldsymbol \omega}}}\) associated to the lattice \(L_{{\boldsymbol \omega}}\) (Theorem~6.2).
In the last section of this paper, the author provides more details in the case \(r = 3.\) Besides tables with values of some of the functions treated in previous sections, the author also provides a brief study of \(g_{1}\) at the second wall \({\mathcal F}_{2}\) and of \(g_{2}\) at the first wall \({\mathcal F}_{1}\) of \({\mathcal F}.\) Drinfeld modular forms; Drinfeld discriminant function; Bruhat-Tits building Modular forms associated to Drinfel'd modules, Drinfel'd modules; higher-dimensional motives, etc., Rigid analytic geometry On Drinfeld modular forms of higher rank. I | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The \(SU(2)\)-invariant selfdual Einstein metric on a four-dimensional Riemannian manifold is given by \(g=F(d\mu^2+\sigma_1^2/w_1^2 +\sigma_2^2/w_2^2+\sigma_3^2/ w_3^2)\). The local classification of metrics of this type was given by \textit{N. J. Hitchin} [J. Differ. Geom. 42, 30-112 (1995; Zbl 0861.53049)]. However, the final form of the metric coefficients related to the Painlevé VI equation in Hitchin's description was rather complicated.
In this paper, the authors give simpler expressions for the same metric by applying the formula for the tau function of the algebraic and geometric solutions of the Schlesinger system. self-dual metric; Einstein metrics; Schlesinger system; theta function; Painlevé VI Babich, MV; Korotkin, DA, Self-dual \(SU(2)\) invariant Einstein metrics and modular dependence of theta-functions, Lett. Math. Phys., 46, 323-337, (1998) Special Riemannian manifolds (Einstein, Sasakian, etc.), Theta functions and curves; Schottky problem, Explicit solutions, first integrals of ordinary differential equations, Approximation procedures, weak fields in general relativity and gravitational theory Self-dual \(SU(2)\)-invariant Einstein metrics and modular dependence of theta functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Consider a finite set of $s\geq 1$ very general lines $\mathcal{L} \subset \mathbb{P}^{3}_{\Bbbk}$, where $\Bbbk$ is an algebraically closed field of characteristic zero. In the paper under review, the authors study the Waldschmidt constants of radical ideals associated with $\mathcal{L}$. Let us recall that if $I \subset R = \Bbbk[x_{0}, \dots, x_{N}]$ is a homogeneous ideal, then the $m$-th symbolic power $I^{(m)}$ is defined as \[ I^{(m)} = R \cap \bigcap_{P \in \text{Ass}(I)}I^{m}R_{P}, \] where the intersection is taken in the ring of fractions of $R$. We define also the initial degree of $I$, namely \[ \alpha(I) = \min \{t: I_{t} \neq 0\}, \] where $I_{t}$ denotes the degree $t$ part of $I$. Finally, the Waldschmidt constant of $I$ is defined as \[ \widehat{\alpha}(I) = \text{lim}_{m \rightarrow \infty} \frac{\alpha(I^{(m)})}{m}. \] The main results of the paper can be formulated as follows. \par Theorem A. Let $\mathcal{L}_{s}\subset \mathbb{P}^{3}_{\Bbbk}$ be a finite set of $s\geq 1$ very general lines and denote by $I_{s}$ the radical ideal associated with $\mathcal{L}_{s}$, then \[ \widehat{\alpha}(I_{s})\geq \lfloor \sqrt{2s-1} \rfloor. \] Theorem B. Let $\mathcal{L}_{s}\subset \mathbb{P}^{3}_{\Bbbk}$ be a finite set of $s$ very general lines such that $s\notin \{4,7,10\}$, and denote by $I_{s}$ the radical ideal associated with $\mathcal{L}_{s}$, then \[ \widehat{\alpha}(I_{s})\geq \lfloor \sqrt{2.5s} \rfloor. \] asymptotic Hilbert function; Chudnovsky conjecture; containment problem; symbolic powers; Waldschmidt constants Configurations and arrangements of linear subspaces, Polynomial rings and ideals; rings of integer-valued polynomials, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Divisors, linear systems, invertible sheaves Lower bounds for Waldschmidt constants of generic lines in \({\mathbb {P}}^3\) and a Chudnovsky-type theorem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The notion of universally saturated morphisms between saturated log schemes was introduced by \textit{K. Kato} [in: Algebraic analysis, geometry, and number theory: proceedings of the JAMI inaugural conference, held at Baltimore, MD, USA, May 16-19, 1988. Baltimore: Johns Hopkins University Press. 191--224 (1989; Zbl 0776.14004)]. In this paper, we study universally saturated morphisms systematically by introducing the notion of saturated morphisms between integral log schemes as a relative analogue of saturated log structures. We eventually show that a morphism of saturated log schemes is universally saturated if and only if it is saturated. We prove some fundamental properties and characterizations of universally saturated morphisms via this interpretation. logarithmic structure; logarithmic scheme; saturated morphism Logarithmic algebraic geometry, log schemes, Schemes and morphisms Saturated morphisms of logarithmic schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A new method of constructing elliptic finite-gap solutions of the stationary Korteweg-de Vries (KdV) hierarchy, based on a theorem due to Picard, is illustrated in the concrete case of the Lamé-Ince potentials \(- s(s+ 1) {\mathcal P}(z),\quad s\in \mathbb{N}\) (\({\mathcal P}(.)\) the elliptic Weierstrass function). Analogous results are derived in the context of the stationary modified Korteweg-de Vries (mKdV) hierarchy for the first time. elliptic finite-gap solutions; Korteweg-de Vries and modified Korteweg-de Vries hierarchy; Lamé-Ince potentials; elliptic Weierstrass function Gesztesy, F.; Weikard, R., Lamé potentials and the stationary (m)KdV hierarchy, Math. Nachr., 176, 73-91, (1995) 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.), Theta functions and curves; Schottky problem Lamé potentials and the stationary (m)KdV hierarchy | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Assuming the base field K is an algebraically closed field of characteristic p \(>0\), the zeta function associated with a regular irreducible prehomogeneous vector space can be defined. These zeta functions satisfy certain functional equations. We are interested in the constants of these equations. N. Kawanaka pointed out that these constants are related to the characters of the generalized Gelfand-Graev representations of finite groups of Lie type. In this paper, we calculate the constants of the five most important cases: \((G'\times GL(m),\quad \rho '\otimes \Lambda_ 1,\quad V(m)\times V(m)),\) \((GL(2m),\quad \Lambda_ 2,\quad V(m(2m-1))),\) \((GL(n),\quad 2\Lambda_ 1,\quad V(n(n+1)/2)),\) \((Sp(n)\times GL(2m),\quad \Lambda_ 1\times \Lambda_ 1,\quad V(2n)\times V(2m)),\) \((SO(n)\times GL(m),\quad \Lambda_ 1\times \Lambda_ 1,\quad V(n)\times V(m)).\) It turns out that these constants are all Gaussian sums or the product of Gaussian sums. constants of functional equation of zeta function; characteristic p; zeta function associated with a regular irreducible prehomogeneous vector space; Gaussian sums Homogeneous spaces and generalizations, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry, Trigonometric and exponential sums (general theory) On zeta-functions associated with prehomogeneous vector spaces and Gaussian sums | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(G_2\) be the smallest exceptional simple linear algebraic group over an algebraic closed field which is not isomorphic to a classical group. Using the middle convolution functor \(MC_{\chi}\) introduced by N. Katz, the authors prove the existence of rigid local systems whose monodromy is dense in \(G_2\). They derive the existence of motives for motivated cycles which have a motivic Galois group of type \(G_2\). Granting Grothendieck's standard conjectures, the existence of motives with motivic Galois group of type \(G_2\) can be deduced, giving a partial answer to a question of Serre. rigid local systems; motivic Galois group Dettweiler, M.; Reiter, S., Rigid local systems and motives of type \(G\)\_{}\{2\}With an appendix by Michale Dettweiler and Nicholas M. Katz., Compos. Math., 146, 929-963, (2010) Arithmetic problems in algebraic geometry; Diophantine geometry, Algebraic cycles, Étale and other Grothendieck topologies and (co)homologies Rigid local systems and motives of type \(G_{2}\) | 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 We study the Hilbert function of certain projective monomial curves. We determine which of our curves are Cohen-Macaulay, and find the Cohen-Macaulay type of those that are Cohen-Macaulay. Cohen-Macaulay curves; Hilbert function; monomial curves Patil, D.P., Roberts, L.G.: Hilbert functions of monomial curves. J. Pure Appl. Algebra 183(1--3), 275--292 (2003) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Special algebraic curves and curves of low genus, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Hilbert functions of monomial curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(m\) be a non-square positive integer and \(K\) the Galois extension generated by \(\sqrt{-1}\) and \(m^{1/4}\) over the rational number field \(\mathbb{Q}\). Then the Galois group of \(K\) over \(\mathbb{Q}\) has one and only one two-dimensional irreducible complex representation. Let \(\theta(\tau,K)\) be the cusp form of weight one obtained from this representation by Weil-Langlands theory. In the present paper, we study expressions of this form and the number-theoretic properties of its Fourier coefficients. Especially we show identities between cusp forms, a special case of quartic reciprocity and ``higher reciprocity law'' of the polynomial \(f(x)=x^4-m\). Further let \(\vartheta(\tau,E)\) be the cusp form of weight two which is the inverse Mellin transform of the \(L\)-function of the elliptic curve \(E\) defined by the equation \(y^2=x^3+4mx\). Then we also offer a congruence mod 4 between \(\theta(\tau,K)\) and \(\vartheta(\tau,E)\) deduced from the quartic residuacity of the integer \(m\). quartic residue; two dimensional irreducible complex representation; cusp form of weight one; Weil-Langlands theory; Fourier coefficients; identities between cusp forms; quartic reciprocity; higher reciprocity law; cusp form of weight two; inverse Mellin transform; L-function of elliptic curve; congruence mod 4 N. Ishii, ''Cusp forms of weight one, quadratic reciprocity and elliptic curves,''Nagoya Math. J.,98, 117--137 (1985). Holomorphic modular forms of integral weight, Congruences for modular and \(p\)-adic modular forms, Langlands-Weil conjectures, nonabelian class field theory, Elliptic curves Cusp forms of weight one, quartic reciprocity and elliptic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{G. Faltings} and \textit{C.-L. Chai} [``Degeneration of abelian varieties'' (1990; Zbl 0744.14031)] proved a certain linear combination of the determinant bundle and relative canonical bundle of a symmetric, relatively ample bundle over an abelian variety is torsion. The present author obtains more exact information about its order, giving a bound which is optimal in many cases. abelian scheme; determinant bundle; Fourier-Mukai transform Polishchuk, A., Determinant bundles for abelian schemes, Compositio Math., 121, 221-245, (2000) Arithmetic ground fields for abelian varieties, Algebraic moduli of abelian varieties, classification, Isogeny, Theta functions and abelian varieties Determinant bundles for Abelian schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The present work is the author's habilitation, concerned with various aspects of the structure of zero-dimensional schemes \({\mathbf X}\) in the projective space \({\mathbb{P}}^d_K\) and the homogeneous coordinate rings of them, \(R = K[X_0, \ldots, X_d]/I_{\mathbf X}\). It is a summary of the author's research over the last eight years published in several mathematical journals, respectively available as preprints. The main technical tools of his investigations are Castelnuovo theory, Hilbert functions, the canonical module of the coordinate ring of \({\mathbf X}\), the Kähler module of differentials, and the canonical ideal.
In the second part the author discusses several conditions for uniformity, among them generic position, uniform position, cohomological uniformity, Cayley-Bacharach schemes and their hierarchy. The knowledge of the Hilbert function of \({\mathbf X}\), its growth behaviour, the relation to the Hilbert function of the canonical module provides important results of \({\mathbf X}\), in particular with respect to the above mentioned uniformity. The author's intention is an interplay between geometrical properties of \({\mathbf X}\) and the algebraic behaviour of \(R\). In particular there is an explicit description of the canonical ideal of \(R\). Furthermore there is also an description of the Kähler module of differentials in the case of a non-reduced \({\mathbf X}\).
Besides of these results the author's applications to several branches of mathematics described in the third part of this habilitation are interesting and original. The subjects he is interested in are the following: level schemes, hyperplane sections of curves, free resolutions, syzygy module of the canonical module, maximal Cayley Bacharach schemes, liaison, zero-dimensional subschemes of the projective plane, determinantal zero-dimensional schemes, coding theory, computer algebra. One of the main themes is an algebraic characterization of the Cayley-Bacharach property [see e.g. \textit{D. Eisenbud, M. Green} and \textit{J. Harris}, Bull. Am. Math. Soc., New Ser. 33, No. 3, 295-324 (1996; Zbl 0871.14024) for a survey]. The author characterizes Gorenstein schemes by the symmetry of the Hilbert function and the Cayley-Bacharach property. Moreover he discusses the question when a subscheme of a Caley-Bacharach scheme is again a Caley-Bacherach scheme. There is a connection of the Cayley-Bacharach property to the property of \(i\)-uniform position. This again is closely related to the Castelnuovo function introduced as the difference function of the Hilbert function of \({\mathbf X}\). One of the main results is a certain estimate shown for the case of the author's cohomological uniformity. A large part of the third section is devoted to the Castelnuovo theory, in particular its behaviour in prime characteristic and the relation to the Cayley-Bacherach property.
As applications the author is interested in the construction of \({\mathbf X}\) with given properties. He describes several methods based on liaison, determinantal schemes, etc. -- Another highlight is the author's study of reduced schemes \({\mathbf X}\) over a field of \(q = p^e\), \(p= \text{prime}\) characteristic of \(K,\) consisting only of \({\mathbb{F}}_q\)-rational points. In this situation there are strong restrictions on the Hilbert function. Some of these \({\mathbf X}\) define linear codes. It is shown that the invariants and geometric properties of \({\mathbf X}\) are closely related to length, dimension and minimal distance of the corresponding codes. In a final section the author summarizes the computer algebra methods for the construction of explicit samples of zero-dimensional schemes. Hilbert function; module of differentials; uniform position; generic position; liaison; coding theory; canonical module; Cayley Bacharach schemes; zero-dimensional subschemes; Castelnuovo function Kreuzer, M.: Beiträge zur theorie nulldimensionalen unterschemata projektiver räume. Regensburger math. Schr. 26 (1998) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Linkage, Linkage, complete intersections and determinantal ideals Contributions to the theory of zero-dimensional subschemes of projective spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article is devoted to the description of an algorithm for subdivision of a plane into nonintersecting domains by a finite set of simple Jordan arcs. Each resultant domain is defined via a set of its boundary arcs and its indicator (a bounded or an unbounded domain) which determines the characteristic domain function. In addition, an algorithm is obtained for implementation of a regularized set operations on domains without cutoffs. It is based on subdividing a plane by common boundaries on subdomains and constructing on this base the result of the operation. For computing the intersection points of the boundary arcs, the Newton method is applied whose square convergence is proven for the case of convex and monotone curves. geometric intersection problem; curve intersection; algorithm; subdivision; Jordan arcs; Newton method; convergence Numerical aspects of computer graphics, image analysis, and computational geometry, Polyhedra and polytopes; regular figures, division of spaces, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Subdivision of a plane and set operations on domains | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In his fundamental paper [``Techniques de construction et théorèmes d'existence en géométrie algébrique. IV: Les schemes de
Hilbert'', Sémin. Bourbaki, exp. 221 (1961; Zbl 0236.14003)], \textit{A. Grothendieck} introduced the so called Hilbert scheme, which parametrizes all projective subschemes of the projective space with fixed Hilbert polynomial. Problems naturally arising in the study of the Hilbert scheme are irreducibility and number of components, dimension and smoothness. For instance, one knows that if \(X\subset \mathbb{P}^r\) is a local complete intersection projective subscheme and \(h^1(X,\mathcal N_{X,\mathbb{P}^r})=0\) (\(\mathcal N_{X,\mathbb{P}^r}=\) normal bundle of \(X\) in \(\mathbb{P}^r\)), then \(X\) is unobstructed, i.e. the corresponding point \([X]\) in the Hilbert scheme is smooth, and in such case the local dimension at \([X]\) is \(h^0(X,\mathcal N_{X,\mathbb{P}^r})\). But, in general, a necessary and sufficient condition for a subscheme to be unobstructed is not known [see also \textit{D. Mumford}, Am. J. Math. 84, 642--648 (1962; Zbl 0114.13106)] and \textit{E. Sernesi} [``Topics on families of projective schemes'', Queen's Pap. Pure Appl. Math. 73 (1986)].
Continuing previous works by \textit{J. O. Kleppe} [``The Hilbert-flag scheme, its properties and its connection with the Hilbert scheme. Applications to curves in \(3-\)space'', Preprint (part of thesis), Univ. of Oslo, March (1981)], \textit{G. Bolondi} [Arch. Math. 53, No. 3, 300--305 (1989; Zbl 0658.14005)], and \textit{M. Martin-Deschamps} and \textit{D. Perrin} [``Sur la classification des courbes gauches'', Astérisque 184--185 (1990; Zbl 0717.14017)], in the paper under review the author exhibits sufficient conditions and necessary conditions for unobstructedness of space curves \(C\subset \mathbb{P}^3\) which satisfy \(_{0}{\text{Ext}}^2_R(M,M)=0\) (e.g. of diameter\((M)\leq 2\)), and computes the dimension of the Hilbert scheme \(H(d,g)\) at \([C]\) under the sufficient conditions. Here \(C\subset \mathbb{P}^3\) denotes an equidimensional, locally Cohen-Macaulay subscheme of dimension one, \(d\) and \(g\) the degree and the arithmetic genus of \(C\subset \mathbb{P}^3\), \(M=\bigoplus_{v}H^1(\mathbb{P}^3, \mathcal I_C(v))\) denotes the Hartshorne-Rao module of \(C\), \(R=k[x_0,x_1,x_2,x_3]\) the polynomial ring over an algebraically closed field \(k\) of characteristic zero, and diameter\((M):=\max\{v\,| \,H^1(\mathbb{P}^3, \mathcal I_C(v))\neq 0\}-\min\{v\,| \,H^1(\mathbb{P}^3, \mathcal I_C(v))\neq 0\}+1\) (when \(H^1(\mathbb{P}^3, \mathcal I_C(v))=0\) for all \(v\), i.e. when \(C\) is arithmetically Cohen-Macaulay, then by \textit{G. Ellingsrud} [Ann. Sci. Éc. Norm. Supér. (4) 8, 423--431 (1975; Zbl 0325.14002)] one already knows that \(C\) is unobstructed).
In the diameter one case, the necessary and sufficient conditions coincide, and the unobstructedness of \(C\) turns out to be equivalent to the vanishing of certain graded Betti numbers of the free minimal resolution of the ideal \(I=\bigoplus_{v}H^0(\mathbb{P}^3, \mathcal I_C(v))\subset R\) of \(C\). The author also gives a description of the number of irreducible components of \(H(d,g)\) which contain an obstructed diameter one curve, and shows that in the diameter one case every irreducible component is reduced. Hilbert scheme; space curve; Buchsbaum curve; unobstructedness; cup-product; graded Betti numbers; ghost terms; linkage; normal module; postulation Hilbert scheme Dan, A.: Non-reduced components of the Noether-Lefschetz locus. Preprint arXiv:1407.8491v2 Parametrization (Chow and Hilbert schemes), Plane and space curves, Linkage, complete intersections and determinantal ideals, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series The Hilbert scheme of space curves of small diameter | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a real coherent analytic space, \({\mathcal O}_X\) the sheaf of analytic functions, \({\mathcal O}(X)\) the ring of global analytic functions. Given an ideal \({\mathfrak a}\subseteq{\mathcal O}(X)\), the zero set of \({\mathfrak a}\) is denoted by \({\mathcal Z}({\mathfrak a})\), the vanishing ideal of \({\mathcal Z}({\mathfrak a})\) is denoted by \({\mathcal I}({\mathcal Z}({\mathfrak a}))\). Nullstellensätze describe the connections between the ideals \({\mathfrak a}\) and \({\mathcal I}({\mathcal Z}({\mathfrak a}))\): When is it true that \({\mathfrak a}={\mathcal I}({\mathcal Z}({\mathfrak a}))\)?
The authors show that, given a real coherent analytic surface \(X\), the equality \({\mathfrak a}={\mathcal I}({\mathcal Z}({\mathfrak a}))\) holds if and only if \({\mathfrak a}\) is a real ideal and is saturated (i.e., \({\mathfrak a}\) is the ideal of global sections of the ideal sheaf generated by \({\mathfrak a}\)). It remains an open question whether the result can be extended to the space \(\mathbb{R}^3\). However, it is shown that the result fails for \(\mathbb{R}^3\) if and only if there is a so-called special irreducible functions that generates a real ideal. Primary ideals and primary decompositions of saturated ideals are among the main tools of the paper. real analytic space; analytic function; zero set; vanishing ideal; Nullstellensatz; primary decomposition Broglia, F; Pieroni, F, The nullstellensatz for real coherent analytic surfaces, Rev. Mat. Iberoam., 25, 781-798, (2009) Real-analytic and semi-analytic sets, Real algebraic and real-analytic geometry, Germs of analytic sets, local parametrization, Sums of squares and representations by other particular quadratic forms The Nullstellensatz for real coherent analytic 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 \(G\) be a simple simply connected algebraic group over an algebraically closed field \(k\) in \(\mathbb C\). The coniveau spectral sequence \(E_2^{\ast,\ast'}\cong H^*(BG;H^{*'}_{{\mathbb Z}/p}) \Rightarrow H^*(BG;{\mathbb Z}/p)\) has for \(E_2\)-term the cohomology of the Zariski sheaf induced from the pre-sheaf \(H^*_{ et}(V;{\mathbb Z}/p)\) for open subsets \(V\) of \(BG\). By using results about cohomological invariants, the authors compute the coniveau spectral sequence for classifying spaces \(BG\). As an illustration, we recall an example from the introduction: There is an epimorphism from \(H^*(BG_2;H^{*'}_{{\mathbb Z}/2})\) onto
\[
\begin{multlined} H^*(BA;H^{*'}_{{\mathbb Z}/2})^{W_G(A)} \oplus (H^*(BA;H^{*'}_{{\mathbb Z}/2})^{W_G(A)}/{ Res}_{H{\mathbb Z}/2})(-1)\\ [2] \!\cong\! {\mathbb Z}/2[c_4,c_6]\otimes ({\mathbb Z}/2\{1,y\}\oplus {\mathbb Z}/2[c_7]\otimes Q(2){\{u_3\}})\end{multlined}
\]
where \(A\cong ({\mathbb Z}/2)^3\), in this case, \(W_G(A)\) is the Weyl group, \(c_i\) are Chern classes, \({ Res}_{H{\mathbb Z}/2}: H^*(BG_2;{\mathbb Z}/2)\to H^*(BA;{\mathbb Z}/2)^{W_G(A)}\) is the restriction map and \((-1)[2]\) is a shift. The differential is given by \(d_2(u_3)=y\) in the coniveau spectral sequence. More general cases are studied, all with the hypothesis that \(G\) has only one conjugacy class of nontoral maximal elementary abelian \(p\)-subgroups \(A\). Motivic cohomology; Milnor invariants; coniveau spectral sequence; exceptional groups Classifying spaces of groups and \(H\)-spaces in algebraic topology, Spectral sequences and homology of fiber spaces in algebraic topology, Linear algebraic groups over the reals, the complexes, the quaternions, Motivic cohomology; motivic homotopy theory, Galois cohomology of linear algebraic groups, Galois cohomology, Loop spaces, Generalized cohomology and spectral sequences in algebraic topology, Hopf algebras (aspects of homology and homotopy of topological groups) Coniveau spectral sequences of classifying spaces for exceptional and spin groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be a number field, and consider a quadratic polynomial \(f_c(x)=x^2+c\), with \(c\in K\), and a point \(a\in K\). Let \(N(c, a)\) denote the number of points \(x\in K\) such that
\[
a\in \left\{f_c(x), f_c(f_c(x)), f_c(f_c(f_c(x))),\dots \right\},
\]
a natural condition when considering the dynamics of \(f_c(x)\). It is relatively clear that \(N(c, a)\) is finite, for any \(c, a\in K\), but it turns out that \(N(c, a)\) is uniformly bounded as \(c\in K\) varies, for a fixed \(a\in K\) [\textit{X. Faber} et al., Math.\ Res.\ Lett.~16, No.~1, 87--101 (2009; Zbl 1222.11086)]. This article examines this bound more closely, or, more specifically, it examines the largest value \(N(c, a)\) attained by infinitely many \(c\in K\), denoted by \(\tilde{\kappa}(a, K)\). The main result is that \(\tilde{\kappa}(a, K)\) is 10 if \(a=-1/4\); it equals 6 or 8 if \(256a^3+368a^2+104a+23=0\); it is 4 if \(a\) comes from a certain finite (but not explicitly known) set \(S\); otherwise it equals 6. arithmetic dynamics; quadratic dynamical systems; arithmetic geometry; preimage; rational points; uniform bound Hutz, B., Hyde, T., Krause, B.: Pre-images of quadratic dynamical systems. Involve (2012, to appear) Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps, Rational points, Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets Preimages of quadratic dynamical systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this work, we introduce the \textit{complexity factor} in the context of self-gravitating fluid distributions for the case of black holes by employing the Newman-Penrose formalism. In particular, by working with spherically symmetric and static AdS black holes, we show that the complexity factor can be interpreted in a natural way at the event horizon. Specifically, a thermodynamic interpretation for the aforementioned complexity factor in terms of a pressure partially supporting a Van der Waals-like equation of state is given. black holes; complexity factor; spectral geometry Black holes, Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.), Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism, Classical and relativistic thermodynamics, Complex multiplication and abelian varieties Complexity factor for black holes in the framework of the Newman-Penrose formalism | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(X\), \(Y\) be spaces and \(f\) a map whose domain \(\mathrm{dom}(f)\) is contained in \(X\) and whose range is \(Y\). Such \(f\) will be called a partial function on \(X\). The closure \(\overline\Gamma(f)\) of the graph of \(f\) in \(X\times Y\) is called the graphoid of \(f\). The graphoid determines a function \(\overline f:X\to\mathcal{P}(Y)\) (the power set of \(Y\)), assigning to a point \(x\in X\), \(\overline f(x)= \{y\in Y\,|\,(x,y)\in\overline\Gamma(f)\}\). This \(\overline f\) is called the graphoid extension of \(f\). The domain, \(\mathrm{dom}(f)\), equals \(\{x\in X\,|\,\overline f(x)\neq\emptyset\}\).
The authors are interested in the graphoids of rational functions of \(k\)-variables, \(f(x_1,\dots,x_k)=\frac{p(x_1,\dots,x_k)}{q(x_1,\dots,x_k)}\), where \(p\) and \(q\) are relatively prime polynomials. The domain of such a map is the open dense subset \(\mathrm{dom}(f)=\mathbb{R}^k\setminus (p^{-1}(0)\cap q^{-1}(0))\) of \(\mathbb{R}^k\). Using \(\overline{\mathbb{R}} =\mathbb{R}\cup\{\infty\}\) to denote the projective real line, one may treat \(\overline{\mathbb{R}}^k\) as a \(k\)-dimensional torus; so \(\mathrm{dom}(f)\) is an open dense subset of \(\overline{\mathbb{R}}^k\). If \(\mathbb{R}(x_1,\dots,x_k)\) denotes the field of such rational functions, then the elements of \(\mathbb{R}(x_1,\dots,x_k)\) will take values in \(\overline{\mathbb{R}}\). A subset \(\mathcal{F}\subset\mathbb{R}(x_1,\dots,x_n)\) is called a rational vector-function.
In this setting, in case \(\mathcal{F}\) is countable, then one can define \(\mathrm{dom}(\mathcal{F})=\bigcap\{\mathrm{dom}(f)\, |\,f\in\mathcal{F}\}\); the latter is a dense \(\mathrm{G}_\delta\)-set in \(\overline{\mathbb{R}}^k\). One may think of \(\mathcal{F}:\mathrm{dom}(\mathcal{F})\to\overline{\mathbb{R}}^\mathcal{F}\) via \(x\mapsto(f(x))_{f\in\mathcal{F}}\). Its graphoid is a closed subset of the compact Hausdorff space \(\overline{\mathbb{R}}^k\times \overline{\mathbb{R}}^\mathcal{F}\), and it has a graphoid extension \(\overline{\mathcal{F}}\) with \(\mathrm{dom}(\overline{\mathcal{F}})= \overline{\mathbb{R}}^k\), i.e., \(\overline{\mathcal{F}} :\overline{\mathbb{R}}^k\to\mathcal{P}(\overline{\mathbb{R}}^\mathcal{F})\).
For uncountable families \(\mathcal{F}\), a different approach is needed, and such is described by the authors. One then similarly obtains \(\overline{\mathcal{F}}:\overline {\mathbb{R}}^k\to\mathcal{P}(\overline{\mathbb{R}}^\mathcal{F})\) as just indicated. The good properties that such a ``set-valued'' function has are listed as (1)--(4) on page 25.
The paper is devoted to:
{ Problem 1.1.} Given a family of rational functions \(\mathcal{F}\subset\mathbb{R}(x_1,\dots,x_k)\), study topological (and dimension) properties of the graphoid \(\overline{\Gamma}(\mathcal{F}) \subset\overline{\mathbb{R}}^k\times\mathbb{R}^\mathcal{F}\) of \(\mathcal{F}\). A precise question: Has \(\overline{\Gamma}(\mathcal{F})\) the topological dimension \(k\)?
The main result of the paper is,
{ Theorem 1.2.} For any family of rational functions \(\mathcal{F}\subset\mathbb{R}(x,y)\), its graphoid \(\overline{\Gamma}(\mathcal{F})\subset\overline{\mathbb{R}}^2\times \overline{\mathbb{R}}^\mathcal{F}\) has covering dimension \(2\). graphoid; graph; rational vector-function; topological dimension; extension dimension; cohomological dimension; Pontryagin space Banakh T., Potyatynyk O., Dimension of graphoids of rational vector-functions, Topology Appl., 2013, 160(1), 24--44 Unicoherence, multicoherence, Dimension theory in algebraic topology, Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants), Real algebraic sets, Real rational functions, Degree, winding number, Topology of special sets defined by functions Dimension of graphoids of rational vector-functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth complex affine variety. Let \(M\) be a smooth projective compactification of
\(X\) by a simple normal crossings divisor \({\mathbf{D}}=D_1\cup\cdots \cup D_k\)
supporting an ample line bundle.
The logarithmic cohomology ring \(H^\ast_{\log}(M,{\mathbf{D}})\)
is a ring that encodes much of the combinatorics and algebraic topology
of the pair \((M,{\mathbf{D}})\).
The authors construct a multiplicative spectral sequence converging to the symplectic
cohomology ring \(SH^\ast(X)\) whose first page is isomorphic to
\(H^\ast_{\log}(M,{\mathbf{D}})\). A key technical ingredient is a morphism introduced in an earlier work of the authors,
[``A log PSS morphism with applications to Lagrangian embeddings'', Preprint, \url{arXiv:1611.06849}].
The authors establish geometric criteria under which the spectral sequence degenerates and
the ring structure of \(SH^\ast(X)\) can be computed.
These results are then applied to give computations and qualitative
results. For example, the authors give a complete topological description of \(SH^\ast(X)\) in the case
where \({\mathbf{D}}\) is a union of sufficiently many generic ample divisors whose homology classes
span a rank one subspace. They also discuss log Calabi-Yau pairs. Finally, they
prove that in many cases symplectic cohomology is finitely generated as a ring. symplectic cohomology; affine variety; multiplicative spectral sequence; Calabi-Yau variety; PSS morphism; logarithmic cohomology Symplectic aspects of Floer homology and cohomology, Lagrangian submanifolds; Maslov index, Generalized geometries (à la Hitchin), Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Calabi-Yau manifolds (algebro-geometric aspects), Calabi-Yau theory (complex-analytic aspects) Symplectic cohomology rings of affine varieties in the topological limit | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 simple algebraic group with Weyl group \((W, S)\) and let \(w \in W\). We consider the descent set \(D(w) = \{ s \in S \mid l(ws) < l(w)\}\). This has been generalized to the situation of the Bruhat poset \(W^J\), where \(J \subset S\). To do this one identifies a certain subset \(S^J \subset W^J\) that plays the role of \(S \subset W\) in the well known case \(J = \emptyset\). One ends up with the descent system \((W^J, S^J)\). On the other hand, each subset \(J \subset S\) determines a projective, simple \(G \times G\)-embedding \(\mathbb{P}(J)\) of \(G\). The case where \(J = \emptyset\) is closely related to the wonderful embedding. One obtains a complete list of all subsets \(J \subset S\) such that \(\mathbb{P}(J)\) is a rationally smooth algebraic variety. In such cases, we determine the Betti numbers of \(\mathbb{P}(J)\) in terms of \((W^J, S^J)\). It turns out that \(\mathbb{P}(J)\) can be decomposed into a union of ``rational'' cells. The descent system is used here to help record the dimension of each cell. Betti numbers; descent systems; H-polynomials; rationally smooth Compactifications; symmetric and spherical varieties, Algebraic monoids, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups The Betti numbers of simple embeddings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Solutions of the quartic Fermat equation in ring class fields of odd conductor over quadratic fields \(K= \mathbb Q(\sqrt{-d})\) with \(-d\equiv 1\pmod 8\) are shown to be periodic points of a fixed algebraic function \(T(z)\) defined on the punctured disk \(0< |z|_2 \le (1/2)\) of the maximal unramified, algebraic extension \(K_2\) of the 2-adic field \(\mathbb Q_2\). All ring class fields of odd conductor over imaginary quadratic fields in which the prime \(p=2\) splits are shown to be generated by complex periodic points of the algebraic function \(T\), and conversely, all but two of the periodic points of \(T\) generate ring class fields over suitable imaginary quadratic fields. This gives a dynamical proof of a class number relation originally proved by Deuring. It is conjectured that a similar situation holds for an arbitrary prime \(p\) in place of \(p=2\), where the case \(p=3\) has been previously proved by the author [Int. J. Number Theory 12, No. 4, 853--902 (2016; Zbl 1415.11063)], and the case \(p=5\) will be handled in Part II [Zbl 1441.11064]. periodic points; algebraic function; 2-adic field; ring class fields; quartic Fermat equation Higher degree equations; Fermat's equation, Elliptic curves over local fields, Complex multiplication and moduli of abelian varieties, Algebraic functions and function fields in algebraic geometry Solutions of Diophantine equations as periodic points of \(p\)-adic algebraic functions. I. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let X be a smooth projective curve. A stable pair (E,\(\phi\)) on X, as defined by \textit{N. J. Hitchin} [Proc. Lond. Math. Soc., III. Ser. 55, 59- 126 (1987; Zbl 0634.53045)], is a vector bundle E on X together with a morphism \(\phi: E\to \Omega_ X\otimes E\) of \({\mathcal O}_ X\)-modules such that for any \(\phi\)-invariant proper subbundle F of E, the inequality \(\mu (F)<\mu (E)\) holds where \(\mu = \)degree/rank. In the quoted paper, Hitchin proved that the set of all isomorphism classes of stable pairs of rank 2 over a compact Riemann surface can be given the structure of a complex manifold which has the coarse moduli property in the analytic category.
The author constructs a coarse moduli scheme within the algebraic category in the following more general set up. Instead of taking (E,\(\phi\)) with \(\phi: E\to \Omega_ X\otimes E\), he considers the more general situation where \(\phi: E\to E\otimes L\) takes values in any fixed line bundle L on X. Furthermore, he proves that for any line bundle L, there exists a coarse moduli scheme M(r,d,L) for (S-equivalence classes of) semistable pairs ((E,\(\phi\)): \(E\to L\otimes E)\) of rank r, degree d, on X. The scheme M(r,d,L) is quasi-projective, and has an open subscheme \(M'\) which is the moduli scheme of stable pairs. stable pairs of rank 2 over a compact Riemann surface; coarse moduli property; moduli scheme of stable pairs 30. Nitsure, Nitin Moduli space of semistable pairs on a curve \textit{Proc. London Math. Soc.}62 (1991) 275--300 Math Reviews MR1085642 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on curves and their moduli, Fine and coarse moduli spaces, Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic) Moduli space of semistable pairs on a curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The monodromy conjecture of Denef and Loeser predicts that the poles of the topological zeta function and related zeta functions associated to a polynomial \(f\) induce monodromy eigenvalues of \(f\). However, not every monodromy eigenvalue can be recovered from a pole. This motivates to generalise the conjecture in order to obtain more poles: one also considers zeta functions associated to a polynomial and a differential form. The authors attach to \(f\) a suitable family of differential forms, such that each pole of the topological zeta function of \(f\) and a form from the family induce a monodromy eigenvalue, and moreover, such that all monodromy eigenvalues are obtained this way. This answers positively a question (regarding the existence of a family of such forms) of \textit{W. Veys} [Adv. Math. 213, No. 1, 341--357 (2007; Zbl 1129.14005)]. monodromy conjecture; monodromy eigenvalues; topological zeta function Cauwbergs, Thomas; Veys, Willem, Monodromy eigenvalues and poles of zeta functions, Bull. Lond. Math. Soc., 49, 2, 342-350, (2017) Singularities in algebraic geometry, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Local complex singularities Monodromy eigenvalues and poles of zeta functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We prove a motivic Landweber exact functor theorem. The main result shows the assignment given by a Landweber-type formula involving the MGL-homology of a motivic spectrum defines a homology theory on the motivic stable homotopy category which is representable by a Tate spectrum. Using a universal coefficient spectral sequence we deduce formulas for operations of certain motivic Landweber exact spectra including homotopy algebraic \(K\)-theory. Finally we employ a Chern character between motivic spectra in order to compute rational algebraic cobordism groups over fields in terms of rational motivic cohomology groups and the Lazard ring. motivic stable homotopy category; Tate spectrum; spectral sequence; algebraic \(K\)-theory; algebraic cobordism groups Naumann, Niko; Spitzweck, Markus; Østvær, Paul Arne, Motivic {L}andweber exactness, Doc. Math.. Documenta Mathematica, 14, 551-593, (2009) Bordism and cobordism theories and formal group laws in algebraic topology, Stable homotopy theory, spectra, Generalizations (algebraic spaces, stacks), Motivic cohomology; motivic homotopy theory, \(K\)-theory of schemes Motivic Landweber exactness | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In \textit{B. Dwork}'s proof of the rationality of the zeta function of a variety over a finite field [see Am. J. Math. 82, 631--648 (1960; Zbl 0173.48501)], the Lefschetz trace formula shown by \textit{A. Grothendieck} [see Sém. Bourbaki 17 (1964/65), Exposé 279 (1966; Zbl 0199.24802)], plays an essential rôle. In order to obtain finer information one likes to split up the total cohomology using algebraic cycles and to describe the Galois representations on the factors. The splitting process is described in terms of correspondences and one wants to have a Lefschetz trace formula for the twist of a correspondence by the Frobenius, or more special, a Lefschetz trace formula for the twist by all sufficiently large powers of a fixed Frobenius. In this context \textit{P. Deligne} has conjectured a certain Lefschetz trace formula in terms of correspondences over a finite field. This is shown in the present paper assuming the resolution of singularities holds. It generalizes L. Illusie's proof in the one-dimensional case [see \textit{A. Grothendieck} and \textit{L. Illusie} in Sémin. Géométrie algébrique 1965-1966, SGA 5, Lect. Notes Math. 589, Exposé III, 73--137 (1977; Zbl 0355.14004)].
For the smooth case of an arbitrary dimension there is an independent proof by \textit{E. Shpiz} [``Deligne's conjecture in the constant coefficient case'', Ph. D. Thesis (Harvard Univ. 1990)]. rationality of the zeta function; Lefschetz trace formula; resolution of singularities R. Pink, On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne, Ann. of Math., 135 (1992), 483--525. Étale and other Grothendieck topologies and (co)homologies, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper it is proved that there exist completely normal spaces that are not real spectra of rings. Real spectra of rings are known to be spectral spaces in the sense of Hochster and the problem of characterizing topologically real spectra has been considered from the beginning of this theory. A property which is quite easy to notice for real spectra, and closely linked to the ordered structures of fields considered in this theory, is the following, called complete normality: whenever points \(x\) and \(y\) are specializations of the same point \(z\) (this means that they belong to the closure of the singleton \(\{z\})\), then either \(x\) is a specialization of \(y\) or \(y\) is a specialization of \(x\).
This paper proves that in order to find a topological description of real spectra, new topological properties of real spectra have to be identified. A completely normal spectral space is constructed which is not the real spectrum of a ring. The example presented is one dimensional which is the first dimension where such a result can be hoped. The construction relies on the Dedekind completion -- through cuts -- of completely dense totally ordered sets and to a construction due to Hausdorff of an \(\eta_2\)-set (where \(\omega_2\) is the second infinite cardinal): it is a set \(Y\) such that for every two subsets \(A\) and \(B\) of \(Y\) such that \(A < B\) with cardinality less than \(\omega_2\), there exist \(y\) in \(Y\) such that \(A < y < B\). completely normal spaces that are not real spectra of rings; spectral spaces Delzell, Charles N.; Madden, James J., A completely normal spectral space that is not a real spectrum, J. Algebra, 169, 1, 71-77, (1994) Real algebraic sets, Real and complex fields, Relevant commutative algebra A completely normal spectral space that is not a real spectrum | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The book presents a collection of known facts and recent developments concerning the Hilbert's sixteenth problem. The main focus is on limit cycles arising from perturbations of Hamiltonian systems and the study of corresponding Abelian and iterated integrals. The book consists of six chapters.
The first chapter is an introduction and presents a way of breaking Hilbert's sixteenth problem in many parts and observing the fact that even such partial problems are extremely difficult to treat. The authors want to point out that both real and complex algebraic geometry would be indispensable for a systematic approach.
The second chapter contains the basic notions on the qualitative theory of differential equations including the limit cycles of polynomial differential equations in \(\mathbb{R}^2\).
The main aim of Chapter 3 is to explain the relation between the zeros of Abelian integrals and the number of limit cycles for corresponding perturbed planar Hamiltonian vector fields. In Chapter 4 the authors explain some fundamental properties of the principal Poincaré-Pontryagin function. In Chapter 5 the space of differential equations in \(\mathbb{C}^2\) which have at least one center singularity is under the consideration. In the last chapter the authors define the corresponding Petrov-Brieskorn type modulus, give a formula for the Gauss-Manin connection of iterated integrals and calculate the Melnikov functions for certain topological cycles in terms of iterated integrals. Hilbert's sixteenth problem; limit cycle; Abelian and iterated integrals; bifurcation; polynomial and analytic vector fields Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory, Bifurcations of limit cycles and periodic orbits in dynamical systems, Structure of families (Picard-Lefschetz, monodromy, etc.), Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) Limit cycles, Abelian integral and Hilbert's sixteenth problem. Paper from the 31st Brazilian mathematics colloquium -- 31\(^{\text o}\) Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, Brazil, July 30 -- August 5, 2017 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 following well known problem can be found, for instance, in a book of A. Leonte and C. Niculescu and it can be solved using inequalities.
Problem. Let \(f,g:\mathbb R\longrightarrow\mathbb R\) be two real functions given by
\[
f(x)=2x^2+2, \;g(x)=x^2+2x+1.
\]
If \(h\) is another polynomial function of degree \(2\) such that
\[
g(x)\leq h(x)\leq f(x),\;\forall\;x\in\mathbb R,
\]
then there exists \(\lambda\in\mathbb R,\;0\leq \lambda\leq1\), such that
\[
h=\lambda f+(1-\lambda)g.
\]
In the paper under review, the author explains (and solves) the above problem using language and results about linear systems of conics in the projective complex plane. linear systems of conics; dimension and base points of a linear system Special algebraic curves and curves of low genus, Divisors, linear systems, invertible sheaves On a property of conics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We derive a blow-up formula for the de Rham cohomology of a local system of complex vector spaces on a compact complex manifold. As an application, we obtain the blow-up invariance of \(E_{1}\)-degeneracy of the Hodge-de Rham spectral sequence associated with a local system of complex vector spaces. local system; twisted de Rham Cohomology; blow-up; Hodge-de Rham spectral sequence Modifications; resolution of singularities (complex-analytic aspects), de Rham cohomology and algebraic geometry, Analytic sheaves and cohomology groups On the blow-up formula of twisted de Rham Cohomology | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The gluing formula for the zeta-determinant of a Laplacian on a compact-oriented Riemannian manifold was proven by \textit{D. Burghelea} et al. in [J. Funct. Anal. 107, No. 1, 34--65 (1992; Zbl 0759.58043)]. In the proof of this gluing formula, there appears a real polynomial whose degree is less than half of the dimension of an underlying manifold. This polynomial is determined by some data on an arbitrarily small collar neighborhood of a cutting compact hypersurface. The constant term of this polynomial appears in the BFK-gluing formula as one ingredient. In [J. Math. Phys. 56, No. 12, 123501, 19 p. (2015; Zbl 1334.58022)], the authors computed the constant term of this polynomial in terms of a warping function when a collar neighborhood is a warped product manifold using methods in [\textit{P. B. Gilkey}, Invariance theory, the heat equation, and the Atiyah-Singer index theorem. Wilmington, Delaware: Publish or Perish, Inc (1984; Zbl 0565.58035)]. In this interesting paper, authors compute this polynomial itself, and the values of a relative zeta function and a zeta function associated to the Dirichlet-to-Neumann operator at zero on a warped product manifold in terms of a warping function. BFK-gluing formula; relative zeta-determinant; Dirichlet-to-Neumann operator; warped product metric; warping function Index theory and related fixed-point theorems on manifolds, de Rham cohomology and algebraic geometry, Determinants and determinant bundles, analytic torsion The polynomial associated with the BFK-gluing formula of the zeta-determinant on a compact warped product manifold | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We observe that there exists a Białynicki-Birula decomposition of the Hilbert scheme \(\text{Hilb}^P_n\) such that the cells are homeomorphic to Gröbner strata of homogeneous ideals with fixed initial ideal. Using such a decomposition, we show that \(\text{Hilb}^P_n\) is singular at a monomial scheme if the corresponding Gröbner stratum is singular at \(J\). Hilbert scheme; Białynicki-Birula decomposition; Gröbner bases Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Polynomial rings and ideals; rings of integer-valued polynomials Computable Białynicki-Birula decomposition of the 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 introduces and studies properties of an invariant, denoted \(\Theta^R_c\), for standard graded complete intersections with an isolated singularity. Let \(R = k[x_0, \dots, x_{n+c-1}]/(f_1, \dots, f_c)\), where \(k\) is a separably closed field, \(\deg x_j = 1\) for each \(j\), and each \(f_i\) is a homogeneous polynomial such that \(f_1, \dots, f_c\) forms a regular sequence. In addition, let \(M\) and \(N\) be finitely graded \(R\)-modules. The author shows that ``if the tensor product of a graded pair \(M, N\) has finite length, then \(\Theta^R_c(M, N) = 0\) if and only if \(\dim M + \dim N \leq \dim R\). Moreover, \(\dim M + \dim N \leq \dim R + 1\) regardless of the value of the invariant \(\Theta^R_c(M, N)\).'' This generalizes work on Hochster's invariant \(\theta^R\) for a ring \(R\) which is the quotient of a regular local ring by a regular element, and in particular work found in the papers \textit{H. Dao} [Math. Res. Lett. 15, No. 2--3, 207--219 (2008; Zbl 1229.13014)], \textit{H. Dao} [``Decency and rigidity over hypersurfaces'', \url{arXiv:math/0611568}, to appear in Trans. Am. Math. Soc.], \textit{Y. Kobayashi} [Math. Jap. 24, 643--655 (1980; Zbl 0434.13009)], and \textit{W. F. Moore}, \textit{G. Piepmeyer}, \textit{S. Spiroff} and \textit{M. E. Walker} [Adv. Math. 226, No. 2, 1692--1714 (2011; Zbl 1221.13027)].
The first section of the paper defines \(\Theta_c^R(M, N)\) and provides a geometric description of this invariant. This establishes that \(\Theta^R_c\) shares many of the same properties as Hochster's theta function. A useful reference for these results is given by \textit{W. F. Moore}, \textit{G. Piepmeyer}, \textit{S. Spiroff} and \textit{M. E. Walker} [Math. Z. 273, No. 3--4, 907--920 (2013; Zbl 1278.13013)].
The second section of the paper is dedicated to an investigation of the expected dimension of the intersection of the modules \(M\) and \(N\). An interesting result (Theorem 2.4) relates \(\Theta_c^R(M, N)\) with ``a generalized Bézout's theorem relating the degrees of the modules, their associated homology modules, and the ambient ring''.
The third section of the paper studies an invariant \(\eta^R_c(M, N)\). This invariant was introduced by Dao and is known to differ from \(\theta^R\). The author also ties \(\eta^R_c\) to a generalized Bézout's theorem.
The paper includes useful examples and references to the literature. Hochster's theta function; complete intersections; Bézout's Theorem Linkage, complete intersections and determinantal ideals, Dimension theory, depth, related commutative rings (catenary, etc.), Syzygies, resolutions, complexes and commutative rings, Complete intersections On invariants of complete intersections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This is a continuation of previous work of \textit{A. S. Tikhomirov} [same conference, Aspects Math. E 25, 183-203 (1994; see the preceding review)] to which we refer for definitions and notations. The main result of the present paper is an explicit formula for the polynomial \(\delta_ 4\) of a nonsingular projective surface \(S\). According to the above mentioned article, the computation reduces to an enumerative problem: the determination of the number of 4-secant planes to a convenient embedding of \(S\) into \(\mathbb{P}^{10}\). Hilbert scheme; Segre class; standard vector bundle; number of 4-secant planes Tikhomirov, A.; Troshina, T., Top Segre class of a standard vector bundle e4 D on the Hilbert scheme hilb4S of a surface S, 205-226, (1994) Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Enumerative problems (combinatorial problems) in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Top Segre class of a standard vector bundle \({\mathcal E}^ 4_ D\) on the Hilbert scheme \(\text{Hilb}^ 4S\) of a surface \(S\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(I\) be a homogeneous ideal in \(R=\mathbb K[x_0,\dots ,x_n]\), such that \(R/I\) is an Artinian Gorenstein ring. A famous theorem of Macaulay says that in this instance \(I\) is the ideal of polynomial differential operators with constant coefficients that cancel the same homogeneous polynomial \(F\). A major question related to this result is to be able to describe \(F\) in terms of the ideal \(I\). In this note we give a partial answer to this question, by analyzing the case when \(I\) is the Artinian reduction of the ideal of a reduced (arithmetically) Gorenstein zero-dimensional scheme \(\varGamma\subset\mathbb P^n\). We obtain \(I\) from the coordinates of the points of \(\varGamma\). Artinian Gorenstein ring; Macaulay inverse system; zero-dimensional scheme Commutative rings of differential operators and their modules, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Parametrization (Chow and Hilbert schemes) Finding inverse systems from coordinates | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 factorize hyperbolic polynomials with quasianalytic coefficients. The author generalizes some results by \textit{K. Kurdyka} and \textit{L. Paunescu} [Duke Math. J. 141, No. 1, 123--149 (2008; Zbl 1140.15006)] on perturbation theory of families of symmetric matrices to the quasianalytic setting. quasianalytic perturbation; hyperbolic polynomials; quasianalytic and arc-quasianalytic functions; polynomially bounded structures; eigenvalues; eigenspaces; symmetric and antisymmetric matrices; spectral theorem; quasianalytic diagonalization Krzysztof Jan Nowak, Quasianalytic perturbation of multi-parameter hyperbolic polynomials and symmetric matrices, Ann. Polon. Math. 101 (2011), no. 3, 275 -- 291. Real-analytic and semi-analytic sets, Semi-analytic sets, subanalytic sets, and generalizations, Eigenvalues, singular values, and eigenvectors, \(C^\infty\)-functions, quasi-analytic functions Quasianalytic perturbation of multi-parameter hyperbolic polynomials and symmetric matrices | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces \(\mathfrak M_{0,n}\) of Riemann spheres with n marked points are multiple zeta values. We do this by introducing a differential algebra of multiple polylogarithms on \(\mathfrak M_{0,n}\) and proving that it is closed under the operation of taking primitives. The main idea is to apply a version of Stokes' formula iteratively to reduce each period integral to multiple zeta values.
We also give a geometric interpretation of the double shuffle relations, by showing that they are two extreme cases of general product formulae for periods which arise by considering natural maps between moduli spaces. moduli spaces; multiple zeta values; iterated integrals; polylogarithms; associators; associahedra F.C.S. Brown, \textit{Multiple zeta values and periods of moduli spaces} \(\mathcal{M}\)\_{}\{0,\(n\)\}(\(\mathbbR\)), \textit{Annales Sci. Ecole Norm. Sup.}\textbf{42} (2009) 371 [math/0606419] [INSPIRE]. Multiple Dirichlet series and zeta functions and multizeta values, Families, moduli of curves (algebraic), Polylogarithms and relations with \(K\)-theory Multiple zeta values and periods of moduli spaces \(\overline{\mathfrak M}_{0,n}\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Twisted Deligne cohomology is the prototype for other twisted differential spectra, and its existence follows from general constructions. For example, a sheaf-theoretic definition of smooth Deligne cohomology was given in [\textit{U. Bunke}, ``Differential cohomology'', Preprint, \url{arXiv:1208.3961}], and a bordism model for the differential extension of ordinary integral cohomology was presented in [\textit{U. Bunke} et al., Ann. Math. Blaise Pascal 17, No. 1, 1--16 (2010; Zbl 1200.55007)]. Since Deligne cohomology has not been explicitly twisted like other differential cohomology theories, in the paper under review the authors aim to twist Deligne cohomology, by using degree one twists of integral cohomology and de Rham cohomology. The main tools are the homotopy sheaves, simplicial presheaves and higher stacks.
In Section 2, the authors deal with twists of integral cohomology at the level of the Eilenberg-Mac Lane spectrum \(H\mathbb{Z}\), and in Section 3 they recall the properties of the Deligne cohomology. Then, in Section 4, twisted integral cohomology and 1-form twisted de Rham cohomology yield compatible twistings of Deligne cohomology. The authors prove that pulling back the universal bundle over a map which classifies a twist, one obtains a bundle \(\mathcal{H}^q \rightarrow M\) over \(M\). The \(\omega\)-twisted Deligne cohomology of degree \(q\) of \(M\) is given by the connected components \(\pi_0\Gamma(M,\mathcal{H}^q)\). In the paper under review the authors use the category of sheaves of chain complexes, but for some results they deal also with twisted differential cohomology within smooth sheaves of spectra.
The properties of basic twisted Deligne cohomology are presented in Section 5, and several examples for the constructions and techniques developed in the previous sections are given in the final part of the paper. Deligne cohomology; differential cohomology; twisted cohomology; local coefficient systems; connections; Cech-de Rham complex; smooth stacks Grady, D.; Sati, H., Higher-twisted periodic smooth Deligne cohomology de Rham theory in global analysis, de Rham cohomology and algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Twisted smooth Deligne cohomology | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we study the tangent spaces of the smooth nested Hilbert scheme \(\mathrm{Hilb}^{n,n-1}(\mathbb A^2)\) of points in the plane, and give a general formula for computing the Euler characteristic of a \(\mathbb T^2\)-equivariant locally free sheaf on \(\mathrm{Hilb}^{n,n-1}(\mathbb A^2)\). Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables \(q\) and \(t\) with non-negative integer coefficients. We call this conjecturally positive polynomial as the ``nested \(q,t\)-Catalan series'', for it has many conjectural properties similar to that of the \(q,t\)-Catalan series. Atiyah-Bott Lefschetz formula; (nested) Hilbert scheme of points; tangent spaces; diagonal coinvariants Can, M.: Nested Hilbert schemes and the nested q,t-Catalan series Parametrization (Chow and Hilbert schemes), Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Nested Hilbert schemes and the nested \(q,t\)-Catalan series | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Denote by \(E\) an Enriques surface, and by \(E^{[n]}\) for \(n\geq 1\), the Hilbert scheme of \(n\) points on \(E\).
Contrary to the case \(n=1\), the first result is to show that the small deformations of \(E^{[n]}\) for \(n\geq 2\) are induced from those of its universal covering space, by computing the dimensions of these deformation spaces.
In the second result, an equivalent condition for an automorphism \(f\) of \(E^{[n]}\) to be natural is given, namely, there exists an automorphism \(g\) of \(E\) such that \(f\) is naturally induced by \(g\). This is analogous to the case of the Hilbert scheme of \(n\) points on \(K3\) surfaces, that is, the automorphism \(f\) is natural if and only if it preserves the exceptional divisor of the Hilbert-Chow morphism of \(E^{[n]}\).
Contrary to the case of \(n=1\) (where the universal covering space being a \(K3\) surface), it is proved that there exists exactly one Enriques surface type quotient for the universal covering space \(X\) of \(E^{[n]}\). The proof is done by a classification of all involutions of \(X\) acting identically on \(H^2(X,\,\mathbb{C})\). Calabi-Yau manifld; Enriques surfaces; Hilbert scheme Calabi-Yau manifolds (algebro-geometric aspects), \(K3\) surfaces and Enriques surfaces, Parametrization (Chow and Hilbert schemes) Universal covering Calabi-Yau manifolds of the Hilbert schemes of \(n\) points of Enriques surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author discusses an approach to the computation of modules of global sections of line bundles on super Riemann surfaces (SUSY curves), e.g. the modules of functions or \(j\)-superdifferentials. The main result is the possibility of nonfreeness of modules of regular \(j\)- superdifferentials in the cases \(j=-1,0,1,2\). The main technique employed is a spectral sequence connected with the filtration by degrees of the nilpotent ideal. The second term of this spectral sequence is calculated. global sections of line bundles; super Riemann surfaces; j- superdifferentials; spectral sequence Hodgkin, L.: Problems of fields on super Riemann surfaces. J. Geom. and Phys.6, 333--348 (1989) Complex supergeometry, Analysis on supermanifolds or graded manifolds, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Supervarieties Problems of fields on super Riemann surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A \(k\)-configuration of type \((d_1, \dots, d_m)\) in \(\mathbb{P}^2\) is a finite subset \(\mathbb{X}\) of \(\mathbb{P}^2\) consisting of \(d_i\) points on each of \(m\) lines \(L_i\) for \(1\leq i\leq m\) such that \(L_i\) contains no point of \(L_j \cap \mathbb{X}\) for \(1\leq j<i\leq m\). It is known that all \(k\)-configurations of type \((d_1, \dots, d_m)\) in \(\mathbb{P}^2\) have the same Hilbert function [cf. \textit{A. V. Geramita}, \textit{P. Maroscia} and \textit{L. G. Roberts}, J. Lond. Math. Soc., II. Ser. 28, 443-452; Zbl 0535.13012)]. The author shows that in fact their homogeneous vanishing ideals have numerically identical minimal graded free resolutions. He applies this result to get the following interesting construction for Gorenstein ideals of codimension three.
Take a \(k\)-configuration \(\mathbb{X} \subseteq \mathbb{P}^2\) of type \((d_1, \dots, d_m)\) which is contained in a complete intersection \(\mathbb{Z}\) of type \((m,d_m +\ell)\) with \(\ell\geq 1\), and let \(Y= \mathbb{Z} \backslash \mathbb{X}\) be the subscheme of \(\mathbb{Z}\) which is residual to \(\mathbb{X}\). Then the sum of the vanishing ideals \(I_{\mathbb{X}} +I_{\mathbb{Y}}\) is a Gorenstein ideal of codimension three whose minimal graded free resolution is explicitly computed by the author. \(k\)-configuration; Hilbert function; Gorenstein ideals of codimension three; vanishing ideals Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Projective techniques in algebraic geometry, Relevant commutative algebra, Enumerative problems (combinatorial problems) in algebraic geometry, Linkage Exposé I B: Remarks on \(k\)-configurations 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 The paper provides a classification of the simple integrable modules of double affine Hecke algebras via perverse sheaves. Let \(\underline G\) be a simple connected simply connected linear algebraic group. Let \(\underline{\text{Lie}}\,\underline G\) denote the Lie algebra of \(\underline G\), let \(\underline{\text{Lie}}\,\underline H\subset\underline{\text{Lie}}\,\underline G\) be a Cartan subalgebra and let \(\underline{\text{Lie}}\,\underline B\subset\underline{\text{Lie}}\,\underline G\) be a Borel subalgebra containing \(\underline{\text{Lie}}\,\underline H\). Let \(\underline\Phi\) be the root system of \(\underline G\) and let \(\Phi^\vee\) be the root system dual to \(\underline\Phi\). Let \(\{\alpha_i:i\in\underline I\}\), \(\{\alpha_i^\vee:i\in\underline I\}\) be the set of simple roots and of simple coroots, respectively. Let \(I:=\underline I\sqcup\{0\}\). Let \(\underline W\), \(W\) be the Weyl group and the affine Weyl group of \(\underline G\). We identify \(\underline I\) (resp. \(I\)) with the set of simple reflections in \(\underline W\) (resp. \(W\)). Let \(s_i\in W\) be the simple reflection corresponding to \(i\in I\). For all \(i,j\in I\), let \(m_{ij}\) denote the order of the element \(s_is_j\) in \(W\). Let \(\underline X\) be the weight lattice of \(\underline\Phi\) and let \(Y^\vee\) be the root lattice of \(\underline\Phi v\). Let \(\{\omega_i:i\in\underline I\}\) be the set of fundamental weights. Consider the lattices \(Y=\bigoplus_{i\in I}\mathbb{Z}\alpha_i\subset X=\mathbb{Z}\delta\oplus\bigoplus_{i\in I}\mathbb{Z}\omega_i\), \(Y^\vee=\bigoplus_{i\in I}\mathbb{Z}\alpha_i^\vee\), where \(\delta\) is a new variable. There is unique pairing \(X\times Y^\vee\to\mathbb{Z}\) such that \((\omega_i:\alpha_j^\vee)=\delta_{ij}\) and \((\delta:\alpha_j^\vee)=0\).
The double affine Hecke algebra \(\mathbf H\) is the unital associative \(\mathbb{C}[q,q^{-1},t,t^{-1}]\)-algebra generated by \(\{t_i,x_\lambda:i\in I\), \(\lambda\in X\}\) modulo the following defining relations:
\[
x_\delta=t,\quad x_\lambda x_\mu=x_{\lambda+\mu}(t_i-q)(t_i+1)=0,
\]
\[
t_it_jt_i\cdots=t_jt_it_j\cdots\text{ if }i\neq j\;(m_{ij}\text{ factors in both products),}
\]
\[
t_ix_\lambda-x_\lambda t_i=0\text{ if }(\lambda:\alpha_i^\vee)=0,\quad t_ix_\lambda-x_{s_i(\lambda)}t_i=(q-1)x_\lambda\text{ if }(\lambda:\alpha_i^\vee)=1,
\]
for all \(i,j\in I\), \(\lambda,\mu\in X\).
One important step of the proof is the construction of a ring homomorphism from \(\mathbf H\) to a ring defined via the equivariant \(K\)-theory of an affine analogue \(\mathcal Z\) of the Steinberg variety. \(\mathcal Z\) is an ind-scheme of ind-infinite type. It comes with a filtration by subsets \({\mathcal Z}_{\leq y}\) with \(y\) in the affine Weyl group \(W\). The subsets \({\mathcal Z}_{\leq y}\) are reduced separated schemes of infinite type, and the inclusions \({\mathcal Z}_{\leq y'}\subset{\mathcal Z}_{\leq y}\) with \(y'\leq y\) are closed immersions. The set \(\mathcal Z\) is endowed with an action of a torus \(A\) which preserves each term of the filtration. For a well-chosen element \(a\in A\), the fixed point set \({\mathcal Z}^a\subset{\mathcal Z}\) is a scheme locally of finite type. Hence there is a convolution ring \(\mathbf K^A({\mathcal Z}^a)\): it is the inductive limit of the system of \({\mathbf R}_A\)-modules \(\mathbf K^A(({\mathcal Z}_{\leq y})^a)\) with \(y\in W\). (Here \({\mathbf R}_A\) means \({\mathbf K}_A(\text{point})\).) The author defines a ring homomorphism \(\Psi_a\colon{\mathbf H}\to\mathbf K^A({\mathcal Z}^a)_a\), where the subscript \(a\) means specialization at the maximal ideal \(J_a\subset{\mathbf R}_A\) associated to \(a\). The map \(\Psi_a\) becomes surjective after a suitable completion of \(\mathbf H\). It is certainly not injective. Using \(\Psi_a\), a standard sheaf-theoretic construction, due to Ginzburg in the case of affine Hecke algebras, provides a collection of simple \(\mathbf H\)-modules. These are precisely the simple integrable modules. -- The paper also give some estimates for the Jordan-Hölder multiplicities of induced modules. simple integrable modules; double affine Hecke algebras; perverse sheaves; linear algebraic groups; Lie algebras; Cartan subalgebras; Borel subalgebras; root systems; affine Weyl groups; simple reflections; pairings; equivariant \(K\)-theory; Jordan-Hölder multiplicities of induced modules Vasserot, Eric, Induced and simple modules of double affine Hecke algebras, Duke Math. J., 126, 2, 251-323, (2005) Hecke algebras and their representations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Lie algebras of linear algebraic groups, Grothendieck groups, \(K\)-theory, etc., Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Induced and simple modules of double affine Hecke 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 In the first part of this article the author proves the following non-solvable base change for Hilbert modular representations:
Theorem 1.1. Let \(F\) be a totally real number field, and let \(\pi\) be a cuspidal
automorphic representation of weight \(k\geq 2\) of \(\text{GL}(2)/F\). Let \(F'\) be a finite solvable extension of a totally real number field containing \(F\). Then there exists a number field \(F''\) containing \(F'\) which is a solvable extension of a totally real field, such that \(F''\) is Galois over \(\mathbb Q\), and such that the representation \(\pi\) admits a base change to \(\text{GL}(2)/F''\). If \(F'\) is a totally real number field, then \(F''\) can be chosen to be a totally real number field.
To show this theorem, the author uses some results from Taylor's papers [\textit{M. Harris, N. Shepherd-Barron} and \textit{R. Taylor}, Ann. Math. (2) 171, No. 2, 779--813 (2010; Zbl 1263.11061)] and [\textit{R. Taylor}, Doc. Math., J. DMV Extra Vol., 729--779 (2006; Zbl 1138.11051)]. Recall that from Langlands and Arthur-Clozel we know that if \(\pi\) is an automorphic representation of \(\text{GL}(n)/F\), where \(F\) is a number field, and \(F'\) is a solvable extension of \(F\), then \(\pi\) admits a base change to \(\text{GL}(n)/F'\).
In the second part of this article, he computes the zeta function of some
``twisted'' quaternionic Shimura varieties in terms of automorphic representations, and as an application of Theorem~1.1 he proves that their zeta function
could be meromorphically continued to the entire complex plane and satisfies a
functional equation. quaternionic Shimura variety; zeta function DOI: 10.5802/afst.1267 Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties Non-solvable base change for Hilbert modular representations and zeta functions of twisted quaternionic Shimura varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors verify the monodromy conjecture [\textit{J. Denef} and \textit{F. Loeser}, J. Am. Math. Soc. 5, No. 4, 705--720 (1992; Zbl 0777.32017)] about topological zeta functions for a large class of singularities that are non-degenerate with respect to their Newton polyhedra, including all such singularities of functions depending on four variables. They explain the novelty of their approach as follows. Thus, it is known that in the case of three variables all singularities close to a non-degenerate one are non-degenerate as well [\textit{A. Lemahieu} and \textit{L. Van Proeyen}, Trans. Am. Math. Soc. 363, No. 9, 4801--4829 (2011; Zbl 1248.14012)]. Next, in the setting of \textit{E. Artal Bartolo} et al. [Quasi-ordinary power series and their zeta functions. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1095.14005)], all singularities close to a quasi-ordinary one are also quasi-ordinary. However, in contrast to these papers, a new phenomenon arises in the four-dimensional case: there are degenerate singularities arbitrarily close to a nonisolated non-degenerate singularity; this is one of the main difficulties of the proof pointed out by the authors.
The paper is divided into several parts. In the introduction, the authors give a detailed review of a number of previously obtained results and the corresponding extensive bibliography, explain the essence of their approach, and formulate the main statements. Then the monodromy conjecture for the topological zeta function and the main properties of Newton polyhedra are discussed, and configurations of faces of the Newton polyhedron that do not ensure the existence of the corresponding pole of the topological zeta function are studied. After that, the authors study face configurations, which, on the contrary, always nontrivially contribute to the multiplicity of the expected monodromy eigenvalue and prove the conjecture for a certain class of non-degenerate singularities in arbitrary dimension. The last section contains a proof of the monodromy conjecture for non-degenerate singularities of functions depending on four variables. In the appendix, it is briefly discussed some basic concepts related to the geometry of the lattice, and the necessary results used in the article. nonisolated singularities; non-degenerate singularities; topological zeta functions; monodromy conjecture; toric varieties; Newton polytopes; non-convenient Newton polyhedra; eigenvalues of monodromy; corners; hypermodular function; lattice geometry Toric varieties, Newton polyhedra, Okounkov bodies, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) On the monodromy conjecture for non-degenerate hypersurfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 paper we introduce a family of functors (called operations) of the category of hypermaps (dessins) preserving the underlying Riemann surface. The considered family of functors include as particular instances the operations considered by \textit{N. Magot} and \textit{A. K. Zvonkin} [ibid. 217, No. 1--3, 249--271 (2000; Zbl 0964.52014)], \textit{D. Singerman} and \textit{R. I. Syddall} [Beitr. Algebra Geom. 44, No. 2, 413--430 (2003; Zbl 1064.14030)], and \textit{E. Girondo} [Exp. Math. 12, No. 4, 463--475 (2003; Zbl 1058.30034)]. We identify a set of 10 operations in the above infinite family which produce vertex-transitive dessins out of regular ones. This set is complete in the following sense: if a vertex-transitive map arises from a regular dessin \(\mathcal{H}\) applying an operation, then it can be obtained from a regular dessin on the same surface (possibly different from \(\mathcal{H}\)) applying one of the 10 operations. The statement includes the classical case when the underlying surface is the sphere. action of a group on a surface; Belyĭ function; dessin; hypermap; map; map covering; orbifold Breda d'Azevedo, A; Catalano, DA; Karabáš, J; Nedela, R, Maps of Archimedean class and operations on dessins, Discret. Math., 338, 1814-1825, (2015) Group actions on combinatorial structures, Compact Riemann surfaces and uniformization, Riemann surfaces; Weierstrass points; gap sequences Maps of Archimedean class and operations on dessins | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an irreducible smooth projective variety of dimension \(n\) over \({\mathbb Z}\) which has an elusive model \({\mathcal X}\) over the field \({\mathbb F}_1\) with one element. The zeta function of \({\mathcal X}\) is \(\zeta_{\mathcal X}(s)=\prod_{i=0}^n(s-i)^{-b_{2i}}\), where \(b_0,\ldots,b_{2n}\) are the Betti numbers of \(X\). The author proves the functional equation
\[
\zeta_{\mathcal X}(n-s)=(-1)^\chi \zeta_{\mathcal X}(s),
\]
where \(\chi=\sum_{i=0}^n b_{2i}\) is the Euler characteristic of \(X_{\mathbb C}\). If \(G\) is a split reductive group scheme of rank \(r\) with \(N\) positive roots, then
\[
\zeta_{\mathcal G}(r+N-s)=(-1)^\chi(\zeta_{\mathcal G}(s))^{(-1)^r}.
\]
\({\mathbb F}_1\)-scheme; zeta function; functional equation Lorscheid, O, Functional equations for zeta functions of \(\mathbb{F}_1\)-schemes, C. R. Math. Acad. Sci. Paris, 348, 1143-1146, (2010) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Varieties over finite and local fields Functional equations for zeta functions of \(\mathbb F_1\)-schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors analyze some particular cases of the Khovanskii's Theorem [\textit{A. G. Khovanskij}, Funct. Anal. Appl. 26, No. 4, 1 (1992; Zbl 0809.13012); translation from Funkts. Anal. Prilozh. 26, No. 4, 57--63 (1992)] in order to give constructive methods to compute effective results on the given sumset structure.
In detail, let \(A\) be a finite subset of \(\mathbb{Z}^d\), the \(h\)-fold sumset of \(A\) is defined by \( hA = \{\texttt{x}_1 + \cdots + \texttt{x}_h : \texttt{x}_i \in A\}. \) Khovanskii proved that there exists a polynomial \(p \in \mathbb{Q}[x]\) of degree at most \(d\) such that \(|hA| = p(h)\) for all \(h>H\), where \(H\) is a suitable integer (called the \textit{phase transition}). Furthermore, if the difference set \(A-A\) generates all of \(\mathbb{Z}^d\) additively, then \(\deg p = d\) and the leading coefficient of \(p\) is the volume of \(\Delta_A\), that is, the convex hull of \(A\).
In this paper, the authors provide an explicit computation of the upper bound on the phase transition when the convex hull of \(A\) is a \(d\)-dimensional simplex, and \(A-A\) generates \(\mathbb{Z}^d\) additively. Moreover, the following interesting results are given:
\begin{itemize}
\item a complete description of the structure of \(hA\), when \(A \subset \mathbb{Z}\), giving an explicit upper bound on the phase transition;
\item a complete description of \(hA\), for all \(h\), in the case that \(A\subset \mathbb{Z}^d\) is small;
\item a description of the structure of \(hA\), providing an explicit upper bound on the phase transition, when \(\Delta_A\) is a simplex;
\end{itemize}
The pivotal idea underlying these arguments is to embed the set \(A\) into a higher-dimensional space: \(\mathcal{C}_A\), the \textit{cone over} \(A\). This object provides information about \(hA\) for all \(h\) simultaneously, thus avoiding the study the structure of \(hA\) individually for each \(h\). Such information is more difficult to be obtained when \(\Delta_A\) is not a simplex.
Finally, the authors make some observations and conjectures to illustrate new opening for further research. Ehrhart theory; iterated sumsets Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Toric varieties, Newton polyhedra, Okounkov bodies, Exact enumeration problems, generating functions, Lattices and convex bodies (number-theoretic aspects), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Khovanskii's theorem and effective results on sumset structure | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The notion of Tannakian category grew out of Grothendieck's theory of motives in his search for a universal cohomology theory for algebraic varieties. A definitely satisfactory theory of motives does not yet exist, the obstruction being Grothendieck's standard conjectures on algebraic cycles. However, for varieties over finite fields one knows, by work of Deligne, Jannsen et. al., that the corresponding category of motives is in fact a (semi-simple, \(\mathbb{Q}\)-linear) Tannakian category. Roughly speaking this means that these motives are the objects of a \(k\)- linear category \({\mathcal C}\) (here e.g. \(k = \mathbb{Q}\), in general \(k\) may be any field) having tensor products, a unit and inverses and satisfying suitable axioms to make it a so-called \(k\)-lineown such that \({\mathcal C}\) is equivalent to the category of finite-dimensional representations in \(k\)-vector spaces of the group \(G\) of tensor automorphisms of the fibre functor \(\omega\).
The general case is much more difficult and it was settled only in the late 1980's by \textit{P. Deligne} in his contribution to the Grothendieck Festschrift [Vol. II, Prog. Math. 87, 111-195 (1990; Zbl 0727.14010)]. In a surprisingly ingenious proof of about fourty pages Deligne showed that for a general Tannakian category \({\mathcal C}\) over the field \(k\) the fibre functors form an affine gerbe over \(\text{Spec} (k)\) in the \textit{fpqc}- topology and \({\mathcal C}\) becomes equivalent to the category of representations of this gerbe.
In the first three sections of the underlying paper these results and related notions are discussed. First, a proof of the neutral case along the lines set out in Deligne's general proof is sketched. The importance of the Barr-Beck theorem giving the equivalence between the categories of certain naturally defined modules and of specific finite type comodules under a \(k\)-coalgebra determined by the fibre functor, thus making possible a co-end construction, is stressed. This coalgebra \(B\) is shown to be a commutative Hopf algebra representing the group functor \(G = \Aut (\omega)\) and \({\mathcal C}\) becomes equivalent to \(\text{Rep}(G)\). Next, the formalism of gerbes and nonabelian second cohomology is extensively discussed. It is shown that a gerbe \({\mathcal G}\) over a site \({\mathcal S}\) with final object \(e\), locally neutralised by an object \(x\) in \({\mathcal G}_ S\) for some covering \(S \to e\), leads to a \((p_ 1^* G, p_ 2^* G)\)-bitorsor \(E = \text{Isom} (p_ 2^*x, p_ 1^*x)\), where \(G = \Aut (x)\) and \(p_ 1, p_ 2 : S \times S \to S\) are the projections, satisfying nice `cocycle' conditions \(\psi\). The pair \((E, \psi)\) is called the bitorsor cocycle. This description determines a transitive groupoid \(\Gamma : (E \twoheadrightarrow S)\). Conversely, one may introduce the notion of torsor under a groupoid \(\Gamma\). These form a stack Tors\((\Gamma)\), and Tors\((\Gamma)\) is a gerbe for transitive \(\Gamma\). In fact, the construction \(\Gamma\to\text{Tors}(\Gamma)\) is quasi-inverse to that which associates to a locally neutralised gerbe \({\mathcal G}\) its bitorsor cocycle (viewed as transitive groupoid \(\Gamma:(E\twoheadrightarrow S))\).
Now the stage is set for the proof of the fact that for a Tannakian category \({\mathcal C}\) the stack \(\text{FIB} ({\mathcal C})\) of fibre functors is indeed a gerbe, i.e. any two fibre functors are locally isomorphic in the \textit{fpqc}-topology (the missing point in Saavedra's proof of the main theorem), or in other words, the associated groupoid \(\Gamma : (\text{Isom} (p_ 2^* \omega, p_ 1^* \omega) \twoheadrightarrow S)\) is transitive in the \textit{fpqc}-topology. The proof of this transitivity is based on Deligne's construction of a tensor product of tensor categories and the fact that this tensor product of two Tannakian categories is itself a tensor category. The proof of the main theorem now follows the lines of the one for neutral Tannakian categories with the coalgebra \(B\) replaced by a coalgebroid \(L\) and the group \(G = \text{Spec} (B) \to \text{Spec} (k)\) replaced by the groupoid \(\Gamma : (E = \text{Spec} (L) \twoheadrightarrow S)\).
The fourth section deals with gerbes in the étale topology. One considers a Tannakian category \({\mathcal C}\) over \(k\) admitting a fibre functor with values in \(K\)-vector spaces, where \(K\) is a finite separable field extension of \(k\). The gerbe \({\mathcal G} = \text{FIB} ({\mathcal C})\) is then a gerbe on \(\text{Spebes} {\mathcal E}\) as group extensions of the form
\[
1 \to G (k^ s) \to {\mathcal E} @> \phi>> \text{Gal} (k^ s/k) \to 1,
\]
split by a local section \(j : \text{Gal} (k^ s/K) \to {\mathcal E}\), where \(k^ s\) is a separable closure of \(k\) and where \(G\) is some algebraic group scheme. Conversely, one recovers the entire bitorsor cocycle from the associated Galois gerbe. As a matter of fact one obtains a bijection between the set of smooth affine gerbes over \(\text{Spec} (k)_{ \text{é}t}\) neutralized over \(\text{Spec} (K)\) and the set of \(K/k\)-Galois gerbes.
The final section gives a very explicit description of a gerbe \({\mathcal G}\), locally neutralized by an object \(x\) in \({\mathcal G}_ S\) and giving a section \(u\) in the induced bitorsor cocycle \(E\), in terms of nonabelian 2-cocycles with values in the \(S\)-group \(G = \Aut (x)\). This leads to a nonabelian \(H^ 2\), not completely identical to Giraud's definition in terms of the band associated to a gerbe. Tannaka-Krein theorem; nonabelian cohomology; Tannakian category; motives; conjectures on algebraic cycles; torsor; bitorsor cocycle; stack; neutral Tannakian categories; algebraic group scheme; bitorsor; gerbe L. BREEN, Tannakian categories, in: Motives, Proc. Symposia pure Math., 55 (I), AMS (1994), pp. 337-376. Zbl0810.18008 MR1265536 Nonabelian homological algebra (category-theoretic aspects), Group schemes, Representation theory for linear algebraic groups, Galois cohomology of linear algebraic groups, Generalizations (algebraic spaces, stacks), (Co)homology theory in algebraic geometry, Monoidal categories (= multiplicative categories) [See also 19D23] Tannakian categories | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a toric Calabi-Yau \(3\)-fold \(X\) the mirror theory lives on a family of curves in \((\mathbb{C}^*)^2\) called the ``mirror curve''. The Eynard-Orantin topological recursion applied to a smooth complex curve produces an infinite tower of free energies \(F_g\) and meromorphic differentials \(W_n^g\). The remodeling conjecture states that the mirror map applied to \(F_g\) and \(W_n^g\) produces the generating functions of the closed and open Gromov-Witten invariants of \(X\) respectively. The open part of the conjecture for \(X=\mathbb{C}^3\) was proved independently by Chen and Zhou. In this note the authors complete the proof for \(\mathbb{C}^3\) by demonstrating that the free energies reproduce the closed invariants given by the celebrated Faber-Pandharipande formula.
The proof relies on methods of Chen and Zhou. The mirror curve to \(\mathbb{C}^3\) is a sphere with \(3\) punctures (pair of pants), and Chen and Zhou expressed its \(W_n^g\) in terms of Hodge integrals. The authors reduce \(F_g\) to Hodge integrals as well, but the computation is more subtle. A subtlety appears also in the relation to matrix models. The Eynard-Orantin recursion applied to the spectral curve of a matrix model exactly reproduces its correlation functions. However, there is a normalization ambiguity in the computation that may lead to discrepancy in the contributions of the constant maps to free energies. The authors show explicitly that this discrepancy is in fact present for the pair of pants, and in contrast to the open part of the conjecture the matrix model machinery can not be used. In conclusion they ask if there is a more direct computation of \(F_g\) that relies on geometry of the pair of pants rather than reduction to Hodge integrals. toric Calabi-Yau; mirror curve; Gromov-Witten invariants; Eynard-Orantin recursion; Hodge integral; spectral curve of a matrix model Bouchard, V; Catuneanu, A; Marchal, O; Sułkowski, P, The remodeling conjecture and the Faber-pandharipande formula, Lett. Math. Phys., 103, 59-77, (2013) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), Relationships between surfaces, higher-dimensional varieties, and physics The remodeling conjecture and the Faber-Pandharipande formula | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Consider a generic point at infinity in the moduli space of connections on a compact Riemann surface. The author shows that the Laplace transform of the family of monodromy matrices has an analytic continuation with locally finite branching. In particular, the convex subset representing the exponential growth rate of the monodromy is a polygon whose vertices are in a subset of points described explicitly in terms of the spectral curve. connection; singular perturbation; turning point; resurgent function; Laplace transform; monodromy; moduli space Simpson, C.: Asymptotics for general connections at infinity. http://www.arXiv.org/abs/math/0311531 Singular perturbations, turning point theory, WKB methods for ordinary differential equations, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation) Asymptotics for general connections at infinity. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 profinite group \(F\) of infinite rank \(m\) is called \textit{quasi-free} if every nontrivial finite split embedding problem for \(F\) has \(m\) distinct proper solutions. Quasi-freeness was introduced in [\textit{D. Harbater} and \textit{K. F. Stevenson}, Adv. Math. 198, No. 2, 623-653 (2005; Zbl 1104.12003)], where the following motivating property was proved: \(F\) is free if, and only if, \(F\) is projective and quasi-free. The main result in that paper was: if \(k((x,y))\) is the Laurent series field in two variables over an arbitrary field \(k\), then its absolute Galois group \(\text{Gal}(k((x,y)))\) is quasi-free (of rank \(\text{card}(k((x,y)))\)). This was used in [\textit{D. Harbater}, J. Reine Angew. Math. 632, 85-103 (2009; Zbl 1192.12004)] to show that, if \(k\) is a separably closed field, then the maximal Abelian extension \(k((x,y))^{ab}\) of \(k((x,y))\) has a free absolute Galois group. Here, \(\text{Gal}(k((x,y))^{ab})\) is quasi-free because \(\text{Gal}(k((x,y)))\) is, thanks to a general result which asserts that the commutator subgroup of a quasi-free group must be quasi-free.
In the present paper, the authors propose the following definition: a profinite group \(F\) of infinite rank \(m\) is called \textit{semi-free} if every nontrivial finite split embedding problem for \(F\) has \(m\) \textit{independent} proper solutions. It is shown that this condition is strictly stronger than quasi-freeness and strictly weaker than freeness. In particular, it can also be combined with projectivity to detect fields with free absolute Galois groups.
Semi-freeness is close enough to quasi-freeness in the sense that some known quasi-free absolute Galois groups turn out to be semi-free indeed. For instance, the authors prove that \(\text{Gal}(k((x,y)))\) is semi-free, for every field \(k\).
On the other hand, the \textit{independence} additional requirement makes possible to deal with some cases of the problem of whether a closed subgroup of a semi-free \(F\) is itself semi-free. The main result of this paper exhibits several situations in which semi-freeness is preserved, although quasi-freeness may not.
Applying their main result, the authors obtain new examples of fields having free absolute Galois groups. These new examples include many field extensions of \(k((x,y))^{ab}\), where \(k\) is a separably closed field. free profinite groups; quasi-free profinite groups; semi-free profinite groups; absolute Galois groups; Laurent series fields; function fields; embedding problems L. Bary-Soroker, D. Haran and D. Harbater, Permanence criteria for semi-free profinite groups, Mathematische Annalen 348 (2010), 539--563. Limits, profinite groups, Separable extensions, Galois theory, Field arithmetic, Coverings of curves, fundamental group, Inverse Galois theory Permanence criteria for semi-free profinite groups. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X^{[n]}\) denote the Hilbert scheme of \(n\) points on a nonsingular projective variety \(X\).
An isomorphism \(g:X \to Y\) of varieties induces an isomorphism \(g^{[n]}: X^{[n]} \to Y^{[n]}\).
\textit{S. Boissière} defined an isomorphism \(\sigma: X^{[n]} \to Y^{[n]}\) to be \textit{natural} if \(\sigma = g^{[n]}\) for some isomorphism \(g:X \to Y\) [Can. J. Math. 64, No. 1, 3--23 (2012; Zbl 1276.14006)].
Using work of \textit{A. Beauville} [J. Differ. Geom. 18, 755--782 (1983; Zbl 0537.53056)], \textit{R. Zuffetti} recently exhibited isomorphisms \(X^{[2]} \cong Y^{[2]}\) that are not natural, with \(X\) and \(Y\) (non-isomorphic) \(K3\) surfaces [Rend. Semin. Mat., Univ. Politec. Torino 77, No. 1, 113--130 (2019; Zbl 1440.14183)].
Defining a variety \(X\) to be \textit{weak Fano} if \(\omega_X^\vee\) is nef and big, the authors prove that if \(X\) is a smooth projective surface that is weak Fano or of general type and \(n\) is an integer, then every automorphism of \(X^{[n]}\) is natural unless \(n=2\) and \(X = C \times D\) is a product of two curves. In the latter case, if \(C,D\) are both rational or both of genus \(g \geq 2\), then there is a unique nonnatural isomorphism of \(X = C \times D\) up to composition with natural automorphisms.
They also prove that every automorphism of \((\mathbb P^n)^{[2]}\) is natural.
The authors also show that if \(X,Y\) are smooth projective surfaces with \(Y\) weak Fano or of general type, then every isomorphism \(X^{[n]} \to Y^{[n]}\) is natural (they assume \(n \geq 3\) if \(Y\) is a product of curves). This can be thought of as an analog to a theorem of \textit{A. Bondal} and \textit{D. Orlov} [Compos. Math. 125, No. 3, 327--344 (2001; Zbl 0994.18007)] and extends results of \textit{T. Hayashi}, who proved this for rational surfaces whose Iitaka dimension of \(\omega_X^\vee\) is at least one [Geom. Dedicata 207, 395--407 (2020; Zbl 1444.14012)]. Hilbert scheme of points; natural isomorphisms; smooth surfaces; automorphisms Parametrization (Chow and Hilbert schemes), Automorphisms of surfaces and higher-dimensional varieties Automorphisms of Hilbert schemes of points on 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 Characteristic elements of the Tits algebra of a real hyperplane arrangement carry information about the characteristic polynomial. We present this notion and its basic properties, and apply it to derive various results about the characteristic polynomial of an arrangement, from Zaslavsky's formulas to more recent results of Kung and of Klivans and Swartz. We construct several examples of characteristic elements, including one in terms of intrinsic volumes of faces of the arrangement. hyperplane arrangement; Möbius function; characteristic polynomial; Tits algebra; characteristic element; intrinsic volumes Configurations and arrangements of linear subspaces, Combinatorial aspects of representation theory, Real algebraic sets Characteristic elements for real hyperplane arrangements | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 the distribution of the traces of the Frobenius endomorphisms of genus \(g\) curves which are quartic non-cyclic covers of \(\mathbb{P}_{\mathbb{F}_q}^1\), as the curve varies in an irreducible component of the moduli space. We show that for \(q\) fixed, the limiting distribution of the traces of Frobenius equals the sum of \(q+1\) independent random discrete variables. We also show that when both \(g\) and \(q\) go to infinity, the normalized trace has a standard complex Gaussian distribution. Finally, we extend these computations to the general case of arbitrary covers of \(\mathbb{P}_{\mathbb{F}_q}^1\) with Galois group isomorphic to \(r\) copies of \(\mathbb{Z}/2\mathbb{Z}\). For \(r=1\) we recover the already known results for the family of hyperelliptic curves. function fields; biquadratic curves; biquadratic covers; number of points over finite fields; arithmetic statistics Curves over finite and local fields, Coverings of curves, fundamental group, Relations with random matrices Statistics for biquadratic covers of the projective line over finite fields. With an appendix by Alina Bucur | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 a digital multisignature scheme, two or more signers are allowed to produce a single signature on a common message, which can be verified by anyone. In the literature, many schemes are available based on the public key infrastructure or identity-based cryptosystem with bilinear pairing and map-to-point (MTP) hash function. The bilinear pairing and the MTP function are time-consuming operations and they need a large super-singular elliptic curve group. Moreover, the cryptosystems based on them are difficult to implement and less efficient for practical use. To the best of our knowledge, certificateless digital multisignature scheme without pairing and MTP hash function has not yet been devised and the same objective has been fulfilled in this paper. Furthermore, we formally prove the security of our scheme in the random oracle model under the assumption that ECDLP is hard. elliptic curve cryptography; certificateless public key cryptosystem; multisignature; map-to-point hash function; bilinear pairing Authentication, digital signatures and secret sharing, Cryptography, Applications to coding theory and cryptography of arithmetic geometry, Data encryption (aspects in computer science), Number theory (educational aspects), Theoretical computer science (educational aspects) A pairing-free certificateless digital multisignature scheme using elliptic curve cryptography | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In diesem Buch werden viele Aspekte der $p$-adischen Analysis angesprochen. Es beginnt mit den Grundlagen über affinoide Algebren und rigid- analytischen Räumen, wobei die Hauptsätze stets nur referiert werden. Anschließend wird die $p$-adische Uniformisierung von Kurven beschrieben. Danach werden analytische Gruppen untersucht und das Bruhat- Tits Gebäude mittels analytischer Räume interpretiert. Im folgenden Kapitel über den Homotopietyp von speziellen analytischen Räumen beschäftigt sich der Autor im wesentlichen mit analytischen Tori, bzw. abelschen Varietäten. Wiederum werden die interessanteren Resultate über $p$-adische Uniformisierung nur zitiert; die Darstellung hat im wesentlichen nur beschreibenden Charakter. In den abschließenden Kapiteln wird nichtarchimedische Spektraltheorie behandelt; Anwendungen dazu werden nicht vorgestellt. Das Buch richtet sich in erster Linie an Experten, die mit den Grundlagen vertraut sind. nonarchimedean spectral theory; Brubat-Tits building; rigid analytic spaces; affinoid algebras V.\ G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Math. Surveys Monogr. 33, American Mathematical Society, Providence 1990. Local ground fields in algebraic geometry, Non-Archimedean analysis Spectral theory and analytic geometry over non-Archimedean fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review concerns the global dimension of the rings of global sections \(D^ \lambda = \Gamma (\mathbb{P}^ n, {\mathcal D}^ \lambda)\) of the sheaves of twisted differential operators \({\mathcal D}^ \lambda\) on \(\mathbb{P}^ n\), which are indexed by \(\mathbb{C}\). It turns out that the answer for \(\text{gldim }D^ \lambda\) can be \(n\), \(2n\), or \(\infty\). These cases arise when \(\lambda \in \mathbb{C} \setminus \mathbb{Z}\), \(n \in \mathbb{Z} \setminus\{ -1, \dots, -n-1\}\), \(n \in \{-1, \dots, -n-1\}\).
For the finite cases the main technique used is the Beilinson-Bernstein localisation theorem in combination with the spectral sequence
\[
H^ q(\mathbb{P}^ n, \text{Ext}^ p({\mathcal F}, {\mathcal G})) \implies \text{Ext}^ n({\mathcal F}, {\mathcal G}).
\]
In the infinite case an ad hoc method is used to produce a module of infinite projective dimension. A complete answer for the global dimension of rings of global sections of twisted differential operators on flag varieties (equivalently, minimal primitive factors of enveloping algebras of semisimple Lie algebras) is known. In detail, suppose that \(\mathfrak g\) is a semisimple Lie algebra with Cartan subalgebra \(\mathfrak h\) and let \(\lambda \in {\mathfrak h}^*\). Associated to \(\lambda\), there is a sheaf of twisted differential operators on the flag variety, \(X\), denoted by \({\mathcal D}^ \lambda\). One writes \(D^{\lambda} := U({\mathfrak g}) \text{ann }M (\lambda) \cong \Gamma (X, {\mathcal D}^ \lambda)\). There is no loss of generality in assuming that \(\lambda\) is antidominant. One says that \(\lambda\) is regular if its stabiliser in the Weyl group is trivial. \textit{T. J. Hodges} and \textit{S. P. Smith} [J. Lond. Math. Soc., II. Ser. 32, 411-418 (1985; Zbl 0588.17009)] showed that if \(\lambda\) is regular then \(\text{gldim }D^ \lambda\) is finite (in fact they found an upper bound and, appealing to an earlier result of Levasseur, pointed out that it is attained). \textit{A. Joseph} and \textit{J. T. Stafford} [Proc. Lond. Math. Soc., III. Ser. 49, 361-384 (1984; Zbl 0543.17004)] showed that \(\text{gldim }D^ \lambda\) is infinite, if \(\lambda\) is singular. \textit{H. Hecht} and \textit{D. Miličić} [Proc. Am. Math. Soc. 108, 249-254 (1990; Zbl 0714.22011)] have also obtained this latter result by showing that the localisation functor \({\mathcal D}^ \lambda \otimes \underline{\phantom m}\) has infinite cohomological dimension.
There do not appear to be analogous general results, for the global dimension of the rings of global sections of twisted differential operators on complete homogeneous spaces, in the literature. global dimension; rings of global sections; sheaves of twisted differential operators; spectral sequence; flag varieties; enveloping algebras of semisimple Lie algebras; Weyl group DOI: 10.1112/blms/24.2.148 Rings of differential operators (associative algebraic aspects), Universal enveloping (super)algebras, Homological dimension in associative algebras, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Arithmetic ground fields for curves, Commutative rings of differential operators and their modules, Sheaves of differential operators and their modules, \(D\)-modules The global dimension of rings of differential operators on projective spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0747.00028.]
The purpose of this paper is to determine explicitly the local structure of the Hilbert scheme of curves in \(\mathbb{P}^ 3\) at certain points. Actually, the main body of this work is devoted to a quite detailed exposition of the various deformation theories of a closed subscheme \(Y\) of a projective scheme \(X = \text{Proj}(S)\), where \(K\) is a field and \(S\) is a graded Noetherian \(K\)-algebra, with \(S_ 0 = K\). The types of deformations are: the deformations of \(Y\) as a subscheme of \(X\), the deformations of the ideal sheaf \({\mathcal I}_ Y\) as a sheaf of \({\mathcal O}_ X\)-modules, the deformations of the ideal \(I(Y)\) as a homogeneous \(S\)-module, and ``conical deformations'' of \(Y\). This exposition builds on previous work by \textit{O. A. Laudal} [in Algebra, algebraic topology and their interactions, Proc. Conf., Stockholm 1983, Lect. Notes Math. 1183, 218-240 (1986; Zbl 0597.14010)] and \textit{J. O. Kleppe} [Math. Scand. 45, 205-231 (1979; Zbl 0436.14004)], but the material is presented here with a view towards explicit calculations. Various conditions are worked out under which these deformation theories are isomorphic. The theory is then applied to the calculation of the deformation theory of particular space curves, yielding explicit local equations for some singular points of the Hilbert scheme and explicit examples of obstructed curves (i.e. corresponding to singular points of the Hilbert scheme) of maximal rank. The same examples were found independently by \textit{Bolondi}, \textit{Kleppe} and \textit{Miró-Roig}.
The question remains whether curves with seminatural cohomology (i.e. curves \(C\) such that for each \(n\), the sheaf \({\mathcal J}_ C(n)\) has at most one nonzero cohomology group) are unobstructed. isomorphic deformation theories; deformation theory of space curves; Hilbert scheme of curves; deformation theories of a closed subscheme Charles H. Walter, Some examples of obstructed curves in \?³, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 324 -- 340. Plane and space curves, Parametrization (Chow and Hilbert schemes), Formal methods and deformations in algebraic geometry, Deformations and infinitesimal methods in commutative ring theory Some examples of obstructed curves in \(\mathbb{P}^ 3\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this paper is to provide a brief introduction to algebraic stacks, and to give several constructions of the moduli stack of Higgs bundles on algebraic curves. It is based in part on lecture notes prepared for the summer school ``The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles'' at the Institute for Mathematical Sciences at the National University of Singapore in July of 2014. The first construction is via a bootstrap method from the algebraic stack of vector bundles on an algebraic curve. This construction is motivated in part by Nitsure's GIT construction of a projective moduli space of semi-stable Higgs bundles, and the authors describe the relationship between Nitsure's moduli space and the algebraic stacks constructed here. The third approach is via deformation theory, where they directly construct the stack of Higgs bundles using Artin's criteria. The authors have tried to provide enough motivation for stacks that the reader is inclined to proceed to the definitions, and a sufficiently streamlined presentation of the definitions that the reader does not immediately stop at that point. The topic are treated in the language algebraic geometry, i.e., schemes. However, one of the objectives is to have a presentation accessible to those working in other fields, particularly complex geometers, and most of the presentation can be made replacing the word scheme with the words complex analytic space (or even manifold) and étale cover with open cover. This is certainly the case up though, and including, the definition of a stack in Section 3. The one possible exception to this rule is the topic of algebraic stacks Section 6, for which definitions in the literature are really geared towards the category of schemes. In order to make the presentation as accessible as possible, in Section 5 and Section 6 the authors provide definitions of algebraic stacks that make sense for any presite; in particular, the definition gives a notion of an algebraic stack in the category of complex analytic spaces. The final sections of this survey (Section 10 and Section 11) study algebraic stacks infinitesimally, with the double purpose of giving modular meaning to infinitesimal motion in an algebraic stack and introducing Artin's criterion as a means to prove stacks are algebraic. Higgs bundles are treated as an extended example, and along the way the authors give cohomological interpretations of the tangent bundles of the moduli stacks of curves, vector bundles, and morphisms between them. In the end they obtain a direct proof of the algebraicity of the stack of Higgs bundles, complementing the one based on established general algebraicity results given earlier. algebraic spaces; stacks; relationships of algebraic curves with integrable systems; vector bundles on curves and their moduli Generalizations (algebraic spaces, stacks), Relationships between algebraic curves and integrable systems, Vector bundles on curves and their moduli, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry An introduction to moduli stacks, with a view towards Higgs bundles on algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this article, the author extends and generalizes the approach to develop a completely geometric Voronoï theory which he began in [J. Reine Angew. Math. 482, 93--120 (1997; Zbl 1011.53035)] where he introduced the concept of generalized systoles. In the present article, the author introduces the notion of nondegenerate point and systematically studies points satisfying particular properties such as perfection and eutaxy whose definitions are motivated by those from the classical Voronoï theory of lattices. Particular emphasis is laid upon the study of onfigurations and of finiteness results. As applications, the author obtains new results on the invariants of Bergé-Martinet and of Hermite-Humbert that are attached to number fields, and on Riemann surfaces.
The theoretical framework of this theory is laid out in \S 1 of the present paper. Let \(E\) be a finite-dimensional \({\mathbb R}\)-vector space, \({\mathcal F}\) a finite set of vectors in \(E\) and \(K\) their convex hull. \({\mathcal F}\) is said to be perfect if it generates \(E\) affinely, and eutactic if the origin is in the affine interior of \(K\). Now let \(V\) be a smooth and connected variety with \({\mathcal C}^1\)-functions \(f_s:V\to{\mathbb R}\) indexed by some set \(C\). \((f_s)_{s\in C}\) is called a system of length functions if for every \(p\in V\), every neighborhood \(U\) of \(p\) and every \(L\in{\mathbb R}\) one has \(f_s >L\) on \(U\) for almost all \(s\in C\). The generalized systole \(\mu (p)\) in a point \(p\in V\) is now defined to be \(\mu (p)=\min_{s\in C}f_s(p)\). Then \(p\in V\) is called perfect resp. eutactic if the family of differentials \((df_s(p))_{s\in S_p}\) is perfect resp. eutactic in the cotangent space \(T^*_pV\), where \(S_p=\{ s\in C; f_s(p)=\mu (p)\}\), and (strictly) extreme if \(\mu\) has a (strict) local maximum in \(p\). \(V\) can be partitioned into so-called minimal classes, where two points \(p,q\in V\) belong to the same class iff \(S_p=S_q\).
Let now \(V\) be equipped with a connection (assumed to be geodesic). A \({\mathcal C}^1\)-function \(f:V\to{\mathbb R}\) is said to be (strictly) convexoïdal if any critical point of \(f\) restricted to a geodesic is a (strict) local minimum. Now suppose in addition that \((f_s)_{s\in C}\) is a family of convexoïdal length functions on \(V\). Then a point \(p\in V\) is said to be nondegenerate if for every geodesic emanating from \(p\) there is at least one \(f_s\) that is strictly convexoïdal on that geodesic in some neighborhood of \(p\). With these definitions, it follows readily that if \(V\) is equipped with a family of convexoïdal length functions, then perfect points are always nondegenerate (Proposition 1.4), and that Voronoï's theorem (i.e. a point is extreme iff it is perfect and eutactic) holds iff every extreme point is nondegenerate iff every extreme point is perfect. This holds in particular whenever the length functions are strictly convexoïdal (Proposition 1.5). Furthermore, the set of points that are eutactic and nondegenerate is discrete, so in particular also the set of points that are perfect and eutactic (Corollary 1.7).
The author then obtains certain somewhat technical results on isolation of points that are eutactic and nondegenerate by which he can prove that if \(V\) is a smooth subvariety of \(P_n\), the space of unimodular positive definite symmetric \(n\times n\) matrices, defined by polynomials with algebraic coefficients in \({\mathbb R}\), then there are only finitely many perfect points for \(\mu^D\) (relative to \(V\)) and they are all algebraic over \({\mathbb Q}\) as are all nondegenerate eutactic points under some additional assumption on \(V\) (Corollary 1.12). Here, \(\emptyset\neq D\subset{\mathbb Z}^n \setminus\{ 0\}\) and \(\mu^D(A)=\min_{s\in D}A[s]\) for \(A\in P_n\). The author points out that the interest in this result lies in the fact that it applies readily to all natural and interesting types of lattices, for example autodual lattices.
In \S 2, the author applies these foundational results in certain situations to obtain finiteness results, notably to \(P_n\) as defined above and connected complete totally geodesic subvarieties thereof, and in particular their \(\rho\)-invariant forms for some representation \(\rho :\Pi\to \text{GL}_n({\mathbb Z})\) for a finite group \(\Pi\) (such \(\rho\)-invariant forms correspond in a natural way to \(\Pi\)-lattices). The main results concern finiteness of minimal classes containing weakly eutactic points and finiteness of perfect or eutatic nondegenerate points (Theorem 1), criteria for nondegeneracy (Theorem 2), the rank of minimal vectors for perfect points (Theorem 3), finiteness of the number of minimal classes modulo a certain action in the hermitian, symmetric or antisymmetric bilinear case (Theorem 4).
Using the methods developed in this paper, the author shows in Theorem 5 that the invariant of Bergé-Martinet satisfies Voronoï's theorem (also generalized to the hermitian case). He then interprets the invariant of Hermite-Humbert for Humbert forms over the ring of integers \({\mathcal O}_L\) in some algebraic number field \(L\). In this new setting, he obtains Voronoï's theorem for a certain \(\mu_L\) which is defined in a way so that the notions of perfection and eutaxy coincide with the classical ones for Humbert forms.
A final application concerns systoles of Riemann surfaces (Theorem 6). Euclidean lattice; geometric Voronoï theory; perfect lattice; eutaxy; systole; length function; convexoïdal; configuration of minimal vectors; Humbert form; invariant of Hermite-Humbert; invariant of Bergé-Martinet; Riemann surface C. Bavard, ''Théorie de Voronoï géométrique. Propriétés de finitude pour les familles de réseaux et analogues,'' Bull. Soc. Math. France 133, 205--257 (2005). Quadratic forms (reduction theory, extreme forms, etc.), Quadratic forms over global rings and fields, Lattices and convex bodies (number-theoretic aspects), Minima of forms, Abelian varieties and schemes, Conformal metrics (hyperbolic, Poincaré, distance functions), Riemann surfaces, Connections (general theory), Global differential geometry Voronoï's geometric theory. Finiteness properties for families of lattices and similar objects | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(KF_n\) be the algebra of non-commutative Laurent polynomials in \(n\) variables over a commutative ring \(K\). For an element \(a\in KF_n\), the constant term \(\mathrm{Tr}(a)\) can be regarded as a ``trace'' of \(a\), and accordingly, there is a ``characteristic polynomial''
\[
P_a(t)=\mathrm{det}(1-ta)=\mathrm{exp}\bigl(-\sum_{k=1}^\infty \frac{\mathrm{Tr}(a^k)}{k}t^k\bigr)\in K[[t]].
\]
For \(K=\mathbb C\), \textit{M. Kontsevich} [``Noncommutative identities'', talk at Mathematische Arbeitstagung 2011, Bonn; \url{arXiv:1109.2469}] recently proved that \(P_a\) is algebraic, that is, \(P_a\in \overline{\mathbb C(t)}\cap\mathbb C[[t]]\subset\overline{\mathbb C((t))}\). His sketch of proof reduced the statement to the series
\[
F_a(t)=\sum_{k=1}^\infty\mathrm{Tr}(a^k)t^k
\]
which is known to be algebraic. As \(P_a'(t)=-\frac{F_a}{t}P_a\), the result then reduces to a known special case of the Grothendieck conjecture on algebraicity. The present paper generalizes Kontsevich's result to ``zeta function'' \(P_a(t)\) where \(a\) is replaced by a square matrix over \(\mathbb Q F_n\). noncommutative formal power series; language; zeta function; algebraic function C. Kassel and C. Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, \textit{Ramanujan J.}, to appear. Noncommutative algebraic geometry, Exact enumeration problems, generating functions, Algebraic theory of languages and automata, Combinatorics on words, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Algebraic functions and function fields in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Algebraicity of the zeta function associated to a matrix over a free group 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 Let G be a group and X be a G-scheme. The q-th equivariant K-group \(K_ q(G,X)\) of X is defined to be the q-th K-group associated with the exact category of G-vector-bundles on X. The aim of this paper is to prove the following Adams-Riemann-Roch formula for elements \(x\in K(G,X):=\oplus_{q\geq 0}K_ q(G,X):\) Let \(f:X\to Y\) be a projective G- morphism of complete intersection and let \(\psi^ j\) be the j-th Adams operation (a certain polynomial in exterior power operations). Then \(f_*(\psi^ j(\theta^ j(\check T_ f)^{-1}\cdot x))=\psi^ j(f_*(x))\). Here \(\theta^ j(\check T_ f)^{-1}\in K_ 0(G,X)\) denotes the Adams multiplier (only depending on j and f) and \(f_*: K(G,X)\to K(G,Y)\) denotes the so-called Lefschetz trace (a generalization of the Euler characteristic).
In the nonequivariant case this theorem is the central part in the proof of the general Grothendieck-Riemann-Roch theorem (see, e.g., \textit{C. Soulé} [Can. J. Math. 37, 488-550 (1985; Zbl 0575.14015)]). In order to define \(\psi^ j\), a natural \(\lambda\)-structure on K(G,X) is constructed. In the affine case this is done by generalizing \textit{H. L. Hiller}'s constructions [J. Pure Appl. Algebra 20, 241-266 (1981; Zbl 0471.18007)] to the equivariant situation and in the general case by means of a generalized Jouanolou trick [\textit{J. P. Jouanolou}, Lect. Notes Math. 341, 293-316 (1973; Zbl 0291.14006)]. The proof of the theorem is based on an equivariant version of the deformation to the normal cone and on an equivariant excess intersection formula. G-scheme; equivariant K-group; exact category of G-vector-bundles; Adams- Riemann-Roch formula; Adams operation; Adams multiplier; Lefschetz trace; deformation to the normal cone; equivariant excess intersection formula B. Köck, Das Adams-Riemann-Roch-Theorem in der höheren äquivarianten \(K\)-Theorie , J. Reine Angew. Math. 421 (1991), 189-217. \(K\)-theory of schemes, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry Das Adams-Riemann-Roch-Theorem. (The Adams-Riemann-Roch theorem) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We prove that the Leray spectral sequence in rational cohomology for the quotient map \(U_{n,d}\to U_{n,d}/G\) where \(U_{n,d}\) is the affine variety of equations for smooth hypersurfaces of degree \(d\) in \(\mathbb P^n(\mathbb C)\) and \(G\) is the general linear group, degenerates at \(E_2\). geometric quotient; hypersurfaces; Leray spectral sequence Peters ( C.A.M. ) , Steenbrink ( J.H.M. ). - Degeneration of the Leray spectral sequence for certain geometric quotients , Moscow Math. J. , Vol. 3 , n^\circ 3 , 2003 , p. 1085 - 1095 . Zbl 1049.14035 Group actions on varieties or schemes (quotients), Hypersurfaces and algebraic geometry, Algebraic moduli problems, moduli of vector bundles, (Co)homology theory in algebraic geometry, Spectral sequences in algebraic topology Degeneration of the Leray spectral sequence for certain geometric quotients | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 Chebyshev curve \(C(a,b,c,\varphi)\) has a parametrization of the form \(x(t)=T_a(t)\); \(y(t)=T_b(t)\); \(z(t)=T_c(t+\varphi)\), where \(a\),\(b\),\(c\) are integers, \(T_n(t)\) is the Chebyshev polynomial of degree \(n\) and \(\varphi\in\mathbb{R}\). When \(C(a,b,c,\varphi)\) is nonsingular, it defines a polynomial knot. We determine all possible knot diagrams when \(\varphi\) varies. When \(a\),\(b\),\(c\) are integers, \((a,b)=1\), we show that one can list all possible knots \(C(a,b,c,\varphi)\) in \(\tilde{\mathcal{O}}(n^2)\) bit operations, with \(n=abc\). We give the parameterizations of minimal degree for all two-bridge knots with 10 crossings and fewer. zero dimensional systems; Chebyshev curves; Lissajous curves; Lissajous knots; polynomial knots; Chebyshev polynomials; minimal polynomial; Chebyshev forms Knots and links in the 3-sphere, Symbolic computation and algebraic computation, Plane and space curves Computing Chebyshev knot diagrams | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We give an explicit formula of the normalized Mumford form which expresses the second tautological line bundle by the Hodge line bundle defined on the moduli space of algebraic curves of any genus. This formula is represented as an infinite product which is a higher genus version of the Ramanujan delta function under the trivialization by normalized abelian differentials and Eichler integrals of their products. Furthermore, this formula gives a universal expression of the normalized Mumford form as a computable power series with integral coefficients by the moduli parameters of algebraic curves. Therefore, one can describe the behavior of this form and hence of the Polyakov string measure around the Deligne-Mumford boundary. normalized Mumford form; moduli space of algebraic curves; Ramanujan delta function; Polyakov string measure Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Arithmetic ground fields for curves, Arithmetic varieties and schemes; Arakelov theory; heights, Local ground fields in algebraic geometry, Theta functions and curves; Schottky problem, String and superstring theories; other extended objects (e.g., branes) in quantum field theory An explicit formula of the normalized Mumford form | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 TS/ST correspondence relates the spectral theory of certain quantum mechanical operators, to topological strings on toric Calabi-Yau threefolds. So far the correspondence has been formulated for real values of Planck's constant. In this paper we start to explore the validity of the correspondence when \(\hslash\) takes complex values. We give evidence that, for threefolds associated to supersymmetric gauge theories, one can extend the correspondence and obtain exact quantization conditions for the operators. We also explore the correspondence for operators involving periodic potentials. In particular, we study a deformed version of the Mathieu equation, and we solve for its band structure in terms of the quantum mirror map of the underlying threefold. topological string; supersymmetric gauge theory; periodic potentials; spectral theory String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Topological field theories in quantum mechanics, Yang-Mills and other gauge theories in quantum field theory, Supersymmetric field theories in quantum mechanics, Toric varieties, Newton polyhedra, Okounkov bodies, \(3\)-folds, Calabi-Yau manifolds (algebro-geometric aspects), General topics in linear spectral theory for PDEs, Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) The complex side of the TS/ST correspondence | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth projective surface over a base field of characteristic \(p\). If \(p=0\), Bogomolov showed that any rank two vector bundle \(E\) on \(X\) with \(\delta(E)=c^ 2_ 1(E)-4c_ 2(E)>0\) is unstable and hence Chern numbers of \(X\) satisfy \(c^ 2_ 1\leq 4c_ 2\) [\textit{F. A. Bogomolov}, Math. USSR, Izv. 13, 499-555 (1979); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 42, 1227-1287 (1978; Zbl 0439.14002)]. --- In general Bogomolov's results do not hold if \(p>0\).
The author proves the following. (1) If \(p=2\) and \(X\) can be lifted to \(W_ 2(k)\), the ring of Witt vectors of length 2, then the above results hold for \(X\). --- (2) If \(X\) is not of general type, then any rank 2 vector bundle \(E\) on \(X\) with \(\delta(E)>0\) is unstable, except possibly for \(X\) properly quasi-elliptic. --- (3) If \(X\) is of general type with minimal model \(X_ 1\), if \(\delta(E)>K^ 2_{X_ 1}/(p-1)^ 2\) and if \(E\) is semistable then \(X\) is purely inseparably uniruled. --- As a corollary it follows that for all surfaces of special type except quasielliptic ones the Kodaira vanishing theorem [\textit{T. Ekedahl}, Publ. Math., Inst. Hautes Étud. Sci. 67, 97-144 (1988; Zbl 0674.14028)] and Reider's analysis [\textit{I. Reider}, Ann. Math., II. Ser. 127, No. 2, 309- 316 (1988; Zbl 0663.14010)] of adjoint linear systems are valid. Finally there are some new results on pluricanonical maps of surfaces of general type. characteristic \(p\); Bogomolov inequality; Chern numbers; vanishing theorem; adjoint linear systems; surfaces of general type Shepherd-Barron, N. I., Unstable vector bundles and linear systems on surfaces in characteristic \(p\), Invent. Math., 106, 2, 243-262, (1991) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Divisors, linear systems, invertible sheaves, Finite ground fields in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Unstable vector bundles and linear systems on surfaces in characteristic p | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We survey recent developments in the classification theory of complex function fields. The subject dates back to the italian school of algebraic geometry at the beginning of the 20th century. In each complex function field of transcendence degree 2, they constructed a so called ``minimal surface'', which is unique for most fields, and they investigated its geometry. By the 1960's, these results were reconsidered and provided with a solid algebraic and analytic basis by the schools of Kodaira, Shafarevich and Zariski.
``Minimal models'' are the higher dimensional analog of ``minimal surfaces''. They were constructed in dimension 3 by Mori and his collaborators in the 1980's, and in dimension 4 by Shokurov in 2000. In 2006, \textit{C. Birkar, P. Cascini, J. McKernan} and \textit{C. Hacon} [Existence of minimal models for varieties of log general type, \url{arXiv:math.AG/0610203}] announced the existence of minimal models for fields of general type of arbitrary dimension. We will survey these developments. complex function fields; minimal surface; minimal models Minimal model program (Mori theory, extremal rays) Recent Advances in the theory of minimal models | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 what follows k denotes the field of real or complex numbers. If \(D,0\subset k^ p\), 0 is the discriminant variety of the versal unfolding of an analytic function germ \(f: k^ n,0\to k,0\), it is of some interest to classify smooth functions h: D,0\(\to k,0.\)
In Commun. Pure Appl. Math. 29, 557-582 (1976; Zbl 0343.58003), \textit{V. I. Arnol'd} classified such germs in the case when they are generic and f is a simple singularity of type \(A_ k\), \(D_ k\), \(E_ k\). Full details were, however, only given in the case of \(A_ k\). In this paper, using the basic vector fields of \textit{K. Saito} [Invent. Math. 14, 123- 142 (1971; Zbl 0224.32011)] tangent to the discriminant D, we give another proof of Arnol'd's results; the method of proof here is fairly unsophisticated. We then generalize these results to cover a wide collection of weighted homogeneous functions, which include the 14 weighted homogeneous unimodal germs, as well as the simple singularities. Furthermore we use these basic fields to prove that the discriminants of the simple elliptic singularities \(\tilde E_ k\), for \(k=6,7,8\), are topologically trivial along the modulus parameter. We also discuss the existence of stable germs on these discriminants. In the appendix we give a self-contained proof that Saito's vector fields are tangent to the discriminant. discriminant variety; versal unfolding; analytic function germ; stable germs; Saito's vector fields DOI: 10.1112/jlms/s2-30.3.551 Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Singularities in algebraic geometry, Local complex singularities Functions on discriminants | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 Chow polytope Ch(X) is a common generalization of the Newton polytope of a hypersurface, and of the matroid polytope of a flat in the projective space. It carries information on asymptotic behavior of X under the action of the complex torus of and depends only on tropicalisation of X.
The paper extends the definition of Chow polytope to abstract tropical cycles in a tropicalised toric varieties. The author gives an example of tropical hypersurface with the same Chow polytope, i.e., the map from varieties to Chow polytopes is not injective. As an application of developed technique the author proves that all tropical varieties of degree one come as Bergman complexes of matroids.
The paper is well structured and all definitions and results are illustrates by good examples. Chow polytope; Chow variety; polytope subdivision; tropical variety; tropical intersection theory; tropical linear space A. Fink, Tropical cycles and chow polytopes. Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry 54(1), 13-40 (2013) , Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Tropical cycles and Chow polytopes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The following theorem is proved: Assume that \(\mathrm{Lie }G_k\) is of dimension \(\leq 1\) and that \(Y_k\) does not arise as the push-forward of a torsor over \(X_k\) under a proper subgroup scheme of \(G_k\). Then, there exist a smooth formal curve \(X\) over \(R\) and a \(G\)-torsor \(Y \to X\) whose special fiber is the \(G_k\)-torsor \(Y_k \to X_k\).
Here \(R\) is a complete local ring with residue field \(k\) of positive characteristic \(p >0\), \(G\) is a finite, flat and of finite presentation, commutative group scheme over \(R\) and \(X_k\) is a smooth curve over \(k\).
This article extends some earlier works of the authors [J. Algebra 318, No. 2, 1057--1067 (2007; Zbl 1135.14036)].
In the acknowledgements, the authors indicate that they `would like to thank M. Raynaud who suggested the problem to us and for his encouragement.'
The interesting feature of the article is the use of the equivariant cotangent complex by \textit{L. Illusie} [Complexe cotangent et déformations. II. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0238.13017)], results on algebraic spaces and schemes by \textit{M. Raynaud} and \textit{L. Gruson} [Invent. Math. 13, 1--89 (1971; Zbl 0227.14010)], the analogue of the stack by \textit{D. Abramovich} et al. [J. Algebr. Geom. 20, No. 3, 399--477 (2011; Zbl 1225.14020); corrigendum ibid. 24, No. 2, 399--400 (2015)] and moduli of Galois \(p\)-covers by \textit{D. Abramovich} and \textit{M. Romagny} [Algebra Number Theory 6, No. 4, 757--780 (2012; Zbl 1271.14032)].
In the last section of the paper under review, the authors show that these techniques can also be applied to the theory moduli of \(p\)-covering of curves. Several interesting examples are given, in particular on relation with Jacobians. lifting of torsors; finite flat group scheme; algebraic curve; cotangent complex Local ground fields in algebraic geometry, Curves over finite and local fields, Coverings of curves, fundamental group, (Equivariant) Chow groups and rings; motives Deformation of torsors under monogenic group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let K be an algebraic function-field in one variable over an algebraically closed constant field k of characteristic \(p\geq 0\), denote by \(g=g_ k\) the genus of K and let \(K_ 0\) be a rational subfield of K. It is a classical result of Hurwitz that if \(k={\mathbb{C}}\) and \(g\geq 2\), then the group Aut(K/\({\mathbb{C}})\) of automorphisms of K over \({\mathbb{C}}\) is finite and the order \(ord(Aut(K/{\mathbb{C}}))\leq 84(g-1).\) The method of proof works over any k of characteristic 0. \textit{H. L. Schmid} [J. Reine Angew. Math. 179, 5-15 (1938; Zbl 0019.00301)] considered the cases in which \(p>0\) and, using work of \textit{F. K. Schmidt}, proved that the group Aut(K/k) is finite. \textit{P. Roquette} [Math. Z. 117, 157-163 (1970; Zbl 0194.353)] showed that if \(p>g+1\geq 3\), then the Hurwitz bound \(ord(Aut K/k)\leq 84(g-1)\) holds also in this case, except in the case \(p\geq 5\), and \(K=k(x,y)\), \(y^ 2=x^ p-x\). Moreover he showed that in that case \(G=Aut(K/k)\) is a central extension of \(C_ 2\) (the cyclic group of order 2) by PGL(2,p).
It is natural to ask which finite groups arise as automorphism groups of function fields. Evidently one requires \(g\geq 2\) and in the case \(k={\mathbb{C}}\) \textit{L. Greenberg} [Discontin. Groups Riemann Surf., Proc. 1973 Conf. Univ. Maryland, 207-226 (1974; Zbl 0295.20053)] showed that every finite group is realizable as the automorphism group of some K/\({\mathbb{C}}\) and that is true in the case \(p>0\) also (for references and a history of the problem, see Chapter 1 of this thesis).
Consider in particular the case when K is hyperelliptic. The author addresses the question as to which finite groups are realizable as some Aut(K/k) in that case and in certain natural generalizations of it (see below). A hyperelliptic function-field K possesses exactly one rational subfield \(K_ 0\) such that \([K:K_ 0]=2\) and there exists a subgroup Z of \(G=Aut(K/k)\) such that Z is in the centre Z(G) and \(Z=Gal(K/K_ 0)\). But then Aut(K/k) is a central extension of Z by a finite subgroup of \(Aut(K_ 0/k)\) and all those groups actually arise as automorphism groups of hyperelliptic function fields [\textit{K. Iwasawa}, Ann. Math., II. Ser. 58, 548-572 (1953; Zbl 0051.266)].
The author gives a complete solution of the problem of determining all those extensions, including their defining equations, and also of the following more general one, which is suggested by the foregoing. A function-field of type \(F[G_ 0| q,p]\) is a function-field (in one variable) over an algebraically closed field of constants k of characteristic \(p\geq 0\) such that: 1) there exist finite subgroups Z and G of Aut(K/k), with Z a subgroup of the centre Z(G) of G, \(Z\neq G\); \(Z\cong C_ q\) where \(q\neq p\) is a prime and \(C_ q\) denotes the cyclic group of order q; \(G/Z\cong G_ 0\); and 2) the fixed field of Z is rational. For fields of that type, G is a central extension of \(Z\cong C_ q\) by a finite subgroup \(G_ 0\) of \(Aut(K_ 0/k)\) and so the first task is to determine all such rational function fields. The group \(G_ 0\) acts on the places of \(K_ 0/k\) and one finds the orbits of \(G_ 0\). The author determines the central extensions of \(C_ q\) by finite subgroups of \(Aut(K_ 0/k)\) in the cases when \(G_ 0\) is an elementary Abelian q-group or a semi-direct product of an Abelian q-group with a cyclic group. He goes on to give a detailed study of function-fields \(F[G_ 0| q,p]\) and in particular of extensions of automorphisms \(\sigma \in Aut(K_ 0/k)\) to K and of the ramification type G(K,G) of K, which is defined as follows. With the foregoing notation, let \(K_ 1\) be the fixed field of G and \(K_ 0\) the fixed field of Z and let \(Q_ 1,...,Q_ r\) be the places of \(K_ 1\) ramified in \(K_ 0/K_ 1\), \(e_ 1,...,e_ r\) their ramification indices in \(K/K_ 1\), then \(T(K,G)=(G;e_ 1,...,e_ r)\). If \(q\neq p\), then the isomorphism type of G is determined completely by T(K,G), but if \(q=p\) then that is no longer the case and he obtains the appropriate analogues.
The thesis is largely self contained, the necessary results concerning group extensions, for example, being included and the greater part of it is concerned with a very detailed discussion of the function-field of type \(F[G_ 0| q,p]\) and their defining equations, for groups G. finite groups; automorphism groups of function fields; hyperelliptic function-field R. Brandt, Über die Automorphismengruppen von algebraischen Funktionenkörpern, PhD thesis, Universität Essen, 1988. Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry On the groups of automorphisms of algebraic function fields. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author studies the geometry of Uhlenbeck partial compactification of orthogonal instanton spaces and the \(K\)-theoretic Nekrasov partition functions. For \(G = \text{Sp}(n)\), a moment map \(\mu: {\mathbf N} \to \text{Lie}(G)\) is constructed where \({\mathbf N}\) is a certain symplectic subspace of the vector space \({\mathbf M}\) consisting of ordinary ADHM quiver representations coming from framed \(\text{SU}(N)\)-instantons with instanton number \(2n\). The main result states that when \(N \geq 4\), the moment map \(\mu\) is flat, and the regular locus \(\mu^{-1}(0)^{\text{reg}}\) is Zariski dense and open in \(\mu^{-1}(0)\). Moreover, if \(N = 4\), then \(\mu^{-1}(0)\) is an irreducible normal variety; if \(N > 4\), then \(\mu^{-1}(0)\) is a reduced variety with \(n +1\) irreducible components. It follows that for \(K = \text{SU}(N, \mathbb R)\) with \(N \geq 5\), the Uhlenbeck partial compactification \(\mathcal U^K_n\) of the moduli space \(\mathcal M^K_n\) of framed \(K\)-instantons over \(S^4\) with instanton number \(n\) is an irreducible normal variety of dimension \(2n(N - 2)\), and its smooth locus is precisely \(\mathcal M^K_n\). Another consequence asserts that the \(K\)-theoretic Nekrasov partition function for any simple classical group other than \(\text{SO}(3, \mathbb R)\) can be interpreted as a generating function of Hilbert series of the instanton moduli spaces.
Section~2 is devoted to the moment map \(\mu\), and proves the main result by assuming both the flatness of \(\mu\) for \(N \geq 4\) and the normality of \(\mu^{-1}(0)\) in the instanton number \(n = 2\) case and \(N \geq 5\). The flatness of \(\mu\) and the normality of \(\mu^{-1}(0)\) for \(N \geq 5\) are then verified in Section~3 and Section~4 respectively. In Section~5, the author applies similar ideas to investigate the ordinary ADHM data and Sp-data. moduli spaces; orthogonal instantons; quiver variety; Hamiltonian reduction; Nekrasov partition function; equivariant \(K\)-group J. Choy, Geometry of the Uhlenbeck partial compactification of orthogonal instanton spaces and the K-theoretic Nekrasov partition functions, arXiv:1606.00707. Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Yang-Mills and other gauge theories in quantum field theory Geometry of Uhlenbeck partial compactification of orthogonal instanton spaces and the \(K\)-theoretic Nekrasov partition 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 See the review in Zbl 0744.12003. Tate module; abelian variety; function field; Galois group; transcendence degree О группах галуа функциональных полей над полями конечного типа над, УМН, 46, 5-281, 163-164, (1991) Separable extensions, Galois theory, Abelian varieties and schemes On Galois groups of function fields over fields of finite type over \(\mathbb{Q}\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 each point \(p\in \left( [0,1]\cap \mathbb Q\right) ^2\) we define and classify a family \(\left( C(p;k)\right) _{k\in \mathbb Z}\) of rational conics passing through \(p\), each one containing a sequence of rational points \(\left( x^k_n,y^k_n\right) _{n\geq -k}\) converging to \(p\) and such that the continued fraction of \(x^k_n\) and \(y^k_n\) are reversal to one another. Moreover the points \(\left( x^k_n,y^k_n,k\right) _{k\in \mathbb Z,n\geq -k}\) are contained in a quartic surface \(Q\)(\(p\)) whose intersection with horizontal planes are conics. continued fraction; palindrome; mirror function; conic Continued fractions, Rational points Palindromic pairs in conics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The purpose of this paper is to construct Newton-Okounkov bodies for graded cancellative torsion-free semigroups \(S\), thanks to the fact that they can be embedded in \(\mathbb{R}^n\), by extending the construction of \textit{K. Kaveh} and \textit{A. G. Khovanskii} [Ann. Math. (2) 176, No. 2, 925--978 (2012; Zbl 1270.14022)]. This is indeed possible, and many of the most important properties of this remain true in the more algebraic context, as Theorem 3.9 ensures; remarkably, the volume of the Newton-Okounkov body associated to \(S\) governs the growth rate of the Hilbert function of \(S\).
Moreover, the authors manage to generalize in Theorem 5.7 the equality between volumes of Newton-Okounkov bodies and volumes of line bundles for the bodies associated to \(S\)-graded algebras.
The paper finishes with an appendix explaining the filtered Newton-Okounkov bodies in the setting of the paper under consideration. Newton-Okounkov body; Hilbert function; semigroup Toric varieties, Newton polyhedra, Okounkov bodies, Commutative semigroups, Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Newton-Okounkov theory in an abstract setting | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(V_{n,d}\subseteq\mathbb P^N\), \(N:=\binom{n+d}{n}-1\), be the order \(d\) Veronese embedding of \(\mathbb P^n\), \(X_{n,d}:= T(V_{n,d})\subseteq \mathbb P^N\) the tangent developable of \(V_{n,d}\) and \(S^{s-1}(X_{n,d})\subseteq\mathbb P^N\) the \(s\)-secant variety of \(X_{n,d}\), i.e. the closure in \(\mathbb P^N\) of the union of all \((s-1)\)-linear spaces spanned by \(s\) points of \(X_{n,d}\). \(S^{s-1}(X_{n,d})\) has expected dimension \(\min\{N,(2n+1)s-1\}\). Catalisano, Geramita and Gimigliano conjectured that \(S^{n-1}(X_{n,d})\) has always the expected dimension, except when \(d=2\), \(n\geq 2s\) or \(d=3\) and \(n=2,3,4\). In this paper we prove their conjecture when \(n=4\) and \(n=5\), \(d\geq 4\), and an asymptotic case of the conjecture for all \(n\geq 6\). tangent developable; secant variety; Veronese variety; fat point; zero-dimensional scheme; postulation Projective techniques in algebraic geometry On the secant varieties to the tangent developable of Veronese 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 Keeping in mind subsequent applications to the spectral theory and to the development of an analog of the Lax-Phillips scattering theory for one-dimensional (discrete and differential) periodic operators, in \textit{B. S. Pavlov} and \textit{S. I. Fedorov} [Leningr. Math. J. 1, No. 2, 447-490 (1990); translation from Algebra Anal. 1, No. 2, 132-168 (1989; Zbl 0722.47011)] Pavlov and the author made an attempt to construct the basic objects of harmonic analysis in the case of a doubly connected domain in the spirit of the book by \textit{N. K. Nikol'skiĭ} [Lectures on the shift operator, ``Nauka'', Moscow (1980; Zbl 0508.47001); English transl. of `Treatise on shift operator. Spectral function theory', Springer-Verlag, Berlin-New York (1986; Zbl 0587.47036)]; later, the author elaborated the same approach in the case of an arbitrary finitely connected domain with nondegenerate boundary components. Together with the theory of Toeplitz operators in such domains [see \textit{K. F. Clancey}, Ill. J. Math. 35, No. 2, 286-311 (1991; Zbl 0806.46060)], as well as with some other studies the author's approach showed that the use of function theory on compact Riemann surfaces (the Schottky doubles of planar domains) may be fruitful. It turned out that along with the Hardy spaces of single-valued functions in a domain, an important role is played by certain spaces of multivalued functions (the \(g\)-dimensional real torus of Hardy spaces \(H_{\vec\lambda}^2\), \(\vec\lambda \in \mathbb R^g/\mathbb Z^g\), where \(g +1\) is the connectivity of the domain under consideration): these functions are said to be modulus-automorphic or character-automorphic. Such spaces arise naturally in connection with factorization problems, invariant subspaces, subnormal operators, extremal polynomials, the Nevanlinna-Pick interpolation, and spectral analysis of almost periodic Jacobian matrices. These spaces can be regarded from various viewpoints. In \textit{J. A. Ball} and \textit{K. F. Clancey} [Integral Equations Oper. Theory 25, No. 1, 35-57 (1996; Zbl 0867.30038)], an explicit correspondence between the torus of character-automorphic Hardy spaces and the torus of spaces associated with the divisors of representing measures in a domain was discussed (the functions belonging to the latter spaces are single-valued, but are allowed to have finitely many poles in the domain). \textit{V. Vinnikov} and \textit{D. Alpay} [C. R. Acad. Sci., Paris, Sér. I 318, , No. 12, 1077-1082 (1994; Zbl 0820.46020)] treat these spaces in a different (but equivalent) way, namely, as Hilbert spaces of differentials of order 1/2.
The author prefers an approach involving multiplicative multivalued functions. The main goal in this paper is to give more or less complete description of the properties of these spaces from the standpoint of harmonic analysis and carry the main results and constructions of the papers by \textit{B. S. Pavlov} and \textit{S. I. Fedorov} [loc. cit.] and \textit{S. I. Fedorov} [Math. USSR, Sb. 70, No. 1, 263-296 (1991); translation from Mat. Sb. 181, No. 6, 833-864 (1990; Zbl 0715.47042) and Math. USSR, Sb. 70, No. 2, 297-339 (1991); translation from Mat. Sb. 181, No. 7, 867-909 (1990; Zbl 0715.47043)] over to the character-automorphic Hardy spaces. harmonic analysis; spectral theory; Lax-Phillips scattering theory; periodic operators; Riemann surfaces; doubly connected domain; Schottky doubles of planar domains; character-automorphic Hardy spaces; torus of spaces; multiplicative multivalued functions S. I. Fedorov, On harmonic analysis in a multiply connected domain and character-automorphic Hardy spaces, Algebra i Analiz 9 (1997), no. 2, 192 -- 240 (Russian); English transl., St. Petersburg Math. J. 9 (1998), no. 2, 339 -- 378. \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables, Harmonic functions on Riemann surfaces, Differentials on Riemann surfaces, Analytic theory of abelian varieties; abelian integrals and differentials On harmonic analysis in a multiply connected domain and character-automorphic Hardy 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 \textit{B. Gross} and \textit{D. Zagier} [Invent. Math. 84, 225-320 (1986; Zbl 0608.14019)]\ proved a formula which relates the first derivative of the \(L\)-function of a modular form \(f\) of weight 2 on \(\Gamma_0 (N)\) and the Néron-Tate height of a Heegner point on the \(f\)-part of the Jacobian \({\mathcal J}_0 (N)\). A \(p\)-adic version of this formula was later found by \textit{B. Perrin-Riou} [Invent. Math. 89, 455-510 (1987; Zbl 0645.14010)].
Now let \(f\) be a modular form of even weight \(2r> 2\). The author proves, under suitable hypotheses, a \(p\)-adic version of the Gross and Zagier formula in this context (Theorem A): the first derivative of a \(p\)-adic \(L\)-function at the central point is related to the \(p\)-adic height of a Heegner cycle. The proof of Theorem A closely follows Perrin-Riou's article. Some arguments are however different, mainly because an archimedean analogue of the Gross and Zagier formula for higher weight modular forms is still lacking. Ideas of \textit{J.-L. Brylinski}'s article [Duke Math. J. 59, 1-26 (1989; Zbl 0702.14016)]\ are of great importance here. The author uses Theorem A and his own generalization of Kolyvagin's method of Euler systems to modular forms of even weight to obtain a (weak) form of the conjecture of Beilinson and Bloch in this situation (Theorem B). modular form; \(p\)-adic \(L\)-function; \(p\)-adic height; Heegner cycle; Euler systems; conjecture of Beilinson and Bloch J. Nekovář, On the \textit{p}-adic height of Heegner cycles, Math. Ann. 302 (1995), no. 4, 609-686. Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, 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 On the \(p\)-adic height of Heegner cycles | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be a number field, \(\mathcal O_K\) its ring of integers, \(\mathcal X\) a projective scheme over \(\text{Spec}\mathcal O_K\) with smooth generic fiber, \(\overline{\mathcal L}\) an ample invertible sheaf on \(\mathcal X\) with positively-curved hermitian metric, and \(\mu\) a strictly positive measure on \(\mathcal X(\mathbb C)\) invariant under complex conjugation. The \(L^2(\mu)\) metrics on \(\Gamma(\mathcal X,\mathcal L^{\otimes n})\) provide hermitian metrics on these \(\mathcal O_K\)-modules for all \(n\in\mathbb N\).
Let \(\Sigma\) be a closed subscheme of \(\mathcal X\) of relative dimension \(0\). The image of the restriction map \(\Gamma(\mathcal X,\mathcal L^{\otimes n})\to \Gamma(\Sigma,\mathcal L^{\otimes n}\bigr| _\Sigma)\) is then given the quotient hermitian metrics, and its resulting Arakelov degree defines the quantity \(h(\Sigma;n)\). The latter may be regarded as an arithmetic Hilbert-Samuel function. This paper first proves an explicit asymptotic formula for \(h(\Sigma;n)\) as \(n\to\infty\), in the case where \(\mathcal X\) is the projective space associated to a hermitian vector sheaf over \(\text{Spec}\mathcal O_K\) and \(\overline{\mathcal L}=\overline{\mathcal O(1)}\).
If one further restricts to \(\mathcal X=\mathbb P^n_K\) and \(\Sigma\) reduced, associated to points \(p_1,\dots,p_l\), then work of \textit{M. Laurent} [in: Approximations diophantiennes et nombres transcendants, C. R. Colloq., Luminy/ Fr. 1990, 215--238 (1992; Zbl 0773.11047)] shows that if \(A_n\) is the map \(K[X_0,\dots,X_N]_n\to K^l\) given by evaluation at the \(p_i\), then \(h\bigl(\bigwedge^l A_n\bigr)=n\sum_{i=1}^l h(p_i) +\chi(\mathcal O_\Sigma)+o(1)\). The present paper shows that \(\chi(\mathcal O_\Sigma)\) is given explicitly in terms of the difference between the scheme \(\Sigma\) and its normalization. arithmetic Hilbert-Samuel function; interpolation matrix Arithmetic varieties and schemes; Arakelov theory; heights, Heights Heights of zero-dimensional subschemes of projective space. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper continues the authors' investigations [see \textit{K. Altmann} and \textit{J. A. Christophersen}, Manuscr. Math. 115, No. 3, 361--378 (2004; Zbl 1071.13008) and \textit{M. Haiman} and \textit{B. Sturmfels}, J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)] on the graph \({\mathcal G}\) of monomial ideals in the polynomial ring \(R=k[x_1,\dots,x_n]\), \(k\) a field. \({\mathcal G}\) is the infinite graph with the monomial ideals in \(R\) as vertex set. Two monomial ideals \(M_1\) and \(M_2\) are connected by an edge if there exists an ideal \(I\) in \(R\) such that the set of all initial monomial ideals of \(I\), with respect to all term orders, is precisely \(\{M_1,M_2\}\). \(I\) is called edge providing in this case. It is well known that \(I,M_1,M_2\) have many invariants in common. Each invariant yields a stratification of \({\mathcal G}\). A first proposition concerns the subgraph \({\mathcal G^r}\) obtained by restriction to artinian ideals of colength \(r\): Each such stratum is a connected component of \({\mathcal G}\). The main result (theorem 8) characterizes edge providing ideals as ``very homogeneous'':
There exists upto multiples a single \(c\in\mathbb Z^n\) such that \(I\) is \(A\)-graded for \(A=\mathbb Z^n/c\mathbb Z\).
This allows to define the Schubert scheme \(\Omega_c(M_1,M_2)=\Omega(M_1,M_2)\) of all \(A\)-homogeneous edge providing ideals connecting \(M_1\) and \(M_2\). A thorough analysis of the settings yields an algorithm that computes, for given \(M_1\) and \(M_2\), the direction \(c\) and \(\Omega(M_1,M_2)\) as affine scheme. More generally, the authors consider \(A\)-homogeneous ideals for arbitrary gradings \(\deg\:\mathbb Z^n\to A\) (including the standard one). Define \(h_I\:A\to \mathbb N\) as the Hilbert function of \(I\), i.e., \(h_I(a),\;a\in A\), is the \(k\)-dimension of the \(a\)-homogeneous part of \(I\). In the above situation, for \(I\in \Omega(M_1,M_2)\) the Hilbert functions of \(I\), \(M_1\) and \(M_2\) coincide. For positive gradings, i.e., \(\mathbb N^n\cap \text{ker}(\deg)=(0)\), this is also the general situation:
\(\Omega_c(M_1,M_2)\not=\emptyset\) implies \(\deg(c)=0\) and hence the Schubert schemes describe an essential part of the multigraded Hilbert scheme \(\text{Hilb}_h\) of all \(A\)-homogeneous ideals with given Hilbert function \(h\). This part is sufficient to detect connectedness: Over \(k=\mathbb R\) or \(k=\mathbb C\), \(\text{Hilb}_h\) is connected if and only if the induced subgraph \({\mathcal G}(\text{Hilb}_h)\) is connected.
Section 4 discusses properties of the Schubert schemes for square-free monomial ideals. The results are more technical and continue the investigations started by \textit{K. Altmann} and \textit{J. A. Christophersen} [loc. cit.]. In particular, it turns out that neighboring square-free ideals are connected by a generalization of the bistellar flip construction [see, e.g., \textit{O. Viro}, Proc. Workshop Differential Geometry Topology, Alghero 1992, World Scientific. 244--264 (1993; Zbl 0884.57015) or \textit{D. Maclagan} and \textit{R. R. Thomas}, Discrete Comput. Geom. 27, No. 2, 249--272 (2002; Zbl 1073.14503)]. The paper ends with a list of open problems about the graph \({\mathcal G}\) and the Schubert schemes. graph of monomial ideals; multigraded Hilbert scheme; Schubert scheme; Gröbner bases; Gröbner degenerations; Stanley-Reisner ideals DOI: 10.1016/j.jpaa.2004.12.030 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Parametrization (Chow and Hilbert schemes), Grassmannians, Schubert varieties, flag manifolds The graph of monomial ideals | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The present work applies Dwork's methods to the p-adic study of exponential sums defined over a finite field of characteristic p. In the first section, we apply the pre-cohomological theory to obtain a general estimate for the p-divisibility of exponential sums of the type \((*)\quad S(\bar f,V,\Psi_ q)=\sum_{x\in V({\mathbb{F}}_ q)}\Psi_ q(\bar f(x))\) where V is an arbitrary affine variety defined over \({\mathbb{F}}_ q\), \(V({\mathbb{F}}_ q)\) denotes the \({\mathbb{F}}_ q\)-rational points of V, \(\bar f\) is a regular function on V, and \(\Psi_ q\) is an arbitrary additive character on \({\mathbb{F}}_ q\). The estimate is given in terms of the dimension of the ambient space, and the degree of polynomials which define \(\bar f\) and V. The work generalizes the result of Katz which gives a best-possible estimate for the sum (*) in the case \(\Psi_ q\) is the trivial character (equivalently, \(\Psi_ q\) is non-trivial but f is the zero function). Thus Katz' estimate is a best-possible estimate for the p-divisibility of N(V), the number of \({\mathbb{F}}_ q\) rational points of V. These questions on p-divisibility trace their origins to a problem posed by Artin and solved by Chevalley and Warning. The results may be interpreted as specifying a sharp lower bound for the ''first slope'' of the Newton polygon of the L-function associated with the sum (*).
A finer measure of the p-adic behavior of this L-function is given by the shape of its Newton-polygon. Dwork showed that for nonsingular projective hypersurfaces the Newton polygon of the zeta function lies over its Hodge polygon. Katz' result showed that the same relationship holds for the first slopes of the two polygons in the case of a nonsingular projective complete intersection; at the same time, he formulated a general conjecture which was subsequently proved by Mazur using crystalline cohomology. In section 2, we prove the analogue for exponential sums of Dwork' result. In particular, we determine a sharp lower-bound for the Newton polygon of the L-function associated with certain exponential sums (*) where V is \({\mathbb{A}}^ n\) or the complement in \({\mathbb{A}}^ n\) of the coordinate hypersurface defined by \(X_ 1X_ 2...X_ n=0\), and where the leading form of f satisfies a nonsingularity hypothesis. One can show that these lower bounds are best-possible, being attained in the case of \(\bar f(x)=\sum^{n}_{i=1}\bar c_ iX^ d_ i\in {\mathbb{F}}_ q[X_ 1,...,X_ n],\) a diagonal form, or in the case of \(\bar f(\)X) a deformation of a diagonal form by lower-degree terms when \(p\equiv 1 (\bmod d)\). In fact, the methods suggest that when \(p\equiv 1 (mod d),\) and \(\bar f\) is a polynomial with regular form \(\bar f_ d\) then the Newton polygons of \(L(\bar f,T)^{(-1)^{n-1}}\) and \(L(\bar f_ d,T)^{(-1)^{n-1}}\) in general coincide. This is in striking contrast, to the behavior of these L-functions when \(p\not\equiv 1 (\bmod d)\). finite ground field; exponential sums; Newton polygon of the L-function Sperber, S., On the \textit{p}-adic theory of exponential sums, Am. J. math., 108, 2, 255-296, (1986) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry, Exponential sums On the p-adic theory of exponential sums | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 a special class of bordered algebraic varieties that are contained in the bidisk \(\mathbb D^2\subset\mathbb C^2\). A non-empty set \(V\subset\mathbb C^2\) is a distinguished variety if there is a polynomial \(p\in\mathbb C[z,w]\) such that \(V = \{(z,w)\in\mathbb D^2 : p(z,w) =0\}\) and such that \({\overline V}\cap\partial(\mathbb D^2) ={\overline V}\cap(\partial\mathbb D)^2\). This condition means that the variety exits the bidisk through the distinguished boundary of the bidisk, the torus. Notice that if \(V\) is a distinguished variety, for each \(z\) in the unit disk \(\mathbb D\), the number of points \(w\) satisfying \((z,w)\in V\) is constant (except perhaps at a finite number of multiple points, where the \(w\)'s must be counted with multiplicity). So the following definition makes sense: A distinguished variety is of rank \((m,n)\) if there are generically \(m\) sheets above every first coordinate and \(n\) above every second coordinate. For positive integers \(m\) and \(n\), let
\[
U =\left(\begin{matrix} A&B\\ C&D \end{matrix} \right):\mathbb C^m\oplus\mathbb C^n\to\mathbb C^m\oplus\mathbb C^n\tag{1}
\]
be an \((m + n)\)-by-\((m + n)\) unitary matrix. The moduli space for distinguished varieties of rank \((m, n)\) is a quotient of the space of \((m + n)\)-by-\((m + n)\) unitaries. Let us write \(\mathcal U_n^m\) to denote the set of \((m + n)\)-by-\((m + n)\) unitaries decomposed as in (1). The main result of this paper is a parametrization of distinguished varieties of rank (2,2). The authors address the question of when two different unitaries in \(\mathcal U_2^2\) give rise to the same distinguished variety. This is equivalent to asking when two rational matrix inner functions are isospectral. The number of open problems are stated. bordered algebraic varieties; matrix valued functions; transfer function; isospectral rational matrix inner functions; unitaries; geometrically equivalent distinguished varieties Jim Agler and John E. McCarthy, Parametrizing distinguished varieties, Recent advances in operator-related function theory, Contemp. Math., vol. 393, Amer. Math. Soc., Providence, RI, 2006, pp. 29 -- 34. Moduli, classification: analytic theory; relations with modular forms, Ideal boundary theory for Riemann surfaces, Theorems of Hahn-Banach type; extension and lifting of functionals and operators, Interpolation between normed linear spaces, Linear operator methods in interpolation, moment and extension problems, Proceedings, conferences, collections, etc. pertaining to functional analysis, Analytic subsets and submanifolds Parametrizing distinguished 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 It is shown that the relative distance in Frobenius norm of a real symmetric order-\(d\) tensor of rank-two to its best rank-one approximation is upper bounded by \(\sqrt{1-(1-1/d)^{d-1}}\). This is achieved by determining the minimal possible ratio between spectral and Frobenius norm for symmetric tensors of border rank two, which equals \((1-1/d)^{(d-1)/2}\). These bounds are also verified for arbitrary real rank-two tensors by reducing to the symmetric case. symmetric tensors; spectral norm; rank-one approximation; rank-two tensors Multilinear algebra, tensor calculus, Norms of matrices, numerical range, applications of functional analysis to matrix theory, Secant varieties, tensor rank, varieties of sums of powers, Real polynomials: analytic properties, etc. Maximum relative distance between real rank-two and rank-one tensors | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Based on the good properties of bilinear pairings on elliptic curves, many good ID-based cryptographic schemes have been proposed. However, in these proposed schemes, the private key generator (PKG) should be assumed to be trusted, while in real environment, this assumption does not always hold. To overcome this weakness, the threshold technology was used to devise a secure ID-based signcryption scheme. Because the threshold technology is adopted not only in the master key management but also in the group signature, the scheme can achieve high security and resist some malicious attacks under a certain threshold. attacks; identity-based cryptography; threshold scheme; signcryption; bilinear pairings S. Duan, Z. Cao, and R. Lu, Robust ID-based threshold signcryption scheme from pairings, Proc. 2004 International Conference on Information Security, Shanghai, China, ACM ISBN: 1-58113-955-1, 2004: 33--37. Authentication, digital signatures and secret sharing, Applications to coding theory and cryptography of arithmetic geometry, Cryptography Robust ID-based threshold signcryption scheme from pairings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study the QRT-maps associated with the family of biquadratic curves \(C_{d} (K)\) with equations
\[
x^2 y^2 - dxy - 1 + K(x^2 + y^2) = 0,
\]
where \(d>0\) and \(K\in \mathbb{R}\). By using the Prime Number Theorem and the geometry of elliptic cubics, they determine the periods of periodic orbits of the dynamical systems defined by these QRT-maps, and prove sensitivity to initial conditions.
The main results of the paper are:
Theorem 1.
(1) For any \(d > 0\) and \(K\neq 0\), the map \(T_d |_{C_{d}(K)}\) is conjugated to a rotation on the unit circle. Moreover, both the levels \(K\neq 0\) for which all the initial conditions give rise to trajectories that fill densely \(C_{d} (K)\) and the levels for which this curve is filled by periodic points (with the same period), are dense. As a consequence, the periodic points for \(T_{d}\) are dense in \(\mathbb{R}^{2}\), and also the non-periodic ones.
(2) For every \(d > 0\), there exists an integer \(N(d)\) such that every integer \(n \geq N(d)\) is a minimal period of some point for the QRT-map \(T_{d}\).
(3) Every integer \(n \geq 3\) is the minimal period for \(T_{d}\) of some initial point for some \(d\).
(4) There is no point of period \(2\), nor real finite fixed point, but it is possible to extend continuously \(T_{d}\) to \((0, 0)\), with image \((0, 0)\).
Theorem 2. The dynamical system associated with the QRT-map \(T_{d}\) is sensitive to initial conditions. More precisely:
(1) For every \(0 < K_1 < K_2\), there exists a constant \(k_d (K_1, K_2) > 0\) such that for every \(K \in [K_1, K_2]\), for every point \(M_{0} \in C_{d} (K)\) and every neighbourhood \(V\) of \(M_{0}\) there exists a point \(M_{1} \in V\) such that \[d\left(T^{n}_{d}(M_{1}), T^{n}_{d}(M_{0})\right)> k_{d}(K_1, K_2)\] for an infinite values of \(n\) (uniform sensitivity);
(2) For \(K < 0\), for every point \(M_{0}\) of \(C_{d} (K)\) there exists a constant \(k_d (M_0)\) such that for every neighbourhood \(V\) of \(M_{0}\) there exists a point \(M_{1} \in V\) such that \[d\left(T^{n}_{d}(M_{1}), T^{n}_{d}(M_{0})\right)> k_{d}(M_{0})\] for infinitely many values of \(n\) (pointwise sensitivity). QRT-maps; periods; integrable discrete systems; pointwise sensitivity; uniform sensitivity Completely integrable discrete dynamical systems, Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.), Dynamical systems involving maps of the circle, Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics, Periodic and quasi-periodic flows and diffeomorphisms, Rational and birational maps, Birational automorphisms, Cremona group and generalizations The periodic orbits of a dynamical system associated with a family of QRT-maps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0588.00015.]
The author discusses the following problem: Let \(p_ i(x_ 1,...x_ n)\) \((i=1,...,m)\) be real homogeneous polynomials of degree \(d_ i\). One wants to get information on the dimension of the spaces \(D_ k\) of homogeneous polynomials f of degree k which are solutions of the system of partial differential equations \(p_ i(\frac{\partial}{\partial x_ 1},...,\frac{\partial}{\partial x_ n})\cdot f=0\quad (i=1,...,m)\). For this purpose the author introduces the sheaf \({\mathcal K}\) of relations between the \(p_ i:\)
\[
0\to \quad {\mathcal K}\to \oplus^{m}_{i=1}{\mathcal O}_{{\mathbb{P}}^{n-1}}(-d_ i)\quad \to^{(p_ 1,...,p_ n)}\quad {\mathcal O}_{{\mathbb{P}}^{n-1}}\quad \to \quad 0.
\]
It turns out that dim \(D_ k=\dim H^ 1({\mathbb{P}}^{n-1},{\mathcal K}(k))\) whenever the polynomials \(p_ 1,...,p_ m\) have no common zero in complex projective space \({\mathbb{P}}^{n-1}\). This relation is used to translate results about the cohomology of reflexive sheaves on \({\mathbb{P}}^{n-1}\) to results about systems of homogeneous partial differential equations. The paper under consideration is a survey of the results contained in the author's articles in Trans. Am. Math. Soc. 279, 125-142 (1983; Zbl 0523.14019) and ''Vector bundles on complex projectives spaces and systems of partial differential equations. I'', Trans. Am. Math. Soc. 298, 537-548 (1986). multivariate splines; cohomology of reflexive sheaves; systems of homogeneous partial differential equations Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Spline approximation, Partial differential equations and systems of partial differential equations with constant coefficients Some applications of algebraic geometry to systems of partial differential equations and to approximation 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 study the problem of placing effective upper bounds for the number of zeroes of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of a given dimension \(n\) having \(m\) singular points. As a function of \(n,m\), this bound turns out to be double exponential in the precise sense explained in the paper. As a corollary, we obtain a solution of the so-called restricted infinitesimal Hilbert 16th problem, an explicit upper bound for the number of isolated zeroes of Abelian integrals which is polynomially growing as the Hamiltonian tends to the degeneracy locus. This improves the exponential bounds recently established by A. Glutsyuk and Yu. Ilyashenko. Fuchsian systems; oscillation; zeros; semialgebraic varieties; effective algebraic geometry; monodromy Binyamini, G.; Yakovenko, S., Polynomial bounds for the oscillation of solutions of Fuchsian systems, Université de Grenoble. Annales de l'Institut Fourier, 59, 2891-2926, (2009) Oscillation, growth of solutions to ordinary differential equations in the complex domain, Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.), Effectivity, complexity and computational aspects of algebraic geometry, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) Polynomial bounds for the oscillation of solutions of Fuchsian 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 See the preview in Zbl 0527.12015. automorphism groups of algebraic function fields; realization of group as Galois group; Galois theory Separable extensions, Galois theory, Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry, Finite automorphism groups of algebraic, geometric, or combinatorial structures, Representations of groups as automorphism groups of algebraic systems Zur Realisierbarkeit endlicher Gruppen als Automorphismengruppen algebraischer Funktionenkörper | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0614.00006.]
The purpose of the paper is to give information on a certain smooth compactification of the space of all morphisms of a given degree from \({\mathbb{P}}^ 1\) to a Grassmann variety. This scheme is the Grothendieck Quot scheme of quotients of a trivial vector bundle on \({\mathbb{P}}^ 1\). We compute the additve and the multiplicative structure of its Chow ring and identify the ample cone and the corresponding projective embeddings. families of rational curves; smooth compactification; Grassmann variety; Quot scheme; Chow ring Strømme, S A, On parametrized rational curves in Grassmann varieties., Lecture Notes in Math, 1266, 251-272, (1987) Grassmannians, Schubert varieties, flag manifolds, Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, Algebraic moduli problems, moduli of vector bundles On parametrized rational curves in Grassmann 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 \(k\) be a field and \(G\) be a finite group acting on the rational function field \(k(x_g : g \in G)\) by \(k\)-automorphisms defined as \(h(x_g) = x_{hg}\) for any \(g, h \in G\). We denote the fixed field \(k(x_g : g \in G)^G\) by \(k(G)\). Noether's problem asks whether \(k(G)\) is rational (= purely transcendental) over \(k\). It is well-known that if \(\mathbb{C}(G)\) is stably rational over \(\mathbb{C}\), then all the unramified cohomology groups \(H_{\mathrm{nr}}^i(\mathbb{C}(G), \mathbb{Q} / \mathbb{Z}) = 0\) for \(i \geq 2\). The first two authors with \textit{B. E. Kunyavskii} [Asian J. Math. 17, No. 4, 689--714 (2013; Zbl 1291.13012)] showed that, for a \(p\)-group of order \(p^5 (p\): an odd prime number), \(H_{\mathrm{nr}}^2(\mathbb{C}(G), \mathbb{Q} / \mathbb{Z}) \ne 0\) if and only if \(G\) belongs to the isoclinism family \(\Phi_{10}\). When \(p\) is an odd prime number, \textit{E. Peyre} [Invent. Math. 171, No. 1, 191--225 (2008; Zbl 1155.12003)] and the authors [J. Algebra 458, 120--133 (2016; Zbl 1348.14032)] exhibit some \(p\)-groups \(G\) which are of the form of a central extension of certain elementary abelian \(p\)-group by another one with \(H_{\mathrm{nr}}^2(\mathbb{C}(G), \mathbb{Q} / \mathbb{Z}) = 0\) and \(H_{\mathrm{nr}}^3(\mathbb{C}(G), \mathbb{Q} / \mathbb{Z}) \ne 0\). However, it is difficult to tell whether \(H_{\mathrm{nr}}^3(\mathbb{C}(G), \mathbb{Q} / \mathbb{Z})\) is non-trivial if \(G\) is an arbitrary finite group. In this paper, we are able to determine \(H_{\mathrm{nr}}^3(\mathbb{C}(G), \mathbb{Q} / \mathbb{Z})\) where \(G\) is any group of order \(p^5\) with \(p = 3, 5, 7\). Theorem 1. Let \(G\) be a group of order \(3^5\). Then \(H_{\mathrm{nr}}^3(\mathbb{C}(G), \mathbb{Q} / \mathbb{Z}) \ne 0\) if and only if \(G\) belongs to the isoclinism family \(\mathrm{\Phi}_7\). Theorem 2. If \(G\) is a group of order \(3^5\), then the fixed field \(\mathbb{C}(G)\) is rational if and only if \(G\) does not belong to the isoclinism families \(\Phi_7\) and \(\Phi_{10}\). Theorem 3. Let \(G\) be a group of order \(5^5\) or \(7^5\). Then \(H_{\mathrm{nr}}^3(\mathbb{C}(G), \mathbb{Q} / \mathbb{Z}) \ne 0\) if and only if \(G\) belongs to the isoclinism families \(\Phi_6, \Phi_7\) or \(\Phi_{10}\). Theorem 4. If \(G\) is the alternating group \(A_n\), the Mathieu group \(M_{11}, M_{12}\), the Janko group \(J_1\) or the group \(P S L_2(\mathbb{F}_q), S L_2(\mathbb{F}_q), P G L_2(\mathbb{F}_q)\) (where \(q\) is a prime power), then \(H_{\mathrm{nr}}^d(\mathbb{C}(G), \mathbb{Q} / \mathbb{Z}) = 0\) for any \(d \geq 2\). Besides the degree three unramified cohomology groups, we compute also the stable cohomology groups. unramified cohomology groups; stable cohomology groups; rationality problems; Noether's problem; unramified Brauer groups; permutation negligible classes; Lyndon-Hochschild-Serre spectral sequence Actions of groups on commutative rings; invariant theory, Rationality questions in algebraic geometry, Cohomology of groups, Brauer groups of schemes, Integral representations of finite groups Degree three unramified cohomology groups and Noether's problem for groups of order 243 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Fix an integer partition \(\lambda\) that has no more than \(n\) parts. Let \(\beta\) be a weakly increasing \(n\)-tuple with entries from \(\{1,\dots,n\}\). The flagged Schur function indexed by \(\lambda\) and \(\beta\) is a polynomial generating function in \(x_1,\dots,x_n\) for certain semistandard tableaux of shape \(\lambda\). Let \(\pi\) be an \(n\)-permutation. The type A Demazure character (key polynomial, Demazure polynomial) indexed by \(\lambda\) and \(\pi\) is another such polynomial generating function. \textit{V. Reiner} and \textit{M. Shimozono} [J. Comb. Theory, Ser. A 70, No. 1, 107--143 (1995; Zbl 0819.05058)] and then \textit{A. Postnikov} and \textit{R. P. Stanley} [J. Algebr. Comb. 29, No. 2, 133--174 (2009; Zbl 1238.14036)] studied coincidences between these two families of polynomials. Here their results are sharpened by the specification of unique representatives for the equivalence classes of indexes for both families of polynomials, extended by the consideration of more general \(\beta\), and deepened by proving that the polynomial coincidences also hold at the level of the underlying tableau sets. Let \(R\) be the set of lengths of columns in the shape of lambda that are less than \(n\). Ordered set partitions of \(\{1,\dots,n\}\) with block sizes determined by \(R\), called \(R\)-permutations, are used to describe the minimal length representatives for the parabolic quotient of the \(n\)th symmetric group specified by the set \(\{1,\dots,n-1\}\setminus R\). The notion of 312-avoidance is generalized from \(n\)-permutations to these set partitions. The \(R\)-parabolic Catalan number is defined to be the number of these. Every flagged Schur function arises as a Demazure polynomial. Those Demazure polynomials are precisely indexed by the \(R\)-312-avoiding \(R\)-permutations. Hence the number of flagged Schur functions that are distinct as polynomials is shown to be the \(R\)-parabolic Catalan number. The projecting and lifting processes that relate the notions of 312-avoidance and of \(R\)-312-avoidance are described with maps developed for other purposes. Catalan number; flagged Schur function; Demazure character; key polynomial; pattern avoiding permutation; symmetric group parabolic quotient Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Exact enumeration problems, generating functions, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Symmetric groups Parabolic Catalan numbers count flagged Schur functions and their appearances as type A Demazure characters (key polynomials) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In birational geometry, by the philosophy of the minimal model program, there are 3 building blocks of varieties: varieties of general type (varieties with \(K\) positive), Calabi-Yau varieties (varieties with \(K\) trivial), and Fano varieties (varieties with \(K\) negative). They are characterized by numerical positivity of the canonical divisors. It is expected that such varieties satisfy certain boundedness or finiteness property. Here a set of projective varieties is bounded if its elements can be realized as fibers of an algebraic family over a base of finite type. For example, for varieties of general type, Hacon, McKernan, and Xu showed that if fix a positive integer \(d\) and a positive rational number \(v\), then the set of \(d\)-dimensional log semi-canonical varieties \(X\) with \(K_X\) ample and \(K_X^d=v\) forms a bounded family [\textit{C. D. Hacon} et al., J. Eur. Math. Soc. (JEMS) 20, No. 4, 865--901 (2018; Zbl 1464.14038)]. For Fano varieties, it was conjectured that, if fix a positive integer \(d\) and a positive real number \(\epsilon\), then the set of \(d\)-dimensional \(\epsilon\)-lc Fano varieties forms a bounded family, which is known as the Borisor-Alexeev-Borisov (BAB) conjecture. The author proves the BAB conjecture in this paper together with a subsequent paper [\textit{C. Birkar}, Ann. Math. (2) 193, No. 2, 347--405 (2021; Zbl 1469.14085)].
In Proposition 7.13, the author provides a criterion for a set of klt Fano varieties to be bounded. To be more precise, given a set \(\mathcal{P}\) of klt Fano varieties of dimension \(d\), if one can find a positive integer \(m\) and positive real numbers \(v, t\) such that for any \(X\in \mathcal{P}\)
\begin{itemize}
\item \(X\) has an \(m\)-complement, that is, there exists \(\Delta\in |-mK_X|\) such that \((X, \frac{1}{m}\Delta)\) is lc;
\item \(|-mK_X|\) defines a birational map;
\item \((-K_X)^d\leq v\);
\item \((X, B)\) is lc for any effective \(\mathbb{R}\)-divisor \(B\sim_\mathbb{R}-t K_X \),
\end{itemize}
then \(\mathcal{P}\) is bounded.
So the author proved the BAB conjecture by showing the existence of such \(m,v,t\) for \(\mathcal{P}\) the set of \(d\)-dimensional \(\epsilon\)-lc Fano varieties.
Section 4 is devoted to the existence of a uniform \(m\) such that \(|-mK_X|\) defines a birational map for any \(\epsilon\)-lc Fano varieties \(X\) of dimension \(d\). The method is to construct isolated non-klt centers and to apply Nadel vanishing to get point separation. Such kind of ideas were used by \textit{U. Angehrn} and \textit{Y.-T. Siu} [Invent. Math. 122, No. 2, 291--308 (1995; Zbl 0847.32035)], \textit{C. D. Hacon} et al. [Ann. Math. (2) 177, No. 3, 1077--1111 (2013; Zbl 1281.14036); Ann. Math. (2) 180, No. 2, 523--571 (2014; Zbl 1320.14023)]. One of the main difficulty here is that to use the \(\epsilon\)-lc condition one need to develop a more presice adjunction theory for non-klt centers (this part is explained in Section 3).
Sections 6--8 are devoted to the existence of a uniform \(m\) such that \(X\) has an \(m\)-complement for any klt Fano varieties \(X\) of dimension \(d\). This result is the most important and technical part of this paper, which is originally conjectured by Shokurov. Note that here \(X\) is only assumed to be klt instead of \(\epsilon\)-lc. As a consequence, it implies that for any klt Fano varieties \(X\) of dimension \(d\), there exists a uniform \(m\) depending on \(d\) such that \(|-mK_X|\neq \emptyset\). For the proof the author introduces the concept of generalized pairs and extend the conjecture in the setting of generalized pairs. Then the author proves the conjecture by induction on dimensions.
Section 9 is devoted to the existence of a uniform \(v\) such that \((-K_X)^d\leq v\) for any \(\epsilon\)-lc Fano varieties \(X\) of dimension \(d\). Fano varieties; complements; linear systems; minimal model program Fano varieties, Minimal model program (Mori theory, extremal rays), Divisors, linear systems, invertible sheaves, Rational and birational maps Anti-pluricanonical systems on Fano varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review presents a new algebraic approach to quantum cluster algebras based on noncommutative ring theory. The paper proposes a general construction of quantum cluster algebra structures on a broad class of algebras. Initial clusters and mutations are constructed in a uniform and intrinsic way, in particular, avoiding any ad hoc constructions with quantum minors.
The main theorem of the paper asserts that every algebra in a very large, axiomatically defined class of quantum nilpotent algebras admits a quantum cluster algebra structure. Furthermore, for all such algebras, the latter equals the corresponding upper quantum cluster algebra.
This theorem has a broad range of applications and the required axioms are easy to verify. Many classical families of algebras fall within this axiomatic class. In particular, an application of this theorem gives an explicit quantum cluster algebra structures on the quantum Schubert cell algebras for all finite dimensional simple Lie algebras. quantum cluster algebras; quantum nilpotent algebras; iterated Ore extensions; noncommutative unique factorization domain K. R. Goodearl and M. T. Yakimov, \textit{Quantum cluster algebra structures on quantum nilpotent algebras}, Memoirs of the American Mathematical Society \textbf{247} (2017). Ring-theoretic aspects of quantum groups, Cluster algebras, Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds Quantum cluster algebra structures on quantum nilpotent algebras | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Gauss conjecture states the existence of infinitely many real quadratic number fields with class number one. The weak Gauss conjecture states the existence of infinitely many number fields having class number one.
In the paper under review, the authors generalize the Gauss conjecture as follows. Consider pairs \((K, S)\) with \(K\) a global field and \(S\) a finite nonempty set of places of \(K\) containing the archimedean ones in the number field case. Fix a pair \((K_0, S_0)\) with the ring of \(S_0\)-integers principal. Then the main conjecture is that there are infinitely many pairs \((K,S)\) such that: (1) all the places of \(S_0\) split completely in \(K\), (2) \(K/K_0\) is of degree two and (3) the ring of \(S\)-integers is principal. The Gauss conjecture is just the case where \(K_0= {\mathbb Q}\) and \(S=\{P_\infty\}\) where \(P_\infty\) is the usual absolute value of the field of rational numbers.
The authors consider several cases of the main conjecture. In particular the case of complex quadratic fields, the case of hyperelliptic function fields and function fields which are Galois extensions of a given one or with certain ramification conditions. Among several other results, they prove that for \(q=4, 9, 25, 49\) or \(169\) there are infinitely many extensions \((K,S)\) of \(({\mathbb F}_q (T), \{P\})\) such that \(P\) splits completely in \(K\), \(K/{\mathbb F}_q(T)\) is a Galois extension and the ring of \(S\)-integers is principal.
From the geometric point of view, they show that if \(X\) is a curve of genus \(g_X\geq 2\) over \({\mathbb F}_q\) such that \({|S|\over g_X - 1} > \sqrt{q} -1\), then the class field tower of \((X, S)\) is finite. A similar result is obtained for Drinfeld modular curves. Finally, they apply classical and Drinfeld modular curves to solve cases of the main problem discussed at the beginning of the paper. Gauss conjecture; modular curves; Drinfeld modular curves; class field tower; congruence function fields; ring of \(S\)-integers; ideal class number; class number Lachaud, G.; Vladut, S.: Gauss problem for function fields, J. number theory 85, No. 2, 109-129 (2000) Arithmetic theory of algebraic function fields, Cyclotomic function fields (class groups, Bernoulli objects, etc.), Class field theory, Finite ground fields in algebraic geometry, Jacobians, Prym varieties, Arithmetic aspects of modular and Shimura varieties, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Curves over finite and local fields Gauss problem for function fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In der Arbeit wird die Vermutung von Tate über die Gleichheit der Dimensionen des Raumes der Galois-Invarianten in der zweiten \(\ell\)- adischen Kohomologiegruppe und des von den Kohomologieklassen von Divisoren aufgespannten Raumes, und der Polordnung in \(s=2\) der entsprechenden L-Funktion für die nichtsingulären Kompaktifizierungen einer Hilbert-Blumenthal-Fläche betrachtet. Insbesondere wird die Vermutung über abelschen Erweiterungen von \({\mathbb{Q}}\) bewiesen. (Die Aussage wird auf eine entsprechende Aussage über Hilbert-Blumenthal- Flächen zurückgeführt.)
Dazu wird die mit der zweiten Schnittkohomologiegruppe gebildete L- Funktion mit einem Produkt von automorphen L-Funktionen identifiziert. Es wird gezeigt, daß eine unendlich-dimensionale automorphe Darstellung \(\pi\) bei der Zerlegung der zweiten (Schnitt-)Kohomologiegruppe unter der Aktion der Heckealgebra einen höchstens eindimensionalen Beitrag zu den Invarianten über einer zyklotomischen Erweiterung liefert, und daß dieser Beitrag genau dann nichttrivial ist, wenn \(\pi\) ''ausgezeichnet'' (im Sinne der Definition 2.7. der Arbeit) ist. Analoges gilt für die Polordnung der entsprechenden L-Funktion von \(\pi\). Schließlich wird der Raum der ''Hirzebruch-Zagier-Zyklen'' konstruiert, der von gewissen algebraischen Zyklen über zyklotomischen Körpern aufgespannt wird (i.w. Transformierte der Diagonalen unter den Heckeoperatoren), und es wird gezeigt, daß der Beitrag von \(\pi\) genau dann einen Hirzebruch- Zagier-Zykel enthält, wenn \(\pi\) ausgezeichnet ist. Es wird auch gezeigt, daß, falls der Beitrag von \(\pi\) Invarianten erst über einer nichtabelschen Erweiterung enthält, so ist \(\pi\) vom CM-Typ. Für den entsprechenden Teil der Kohomologie bleibt die Frage nach der Gültigkeit der Tate'schen Vermutungen ungeklärt [Vgl. aber: \textit{C. Klingenberg}, Dissertation (Bonn)]. Tate conjecture; compactification of Hilbert-Blumenthal surface; L- function; algebraic cycles; Hirzebruch-Zagier cycles Harder, G.; Langlands, R. P.; Rapoport, M., Algebraische Zyklen auf Hilbert-Blumenthal-Flächen, J. Reine Angew. Math., 0075-4102, 366, 53-120, (1986) Cycles and subschemes, Special surfaces, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Algebraische Zyklen auf Hilbert-Blumenthal-Flächen | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, the author studies a special holomorphic family \((M,\pi,R)\) of closed Riemann surfaces of genus two over a four-punctured torus \(R\), which is a kind of Kodaira surface in the sense of \textit{K. Kodaira} [J. Anal. Math. 19, 207--215 (1967; Zbl 0172.37901)]. This family has been previously considered by \textit{G. Riera} [Duke Math. J. 44, 291--304 (1977; Zbl 0361.32014)]. The author gives two explicit equations that define this family, that involve elliptic functions, and he shows that there are exactly two holomorphic sections of this family, which he determines explicitely. holomorphic family; function field; Kodaira surface; Teichmüller space; Diophantine equation; Riemann surfaces; holomorphic section Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Families, moduli of curves (analytic), Curves of arbitrary genus or genus \(\ne 1\) over global fields, Teichmüller theory for Riemann surfaces A remark on holomorphic sections of certain holomorphic families of Riemann surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(H_{d,g}\) denote the Hilbert scheme of locally Cohen-Macaulay curves in \(\mathbb{P}^3\). For any \(d>4\) and \(g\leq {d-3\choose 2}\), \(H_{d,g}\) has two well-understood irreducible families: There is a component \(E\subset H_{d,g}\) corresponding to extremal curves [see \textit{M. Martin-Deschamps} and \textit{D. Perrin}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, No. 6, 757-785 (1996; Zbl 0892.14005), these are the curves with maximal Rao function] and \(S\), the family of subextremal curves [see \textit{S. Nollet}, Manuscr. Math. 94, No. 3, 303-317 (1997; Zbl 0918.14014), these have the next largest Rao function]. In this short note we show that \(S\cap E\neq \emptyset\) in \(H_{d,g}\) by constructing an explicit specialization (proposition 1). Our construction also works for ACM curves of genus \(g= {d-3\choose 2}+1\) (remark 2) and hence \(H_{d,g}\) is connected for \(g> {d-3\choose 2}\) (corollary 3). Hilbert scheme Nollet, S., A remark on connectedness in Hilbert scheme, Communications in Algebra, 28, 5745-5747, (2000) Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) A remark on connectedness in Hilbert schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this book written by the well known Soviet mathematicians Yu. I. Manin and A. A. Panchishkin is to inform about classical methods and results in number theory and to present recent achievements in number theory associated with applications of the theory of automorphic forms and automorphic representations in number theory.
Section I is entitled ``Problems and methods''. In chapter 1 (``Elementary number theory'') the authors deal with well-known facts from elementary number theory such as decomposition of any natural number into prime factors, the Euclidean algorithm, Chinese remainder theorem, Gauss' quadratic reciprocity law and such problems of analytic number theory as the distribution of prime numbers in natural sequences. Also the authors discuss problems arising in the theory of representation of integers by quadratic forms, Diophantine approximations of irrational numbers and connections with the theory of continued fractions.
In chapter 2 (``Selected modern problems in elementary number theory'') the authors discuss such computational problems in number theory as the search for a prime divisor of a given large natural number and their applications to asymmetric coding, reliable tests (for primality). Also the authors give a sketch of a proof of the irrationality of \(\zeta\) (3) presented by Apéry in 1978.
Section II is called ``Ideas and Theories''. In chapter 1 (``Induction and Recursion'') the authors explain some methods of logic, namely induction and recursion in their connections with number theory. The authors show that the notions of Diophantine sets, partially recursive functions, recursively enumerable sets, algorithmic unsolvability play an important role in number theory. The authors emphasize the importance of Gödel's incompleteness theory.
In chapter 2 (``Arithmetic of algebraic numbers'') the authors communicate some facts about Galois extensions of number fields, actions of the Frobenius elements, decomposition of prime ideals in field extensions, Hilbert's norm residue symbol and its properties, Artin's reciprocity map, and Galois cohomology.
In chapter 3 (``Arithmetic of algebraic varieties'') the authors deal with problems of solvability of Diophantine equations. They consider Diophantine problems connected with problems of algebraic geometry, discuss the importance of such notions from algebraic geometry as schemes of finite type over rings, cycles and divisors on varieties and schemes, regular differential forms on varieties, linear space of the divisor, heights associated with the divisor in number theory. The authors give a sketch of Faltings' approach to the Tate conjecture about isogenies of abelian varieties.
In chapter 4 (``Zeta-functions and modular forms'') the authors introduce the reader into an extremely wide area of problems arising from the connections between the theory of zeta- and L-functions and the theory of automorphic forms and automorphic representations. The authors begin with a definition of zeta-functions of an arithmetic scheme and discuss the famous Deligne result about the ``Weil conjecture'' for the zeta-function attached to the scheme over a finite field, give an example of application of the Deligne estimate to problems of estimates of trigonometric sums. The authors briefly present the theory of L-functions associated with rational representations of Galois groups, Artin formalism, the theory of Hecke characters and Tate theory. One of the most important part of the modern theory of L-functions connects with the theory of automorphic forms. The authors give a sketch of the theory of modular forms and discuss the problems of representability of Dirichlet series in terms of their Euler product. Discussing connections between modular forms and Galois representations the authors present important conjectures of number theory such as Taniyama-Weil conjecture, Birch- Swinnerton-Dyer conjecture, Ramanujan-Petersson conjecture, Artin conjecture, Sato-Tate conjecture, Serre conjecture. In conclusion of this chapter the authors discuss Langlands' functoriality principle.
The list of references contains 457 titles.
The present book is written at a high mathematical level and can be considered as a good introduction to number theory. Diophantine sets; recursively enumerable sets; Arithmetic of algebraic varieties; Diophantine problems; Tate conjecture; isogenies of abelian varieties; Zeta-functions; L-functions; automorphic forms; automorphic representations; arithmetic scheme; Weil conjecture; modular forms; Galois representations Research exposition (monographs, survey articles) pertaining to number theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory, Discontinuous groups and automorphic forms, Arithmetic algebraic geometry (Diophantine geometry), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Abelian varieties and schemes, Arithmetic problems in algebraic geometry; Diophantine geometry, Connections of number theory and logic, Algebraic number theory: global fields, Elementary number theory Introduction to number theory | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.