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The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Resorting to the Lenard recursion scheme, we derive the TD hierarchy associated with a \(2 \times 2\) matrix spectral problem and establish Dubrovin-type equation in terms of the introduced elliptic variables. Based on the theory of algebraic curve, all the flows associated with the TD hierarchy are straightened under the Abel-Jacobi coordinates. An algebraic function \(\phi \), also called the meromorphic function, carrying the data of the divisor is introduced on the underlying hyperelliptic curve \(\mathcal {K}_{n}\). The known zeros and poles of \(\phi \) allow to find theta function representations for \(\phi \) by referring to Riemann's vanishing theorem, from which we obtain algebro-geometric solutions for the entire TD hierarchy with the help of asymptotic expansion of \(\phi \) and its theta function representation. TD hierarchy; algebro-geometric solutions [34] Geng X. G., Zeng X. and B. Xue, Algebro-geometric solutions of the TD Hierarchy, Math. Phys. Anal. Geom. \textbf{16} (2013), 229-251. Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Series solutions to PDEs, Relationships between algebraic curves and integrable systems, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Algebro-geometric solutions of the TD hierarchy | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(F : (\mathbb C^n,0)\to(\mathbb C^m,0)\) be a holomorphic mapping defined in a neighborhood \(0\in\mathbb C^n\). The local Łojasiewicz exponent of \(F\), denoted by \(\mathcal L_0(F)\), is defined to be the infimum of the set of all exponents \(v\in\mathbb R\) in the Łojasiewicz inequality \(|F(z)|\geq C|z|^v\) as \(|z|\to 0\) for some constant \(C > 0\).
In this paper, the authors consider the case when \(F : (\mathbb C^n , 0)\to (\mathbb C^m , 0)\) is a polynomial mapping. They give an effective algorithm computing the dimension of the germ \(V (F ) = F^{-1}(0)\) at the origin and if this dimension is zero, they compute the local Lojasiewicz exponent \(\mathcal L_0(F)\).
More precisely, the main result of this paper is the following. Let \(F :(\mathbb C^n,0)\to(\mathbb C^m, 0)\) be a polynomial mapping of degree \(d = \deg F\). Denote by \(\mathbb L(m, n)\) the set of all linear mappings \(\mathbb C^m\to\mathbb C^n\). For \(q\in\{0,\dots, n\}\), \(M\in\mathbb L(n, q)\) and \(L\in\mathbb L(m+q, n)\), define
\[
H_{L,M}(z) = L(F(z),M(z)) + (z_1^{d^n+1},\dots,z_n^{d^n+1}).
\]
Let \(\Phi_q :\mathbb L(m+q, n)\times\mathbb L(n, q)\times\mathbb L(n, 1)\times\mathbb C^n\to\mathbb L(m+q, n)\times\mathbb L(n, q)\times\mathbb L(n, 1)\times\mathbb C^n\times\mathbb C\) be a mapping given by the equation
\[
\Phi_q (L, M, N, z) = (L, M, N, H_{L,M} (z), N (z)).
\]
The mapping \(\Phi_q\) is proper and its image is an algebraic set of pure dimension \((m + q)n + nq + 2n\). So there exists a polynomial \(P_q\in\mathbb C[L, M, N, y, t]\) of the form
\[
P_q (L, M, N, y, t) = \sum_{j=0}^p P_{q,j}(L, M, N, y)t^j
\]
such that \(P_{q,p} \neq 0\) and the zero-set \(V (P_q)\) of \(P_q\) is the image of \(\Phi_q\). There exists \(r\) with \(0\leq r < p\) such that \(\mathrm{ord}_y P_{q,j} > 0\) for \(j = 0,\dots, r\), and \(\mathrm{ord}_y P_{q,r+1} = 0\).
Set
\[
\Delta'(P_q)=\min_{j=0}^r \frac{\mathrm{ord}_yP_{q,j}}{r+1-j}.
\]
Then
\[
\dim_0 V (F) = \min\{q :\frac{1}{\Delta'(P_q)}<d^n+1\}.
\]
If \(F\) is finite at \(0\), then
\[
\mathcal L_0(F)=\frac{1}{\Delta'(P_0)}.
\]
Łojasiewicz exponent; effective formulas; germ of algebraic set; dimension [9]T. Rodak and S. Spodzieja, Effective formulas for the local Łojasiewicz exponent, Math. Z. 268 (2011), 37--44. Effectivity, complexity and computational aspects of algebraic geometry, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables, Singularities in algebraic geometry Effective formulas for the local Łojasiewicz exponent | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 explicate the combinatorial/geometric ingredients of Arthur's proof of the convergence and polynomiality, in a truncation parameter, of his noninvariant trace formula. Starting with a fan in a real, finite dimensional, vector space and a collection of functions, one for each cone in the fan, we introduce a combinatorial truncated function with respect to a polytope normal to the fan and prove the analogues of Arthur's results on the convergence and polynomiality of the integral of this truncated function over the vector space. The convergence statements clarify the important role of certain combinatorial subsets that appear in Arthur's work and provide a crucial partition that amounts to a so-called nearest face partition. The polynomiality statements can be thought of as far reaching extensions of the Ehrhart polynomial. Our proof of polynomiality relies on the Lawrence-Varchenko conical decomposition and readily implies an extension of the well-known combinatorial lemma of Langlands. The Khovanskii-Pukhlikov virtual polytopes are an important ingredient here. Finally, we give some geometric interpretations of our combinatorial truncation on toric varieties as a measure and a Lefschetz number. combinatorial truncation; Arthur trace formula; convex polytopes; toric varieties Spectral theory; trace formulas (e.g., that of Selberg), Toric varieties, Newton polyhedra, Okounkov bodies On combinatorics of the Arthur trace formula, convex polytopes, and toric varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The present paper is a natural sequel in a series of publications by the author. The theme centers around the method of studying the geometry of cycles on a projective manifold \(X\) by means of their Chow transforms. In this manner the transform of a cycle becomes a divisor in a Grassmannian, and moreover one has also an integral transformation for currents on \(X\), which extends the one for effective cycles. \newline We find here results connected with the Hodge conjecture, they are the continuation of the paper [C. R., Math., Acad. Sci. Paris 348, No. 11--12, 625--628 (2010; Zbl 1194.14015)].
More specifically, the author's summary reads: ``We prove that a closed differential form of bidimension \((p,p)\) on a projective manifold is cohomologous to an algebraic cycle with complex coefficients if and only if it is a weak limit of such cycles. This allows us to approach the problem of the algebraicity of cohomology classes. Using the characterization of currents associated with algebraic cycles by the Chow transform, the obstructions are reduced to an orthogonality condition with certain smooth functions on the Grassmannian, which are in general merely images of distributions by a suitable explicitly defined linear differential operator. These distributions are of order less than \(k\). This forces a convergence in the space of \(\mathcal C^k \) functions, which is achieved, when the cohomology class is rational, thanks to the constructibility of the Bernstein polynomial.'' algebraic cycles; currents, Chow transform Méo, M., Chow forms and Hodge cohomology classes, C. R. Acad. Sci. Paris, Ser. I, 352, 339-343, (2014) Algebraic cycles, Transcendental methods, Hodge theory (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Currents in global analysis Chow forms and Hodge cohomology 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 This is an important and pleasing paper. It introduces a basic integral formalism into the theory of rigid analytic modular forms for function fields. The results work in complete generality (see remark below) though we will just describe them in the simplest case of \({\mathbf A}=\mathbb F_ r[T]\), \(r=p^ m\), \({\mathbf k}=\mathbb F_ r(T)\) and \({\mathbf K}={\mathbf k}_ \infty\); so \({\mathbf A}\subset {\mathbf K}\) is discrete and co-compact. Let \(\Gamma=\text{GL}_ 2({\mathbf A})\) and let \(\Omega=\overline{\mathbf K}-{\mathbf K}\) for some fixed choice of algebraic closure of \({\mathbf K}\). The space \(\Omega\) can be given the structure of a connected rigid analytic space (i.e., a non-archimedean ringed-space where one can use analytic continuation). The group \(\Gamma\) acts discretely on \(\Omega\) and the function \(q(z)=\sum_{a\in{\mathbf A}}(z-a)^{-1}=1/{\mathbf e}(z)\) is a uniformizing element at infinity. The classical definitions of ``modular form'' and ``cusp form'' carry over in a straightforward fashion to \(\{\Omega,\Gamma\}\).
Let \(f(z)\) be a cusp form of weight \(k\) for \(\Gamma\). The author associates to \(f\) a measure \(\mu_ f\) (in the sense of Vishik, Amice and Vélu -- so one cannot integrate all continuous functions but rather those that are locally analytic, which suffices for most applications). These measures have symmetries which reflect the defining \(\Gamma\)-invariance properties of the cusp form. A major result of the paper is an isomorphism between the space of such measures and the space of the cusp forms of weight \(k\). One recovers the cusp form from the measure by the Poisson kernel formula:
\[
f(z)=\int_{{\mathbb P}^ 1({\mathbf K})}{1\over z-x} \,d\mu(x).
\]
The author also gives an integral for the \(q\)-expansion coefficients of the form \(f(z)=\sum b_ jq^ j\): \(b_ j=\int_{{\mathbf K}/{\mathbf A}}{\mathbf e}(x)^{j-1}\,d\mu(x).\) (This can be proved in the following fashion which works, unlike the author's proof, in complete generality: By the remark following Lemma 13 of the paper, we can write \(f(z)=\int_{\mathbf K}{1\over z-x}\, d\mu(x).\) By the symmetry properties of \(\mu\), this equals
\[
\begin{aligned} \int\limits_{{\mathbf K}/{\mathbf A}}\sum_{a\in A}{1 \over z-x+a}\,d\mu(x)&=\int {1\over{\mathbf e}(z-x)}\,d\mu(x),\\ &=\int{1\over {\mathbf e}(z)-{\mathbf e}(x)}\,d\mu(x),\end{aligned}
\]
as \({\mathbf e}(z)\) is additive. Now express the integrand in terms of \(q(z)={\mathbf e}(z)^{- 1}\) and expand via the geometric series. The existence and integral for \(q\)-expansions follows immediately and quite formally.) If we view \(d\mu(x)\) as `morally' \(f(x)dx\) and note that \(dq(x)=-q(x)^ 2\,dx\), the analogy with Cauchy's Theorem becomes apparent upon substituting the \(q\)-expansion for \(f\) in the integral!
Let \({\mathcal U}_ 1\subset {\mathbf K}\) be the group of 1-units. Let \(s\in\mathbb Z_ p\) and \(L_ f(s)=\int_{{\mathcal U}_ 1}u^ s{d\mu(x)\over u}\). Then using the symmetry properties of \(\mu\) one can show that \(L_ f(s)=\pm L_ f(k-s)\) (where the exact sign can be found). The relationship between \(L_ f(s)\) is given by a Fourier transform [see the reviewer, J. Algebra 146, No. 1, 219--241 (1992; Zbl 0758.11030)]. These Fourier transforms arise naturally when one wants to understand the \(L\)-series of Drinfeld modules; functions naturally defined on the space \(\overline{\mathbf K}^*\times \mathbb Z_ p\). To define the F. T. one needs to pick a uniformizing power series. The power series needed to compute \(L_ f\) is the logarithm of \({\mathbf e}(z)\); for Drinfeld modules one does not quite yet know what to expect, but the logarithm does not seem to be the correct choice. It may be that there is some sort of deformation theory that allows one to ultimately relate the two type of \(L\)-series. This would allow one to use Teitelbaum's measures to study \(L\)-series of Drinfeld modules also.
There is another fascinating area raised by the existence of these measures. Let \(E\) be an elliptic curve over \(\mathbf k\) with split multiplicative reduction at \(\infty\). One knows by Drinfeld that \(E\) is isogenous to a factor of the Jacobian of some moduli curve. Thus we deduce a differential form associated to \(E\to\) a cusp form \(\to\) a measure associated to \(E\to\) possibility of a characteristic \(p\) \(L\)-series for \(E\).
On the other hand, E.-U. Gekeler has used this work to establish the non- existence of cusp forms of weight 1 in the case of nontrivial level structure. One takes the \(p\)-th power of such a form and then applies Teitelbaum's results to show that the corresponding measure must be trivial. While disappointing, this result is, perhaps, not surprising in the following sense: Classical theory leads to the expectation that a form of weight one ``should'' correspond to a representation of a finite Galois group \(G\). However, with a little thought, one sees that these representations should really be defined in characteristic 0 (else, if \(p\mid\text{Ord}(G)\), one loses control of local factors).
In any case, much work needs to be done before one has exhausted the potential raised by the exciting paper being reviewed! Drinfeld modules; L-series; elliptic curve; q-expansion coefficients; Poisson kernel formula; space of cusp forms; non-archimedean measure; rigid analytic space; rigid analytic modular forms for function fields; Mellin transform J. T. Teitelbaum, The Poisson kernel for Drinfeld modular curves , J. Amer. Math. Soc. 4 (1991), no. 3, 491-511. JSTOR: Drinfel'd modules; higher-dimensional motives, etc., Arithmetic theory of algebraic function fields, \(p\)-adic theory, local fields, Local ground fields in algebraic geometry, Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Formal groups, \(p\)-divisible groups, Arithmetic theory of polynomial rings over finite fields The Poisson kernel for Drinfeld modular curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper concerns the controlled parametrizations of compact semi-algebraic sets. The main result is a generalization, to the analytic case, of the following theorem (Yomdin, Gromov). For any natural \(k\) and any compact semi-algebraic set \(A\subset I^{n}\subset \mathbb{R}^{n}\) (\(I:=[-1,1]\)) there exists a subdivision of \(A\) into semi-algebraic pieces \(A_{j},\) \( j=1,\dots ,N,\) together with algebraic \(C^{k}\)-mappings \(\psi _{j}:I^{n_{j}}\rightarrow A_{j}\) such that \(\psi _{j}\) are onto, homeomorphic on the interior of \(I^{n_{j}}\) and \(\psi _{j}(x)-\psi _{j}(0)\) are uniformly bounded by \(1\) with the all derivatives up to the order \(k.\) Moreover \(N\) depends only on \(k\) and the degrees and the number of equations and inequalities defining \(A.\) The analytic case differs in\ that the \(\psi _{j}\) are real analytic, uniformly bounded \ with the all derivatives but after removing from \(A\) a finite number \(M\) of open boxes of size at most \(2\delta \) (\(\delta \) is a given positive real number). Moreover \(M\) and \(N\) depends only on \(k\) and the degrees and the number of equations and inequalities defining \(A\) times the factor \(\log _{2}(\frac{1}{\delta }).\) semi-algebraic set; analytic reparametrization; analytic complexity Yomdin, Y, Analytic reparametrization of semialgebraic sets, J. Complex., 24, 54-76, (2008) Triangulation and topological properties of semi-analytic and subanalytic sets, and related questions, Semialgebraic sets and related spaces Analytic reparametrization of semi-algebraic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathcal{P} = \{P_{1},\dots, P_{s}\} \subset \mathbb{P}^{2}_{\mathbb{K}}\) be a finite set of \(s>0\) distinct points on the projective plane defined over an algebraically closed field of characteristic zero. Then we define
\[
I = I(\mathcal{P}) = I(P_{1}) \cap \dots \cap I(P_{s}),
\]
where \(I(P_{i})\) is the defining ideal of point \(P_{i}\). The \(m\)-th symbolic power of \(I\) is defined to be the ideal
\[
I^{(m)} = I(P_{1})^{m} \cap \dots \cap I(P_{s})^{m}.
\]
The famous Nagata-Zariski theorem tells us that \(I^{(m)}\) corresponds the set of forms vanishing along \(P_{i}\)'s with multiplicities \(\geq m\). Moreover, for \(I\) we define \(\alpha(I)\) to be the least degree of a minimal generator of \(I\). Now we can define the Waldschmidt constant of \(\mathcal{P}\) as
\[
\widehat{\alpha}(I(\mathcal{P})) = \lim _{m \rightarrow \infty} \frac{\alpha(I^{(m)})}{m}.
\]
Using some properties of symbolic powers and Fekete's lemma one can show that the above limit exists and
\[
\widehat{\alpha}(I(\mathcal{P})) = \inf_{m\geq 1} \frac{\alpha(I^{(m)})}{m}.
\]
In the present note the authors provide a complete characterization of configurations of points \(\mathcal{P}\) on the projective plane having \(\widehat{\alpha}(I(\mathcal{P})) = 2\).
Theorem A. Let \(I(\mathcal{P})\) be the radical ideal of a finite set of points \(\mathcal{P}\) in \(\mathbb{P}^{2}_{\mathbb{K}}\). Then \(\widehat{\alpha}(I(\mathcal{P})) = 2\) if and only if \(\mathcal{P}\) {\parindent=6mm \begin{itemize}\item[a)] consists of \(n\geq 4\) points, contained in a smooth conic, or; \item[b)] consists of \(r+s\) points, where \(s,r \geq 2\), and \(r\) points out of it lie on a line \(L_{1}\), and \(s\) points out of it lie on another line \(L_{2}\), or; \item[c)] consists of \(r+s+1\) points, where \(r,s\geq 2\), and \(r\) points out of it lie on a line \(L_{1}\), and \(s\) points out of it lie on another line \(L_{2}\), and the last point is the intersection point of these two lines, or; \item[d)] consists of \(6\) points given by the singular locus of a star configuration of \(4\) lines.
\end{itemize}} An important tool which allowed to obtain the above classification is the following result/construction.
Theorem B. For every integer \(d\), there exists a configuration of points \(\mathcal{P}\) in \(\mathbb{P}^{2}_{\mathbb{K}}\), such that for every positive integer \(m\) one has \(\alpha(I(\mathcal{P})^{(m)}) = dm\). In particular, \(\widehat{\alpha}(I(\mathcal{P})) = d\).
The idea behind this construction is quite easy. For a positive integer \(d\) one considers a partition \(d = d_{1} +\dots+ d_{k}\). For each \(i\) let \(C_{i} \subset \mathbb{P}^{2}_{\mathbb{K}}\) be a reduced and irreducible curve of degree \(d_{i}\), and for every \(i\) one imposes \(\mathcal{P}_{i}\) to be a set of \(d\cdot d_{i}\) distinct simple points on \(C_{i}\) such that \(Z_{i} \cap Z_{j} = \emptyset\) for \(i\neq j\). Then our set \(\mathcal{P}\) in Theorem B is defined to be \(\mathcal{P} = \bigcup_{i=1}^{k} \mathcal{P}_{i}\). symbolic powers of ideals; configurations of points; Waldschmidt constants; curve arrangements Mosakhani, M.; Haghighi, H., On the configurations of points in \({\mathbb{P}}^2\) with the Waldschmidt constant equal to two, J. Pure Appl. Algebra, 220, 3821-3825, (2016) Ideals and multiplicative ideal theory in commutative rings, Configurations and arrangements of linear subspaces, Polynomial rings and ideals; rings of integer-valued polynomials, Projective techniques in algebraic geometry, Arrangements of points, flats, hyperplanes (aspects of discrete geometry) On the configurations of points in \(\mathbb{P}^2\) with the Waldschmidt constant equal to two | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems One can begin with author's abstract:
``We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space \(G/\Gamma\) of a semisimple algebraic group \(G\). We define two families of algebraic subvarieties of the associated partial flag variety \(G/\Gamma\), which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of \(m\times n\) real matrices whose image is not contained in any subvariety coming from these two families, Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved. The proof combines geometric invariant theory, Ratner's theorem on measure rigidity for unipotent flows, and linearization technique.''
It is noted that many problems in number theory can be recast in the language of homogeneous dynamics. A survey is devoted to this fact, to the main problems of this research, and to the motivation of the present investigations. Notions that are useful for proving the main statements are recalled and explained.
The main results and several auxiliary statements are proven with explanations. Applications of the main results and also connections between these results and known researches are noted. geometric invariant theory; homogeneous spaces; equidistribution; Ratner's theorem; Dirichlet's theorem; Diophantine approximation Discrete subgroups of Lie groups, Geometric invariant theory, Metric theory Equidistribution of expanding translates of curves and Diophantine approximation on 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 A classical way to study a finite set of points in a projective space \(\mathbb{P}^r\) over an algebraically closed field is to look at relations between its Castelnuovo function (i.e. the first difference function of the Hilbert function of its homogeneous coordinate ring) of geometric properties of the point set. In this extended abstract (containing no proofs), the authors refine that method as follows.
Given a sequence \(X= (P_1,\dots, P_s)\) of points in \(\mathbb{P}^r\), let \(S= (d_1,\dots, d_s)\in \mathbb{N}^s\) be defined by \(d_1=0\) and \(d_k=\) least degree of a hypersurface separating \(P_k\) from \(P_1,\dots, P_{k-1}\) for \(k>1\). Then the multiplicity sequence \(\gamma_S(n)= \#\{i\mid d_i=n\}\) equals the Castelnuovo function of \(X\) and does not depend on the order of the points. Hence it makes sense to study which sequences \(S\) are realizable and to try to classify all point sets \(X\) with given Castelnuovo function \(\Delta HF_X\) according to their sequences \(S\).
Here the authors announce some steps in this direction by examining the effect of neighbour transposition in the point sequence on the degree sequence. They discover that a set \(X\) gives rise to only one non-decreasing sequence \(S\) if and only if \(X\) is in uniform position. Moreover, maximal growth of the Castelnuovo function is shown to correspond to sequences of the form \(S= (0,1,\dots, n+h, S')\) with non-decreasing sequence \(S'\). All results are amply illustrated by examples. 0-dimensional scheme; Hilbert function; geometric properties of the point set; Castelnuovo function Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Cycles and subschemes A new approach to the Hilbert function of a 0-dimensional projective 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 Consider a smooth scheme \(X\) over a perfect field \(k\) of characteristic \(p>0\). Berthelot's theory of overconvergent isocrystals on \(X\) and the corresponding rigid cohomology of such isocrystals often lead to infinite dimensional cohomology spaces. Here the case of a smooth curve \(X/k\) is treated in detail. A fairly general sufficient condition for an overconvergent isocrystal on \(X\) to have finite dimensional cohomology is given. In such cases one can also give a proof of Poincaré duality. The paper consists of an introduction to set the stage, then three parts treating in detail the necessary functional analysis over discretely valuated fields, local duality, global duality and finiteness with applications to quasi-unipotent isocrystals on smooth curves.
The introduction gives some philosophy on quasi-unipotent isocrystals and their relation to Grothendieck's local monodromy theorem. It seems reasonable to expect that overconvergent isocrystals ``of geometric origin'' are quasi-unipotent.
Part I may be considered as a very wellcome and readible overview of non-archimedean functional analysis. Starting from general definitions, notions such as local convexity, balance, boundedness and linear compactness are discussed. Also Banach, Fréchet, dual-of-Fréchet and Montel spaces enter the stage, as well as \(LF\)-spaces, strictness and duality, in particular, a \(LF\)-space is bornological and barreled, the Banach-Steinhaus theorem holds and a map from an \(LF\)-space to a locally convex space is continuous if and only if it is bounded. A locally convex space which is Fréchet and dual-of-Fréchet is necessarily finite dimensional.
In part II local duality is treated. Let \(K\) be a discretely valued field. A topological \(K\)-algebra isomorphic to the algebra \(A\) of Laurent series convergent in some annulus \(r<|x|<1\) is called a \textit{local algebra}. Let \(\Omega_A^1\) be the free topological \(A\)-module of rank one with basis \(dx/x\) and let \(M\) be a finite free \(A\)-module. Then one has a pairing \(M\times(M^\vee\otimes\Omega^1_A)\rightarrow K\) given by \((m,m^\vee\otimes dx/x)\mapsto\text{Res} m^\vee(m)\otimes dx/x\), where \(\text{Res}\) denotes the residue at \(x=0\). As a first result one may state: Any free \(A\)-module \(M\) of finite type is a Montel space for which the map \(M\rightarrow(M^\vee\otimes\Omega^1_A)^\prime_s\) induced by the pairing above, is a topological isomorphism. Here \((\ldots)^\prime_s\) denotes the strong dual. For an \(A\)-module \(M\) one has the usual notion of \textit{connection} \(\nabla:M\rightarrow M\otimes\Omega^1_A\), and for an \(A\)-module with connection \((M,\nabla)\) one defines the cohomology \(H^0(M):=\text{Ker} \nabla\) and \(H^1(M):=\text{Coker} \nabla\). The pairing above induces for any free \(A\)-module of finite type with connection \((M,\nabla)\) a pairing in cohomology \(H^i(M)\times H^{1-i}(M^\vee)\rightarrow K\). A connection \(\nabla\) on \(M\) induces a dual connection \(\nabla^\vee\) on \(M^\vee\). One calls an \(A\)-module \(M\) \textit{strict} if the connection \(M\rightarrow M\otimes\Omega^1_A\) is a strict map of topological vector spaces. Similarly for \(M^\vee\). One has the result:
For any free \(A\)-module of finite type with connection \((M,\nabla)\) the following are equivalent:
(i) \(M\) is strict;
(ii) \(M^\vee\) is strict;
(iii) \(H^1(M)\) is finite dimensional and separated.
Furthermore for strict \(M\), the cohomogical pairing above is a perfect pairing between finite dimensional \(K\)-vector spaces. A connection \(\nabla\) on a finite free \(A\)-module \(M\) is called \textit{unipotent} if \((M,\nabla)\) is a successive extension of trivial rank one objects \((A,d)\). Then an \(A\)-module with unipotent connection is strict.
Let \(X/k\) be a smooth affine \(k\)-curve and let \(X\hookrightarrow\overline X\) be a smooth projective embedding. Let \(X\hookrightarrow\overline X\) lift to a morphism \(\mathfrak X\hookrightarrow\overline{\mathfrak X}\) of formally smooth \(R\)-schemes, where \(R\) is the discrete valuation ring of \(K\). For a strict neighborhood \(V\) of the tube \(]X[\) of \(X\) let \(A_V:=\Gamma(V,\mathcal O_V)\). For a cofinal set of strict neighborhoods \(V\) of \(]X[\), put \(\displaystyle{A^\dagger_X:=\lim_{\longrightarrow\atop V}A_V}\). For \(a\in D=\overline{X}-X\) define \(\displaystyle{A(a):=\lim_{\longrightarrow\atop V}\Gamma(V\cap]a[,\mathcal O_{\mathfrak X}^{\text{an}})}\) and let \(\displaystyle{A_X^{\text{loc}}:=\bigoplus_{a\in D}A(a)}\). Define \(A_X^{\text{qu}}\) by the exactness of the sequence \(0\rightarrow A^\dagger_X\rightarrow A_X^{\text{loc}}\rightarrow A_X^{\text{qu}}\rightarrow 0\). Define \(\displaystyle{\Omega_{A^\dagger}^1:=\lim_{\longrightarrow\atop V}\Gamma(V,\Omega^1_V)}\). Then, for a locally free \(A^\dagger\)-module \(M\) one has the \textit{global pairing} \((M\otimes A^{\text{loc}}_X)\times(M^\vee\otimes A^{\text{loc}}_X)\rightarrow K\), with \(\displaystyle{((m_a),(\omega_a))\mapsto\langle(m_a),(\omega_a) \rangle_X:=\sum_{a\in D}\langle m_a,\omega_a\rangle_a}\), where \(\langle , \rangle_a\) denotes the local pairing as above. The following result summarizes the global duality: With notations as above, for a coherent locally free \(A^\dagger\)-module \(M\), the natural topology on \(M\) is dual-of-Fréchet, \(M\otimes_{A^{\dagger}}A^{\text{qu}}\) is Fréchet, and the maps \(M\rightarrow(M^\vee\otimes\Omega^{\text{qu}})^\prime_s\) and \(M\otimes\Omega^{\text{qu}}\rightarrow(M^\vee)^\prime_s\) are topological isomorphisms. The above theory may now be worked out for overconvergent isocrystals on a curve \(X/k\). One may define cohomology with compact support \(H^1_c(X,M)\), \(i=0,1,2\), where \((M,\nabla)\) is an overconvergent isocrystal on \(X/K\) with connection, and relate it to de Rham cohomology via an exact sequence. One also has parabolic cohomology \(H^i_p(X,M)\). The main result of the paper can now be stated:
For a smooth affine curve \(X/k\) and a strict overconvergent isocrystal \(M\) on \(X/K\), the \(K\)-vector spaces \(H^i(X,M)\), \(H^i_c(X,M)\) and \(H^i_p(X,M)\) are finite dimensional and there are perfect pairings \(H^i_c(X,M)\times H^{2-i}(X,M^{\vee})\rightarrow K\) and \(H^1_p(X,M)\times H^1_p(X,M^{\vee})\rightarrow K\) which can be given explicitly. In the final section the case of quasi-unipotent isocrystals is studied as an analogue of Deligne's Weil II article [\textit{P. Deligne}, Publ. Math., Inst. Hautes Étud. Sci. 52, 137-252 (1980; Zbl 0456.14014)] for \(\ell\)-adic étale cohomology related to Grothendieck's local monodromy theorem. Notions of weight and pureness can be introduced. Several interesting analogues can be formulated. As a final result we mention (as a first step to the Riemann hypothesis):
For a quasi-unipotent isocrystal \((M,\Phi)\) on the curve \(X/K\), \(\imath\)-pure of weight \(\beta\), the weights of Frobenius on \(H^1_c(X,M)\) are strictly less than \(\beta+2\). strict overconvergent isocrystal; non-archimedean functional analysis; discrete valuation; Fréchet space; Montel space; characteristic \(p\); dagger algebra; connection; quasi-unipotent isocrystal; global duality; Riemann hypothesis R.~Crew, \emph{Finiteness theorems for the cohomology of an overconvergent isocrystal on a curve}, Ann. Sci. Ecole Norm. Sup. (4) \textbf{31} (1998), no.~6, 717--763. DOI 10.1016/S0012-9593(99)80001-9; zbl 0943.14008; MR1664230 \(p\)-adic cohomology, crystalline cohomology, Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis, Analytic algebras and generalizations, preparation theorems Finiteness theorems for the cohomology of an overconvergent isocrystal 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 main part of this paper consists in developing a version of motivic integration in real algebraic geometry which has usual good properties like a change of variables formula. Using it, the authors obtain some inverse mapping theorems for arc-analytic maps, namely:
Suppose that \((X_1, x_1)\) and \((X_2, x_2)\) are two germs of real algebraic sets and suppose that \(f\colon (X_1, x_1) \to (X_2, x_2)\) is a germ of a semialgebraic, generically arc-analytic homeomorphism. (Arc analytic means that the composition of a real analytic arc \((-1,1) \to X_1\) with \(f\) is again analytic.) Under the assumption that the motivic measure of the arcs of \(X_i\) centered at \(x_i\) are equal for \(i=1,2\), the authors obtain the following: (a) If the Jacobian determinant of \(f\) is bounded from below, then the inverse \(f^{-1}\) is also arc analytic and also has Jacobian determinant bounded from below. (b) If \(f^{-1}\) Lipschitz with respect to the inner distance, then so is \(f\).
The notion of motivic integration is of ``geometric style'': Given an algebraic set \(X\) over \(\mathbb{R}\) (possibly singular), a notion of measurable subsets \(A\) of the arc space \(\mathcal{L}(X)\) is introduced, and for such \(A\), a motivic measure \(\mu(A)\) is defined, which takes values in the completion \(\widehat{\mathcal M}\) of the localization \(\mathcal M = K_0(\mathcal{AS})[\mathbb{L}^{-1}]\) of some Grothendieck ring \(K_0(\mathcal{AS})\) (where as usual \(\mathbb{L}\) is the class in \(K_0(\mathcal{AS})\) of the affine line).
The Grothendieck ring \(K_0(\mathcal{AS})\) used here is the one obtained from arc symmetric sets, which form a sub-class of the semi-algebraic sets (but contain all algebraic sets). Note that the Grothendieck ring of all semi-algebraic sets is almost trivial, whereas \(K_0(\mathcal{AS})\) contains a reasonable amount of information. Moverover, \(K_0(\mathcal{AS})\) is well-understood.
As in the complex world, one would like to define the measure of a set \(A \subset \mathcal{L}(X)\) via its images \(\pi_m(A)\)
in the jet spaces \(\mathcal{L}_m(X)\). However, when \(X\) is singular, even in the case \(A = \mathcal{L}(X)\), \(\pi_m(A)\) might not be arc symmetric. The authors overcome this difficulty by removing a small neighbourhood of the singular locus \(\mathcal{L}(X_{\mathrm{sing}})\) and letting that neighbourhood become smaller and smaller. arc-analytic map; inverse mapping theorem; arc-symmetric sets; motivic integration; arc space; blow-Nash map; inner distance; semi-algebraic sets; semi-algebraic maps; real algebraic geometry; germs Nash functions and manifolds, Lipschitz (Hölder) classes, Arcs and motivic integration, Singularities in algebraic geometry Arc spaces, motivic measure and Lipschitz geometry of real algebraic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathsf{A} = \{ a_1, \ldots, a_m \}, m \in \mathbb{N} \), be measurable functions on a measurable space \((\mathcal{X}, \mathfrak{A})\). If \(\mu\) is a positive measure on \((\mathcal{X}, \mathfrak{A})\) such that \(\int a_i d \mu < \infty\) for all \(i\), then the sequence \((\int a_1 \text{d} \mu, \ldots, \int a_m \text{d} \mu)\) is called a truncated moment sequence. By Richter's Theorem each truncated moment sequence has a \(k\)-atomic representing measure with \(k \leq m\). The set \(\mathcal{S}_{\mathsf{A}}\) of all truncated moment sequences is the truncated moment cone. The aim of this paper is to analyze the various structures of the truncated moment cone. The main results concern the facial structure (exposed faces, facial dimensions) and lower and upper bounds of the Carathéodory number (that is, the smallest number of atoms which suffices for all truncated moment sequences) of the convex cone \(\mathcal{S}_{\mathsf{A}} \). In the case when \(\mathcal{X} \subseteq \mathbb{R}^n\) and \(a_i \in C^1(\mathcal{X}, \mathbb{R})\), the differential structure of the moment map and regularity/singularity properties of moment sequences are analyzed. The maximal mass problem is considered and some applications to other problems are sketched. truncated moment problem; moment cone; atomic measure; Carathéodory number; positive polynomial; maximal mass The multidimensional truncated moment problem: the moment cone | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review represents a continuation of the authors' work on the interpolation problem posed by a finite number of general points with weights in the projective plane, i.e., the problem of determining the dimension of the linear system \({\mathcal L}_d(p_1^{m_1},\dots,p_n^{m_n})\) of plane curves of degree \(d\) with prescribed multiplicity at prescribed general points in the projective plane. By the Riemann-Roch Theorem, such a linear system has the expected dimension if \(H^1(S,{\mathcal L})=0\), where \({\mathcal L}\) is the line bundle corresponding to \({\mathcal L}_d(p_1^{m_1},\dots,p_n^{m_n})\) on the blow-up \(S\) of the plane at the prescribed points.
Segre's Conjecture states that if the linear system \(|{\mathcal L}|\) is reduced, meaning it is nonempty and a general curve in \(|{\mathcal L}|\) has at most isolated singularities, then \(H^1(S,{\mathcal L})=0\). The paper's main result is a proof of Segre's Conjecture for the case of all weights being at most three. The proof is essentially a deformation theory argument. \(H^1(S,{\mathcal L})\) is shown to be an obstruction space for deformations of a general element of \(|{\mathcal L}|\) as the points \(p_i\) vary. However, as the \(p_i\) are general, there are no obstructions, and one can conclude \(H^1(S,{\mathcal L})=0\). Divisors, linear systems, invertible sheaves, Plane and space curves, Rational and ruled surfaces A deformation theory approach to linear systems with general triple points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The purpose of the paper is to describe some properties of Riemann- Hilbert map (a map from the set of systems of differential equations to the set of representations of the fundamental group) which reflect on its essentially transcendental nature. The technique is base on Kuo-Tsai Chen expansion of the solution of a system of differential equations as a sum of iterated integrals. Let X be a compact Riemann surface of genus g.
Theorem 1. Suppose X is defind over \(\bar Q.\) The standard conjecture holds for regular singular systems of equations of rank one on X.
Theorem 2. The standard conjecture holds for irreducible regular singular systems of rank two on \(P^ 1-\{0,1,\infty \}.\)
Theorem 3. Suppose \(s_ 1,...,s_ n\) are \(\bar Q\) rational points in \(P^ 1\). The conjectures of absoluteness I and II, and the conjecture of variation of Hodge structure, hold for unipotent regulare singular systems of equations of rank 3 on \(P^ 1-\{s_ 1,...,s_ n\}.\)
Theorem 4. The Laplace transform of the integral
\[
\int_{\Delta_ k\gamma}b_{i_ ki_{k-1}}(z_ k)...b_{i_ 2i_ 1}(z_ 2)b_{i_ 1i_ 0}(z_ 1)e^{t(g_{i_ k}(Q)-g_{i_ k}(z_ k)+...+g_{i_ 0}(z_ 1)-g_{i_ 0}(P))}
\]
has an extension with locally finite regular singularities.
In theorem 5 the author states a convergence result for adding up the expansions even if the series does not terminate. Riemann-Hilbert correspondences; representations of the fundamental group; Kuo-Tsai Chen expansion; iterated integrals Carlos Simpson, Transcendental aspects of the Riemann-Hilbert correspondence. \textit{Illinois J. of Math. }34 (1990), 368--391. Ordinary differential equations in the complex domain, Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc., Riemann surfaces; Weierstrass points; gap sequences Transcendental aspects of the Riemann-Hilbert 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 Taking as a model the completed theory of vector space endomorphisms, the present text aims at extending this theory to endomorphisms of finitely generated projective modules over a general commutative ring; now analogous results often require totally different methods of proof.
The first important result is a structure theorem for such modules when the characteristic polynomial of the endomorphism is separable. The second topic deals with the minimal polynomial, whose mere existence is shown to require additional hypotheses, even over a domain. In the third topic we extend the classical notion of 'cyclic modules' as the modules which are invertible over the ring of polynomials modulo the characteristic polynomial. {
} Regarding the diagonalization of endomorphisms, we show that a classical criterion of being diagonalizable over some extension of the base field can be transferred nearly verbatim to rings, provided that diagonalization is expected only after some faithfully flat base change. Many results that hold over a field, like the fact that commuting diagonalizable endomorphisms are simultaneously diagonalizable, hold over arbitrary rings, with this extended meaning of diagonalization. The Jordan-Chevalley-Dunford decomposition, shown as a particular case of the lifting property of étale algebras, also holds over rings.{
} Finally, in several reasonable situations, the eigenspace associated with any root of the characteristic polynomial is shown to be given a more concrete description as the image of a map. In these situations the classical theory generalizes to rings. projective modules; characteristic polynomials; eigenspaces; étale algebras; Jordan decomposition Projective and free modules and ideals in commutative rings, Algebraic systems of matrices, Relevant commutative algebra, Local structure of morphisms in algebraic geometry: étale, flat, etc., Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra, Linear transformations, semilinear transformations, Eigenvalues, singular values, and eigenvectors, Canonical forms, reductions, classification On the structure of endomorphisms of projective modules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(g\): \(U\to \mathbb{R} \) denote a real analytic function on an open subset \(U\) of \(\mathbb{R} ^n\), and let \(\Sigma \subset \partial U\) denote the points where \(g\) does not admit a local analytic extension. We show that if \(g\) is semialgebraic (respectively, globally subanalytic), then \(\Sigma\) is semialgebraic (respectively, subanalytic) and \(g\) extends to a semialgebraic (respectively, subanalytic) neighbourhood of \(\overline{U}\backslash\Sigma\). (In the general subanalytic case, \(\Sigma\) is not necessarily subanalytic.) Our proof depends on controlling the radii of convergence of power series \(G\) centred at points \(b\) in the image of an analytic mapping \(\varphi\), in terms of the radii of convergence of \(G\circ\widehat{\varphi}_a\) at points \(a\in\varphi^{-1}(b)\), where \(\widehat{\varphi}_a\) denotes the Taylor expansion of \(\varphi\) at \(a\). E. Bierstone, Control of radii of convergence and extension of subanalytic functions, Preprint, University of Toronto (2001). Zbl1082.32002 MR2045414 Semi-analytic sets, subanalytic sets, and generalizations, Analytical algebras and rings, Semialgebraic sets and related spaces, Power series rings, Holomorphic functions of several complex variables Control of radii of convergence and extension of subanalytic 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 \({\mathbb F}_q\) be a finite field of characteristic \(p\). For a function field \(F\) over \({\mathbb F}_q\), \(g(F)\) denotes the genus of \(F\) and \(N(F)\) the number of rational places in \(F\). A tower of function fields is an infinite sequence \({\mathcal F}=\big(F_i\big)_{i\geq 0}\) of function fields \(F_i\) over \({\mathbb F}_q\) such that \(F_0\subseteq F_1\subseteq F_2\subseteq\ldots \), \(F_{i+1}/F_i\) is separable for all \(i\) and \(g(F_i)\xrightarrow[i\to\infty]{}\infty\). Call \(\lambda({\mathcal F}) :=\lim\limits_{i\to\infty} N(F_i)/g(F_i)\) the {\em limit} of the tower. We have \(0\leq \lambda({\mathcal F})\leq \sqrt q-1\). The tower \({\mathcal F}\) is called {\em asymptotically good} if \(\lambda({\mathcal F})>0\). Let \(\sigma({\mathcal F}):=\liminf\limits_{i\to\infty}s(F_i)/g(F_i)\) be the asymptotic \(p\)-rank of \({\mathcal F}\), where \(s(F_i)\) is the \(p\)-rank of \(F_i\). We have \(0\leq \sigma({\mathcal F}) \leq 1\).
The aim of this paper is to construct asymptotically good towers \({\mathcal F}\) with \(\sigma({\mathcal F})\) as small as possible. In Section 4 the authors construct asymptotically good towers over quadratic fields \({\mathbb F}_q\), that is \(q\) is a square, whose asymptotic \(p\)-rank is small.
Let \({\mathcal G}:=(G_i)_{i\geq 0}\) be the optimal tower introduced by \textit{A. Garcia} and \textit{H. Stichtenoth} [J. Number Theory 61, No. 2, 248--273 (1996; Zbl 0893.11047)]. That is: \(G_1:= {\mathbb F}_q(x_1)\) is a rational function field, \(G_0:= {\mathbb F}_q(x_0)\) with \(x_0=x_1^l+x_1\) and for \(i\geq 1\), \(G_{i+1}=G_i(x_{i+1})\) with \(x_{i+1}^l+x_{i+1}=\frac{x_i^l}{ x_i^{l-1}+1}\), where \(q=l^2\). Let \(E:=G_0(y)={\mathbb F}_q (x_0,y)\) with \(y^m=x_0\). Let \({\mathcal E}=E\cdot {\mathcal G}=\big(E_i\big)_{i\geq 0}\) be the composite of the function field \(E\) and the tower \({\mathcal G}\). The main result of the paper is that \(\lambda({\mathcal E})=(l-1)\frac{\gcd(l+1,m)}{m}\) and \(\sigma({\mathcal E})=\frac 1m\).
It is also shown that for any \(\varepsilon >0\), there exists a constant \(B>0\), depending on \(q\), and an asymptotically good tower \({\mathcal F}=\big(F_i\big)_{i\geq 0}\) over \({ \mathbb F}_q\) such that \(\sigma({\mathcal F})<\varepsilon\) and \(|\mathrm{Aut}(F_i)|\geq B\cdot g(F_i)\) for all \(i\geq 0\). That is, there exist function fields over \({\mathbb F}_q\) having large genus, which have simultaneously many rational points, many automorphisms and small \(p\)-rank. towers of function fields; rational places; genus of a function field; automorphisms of function fields; \(p\)-rank Arithmetic theory of algebraic function fields, Curves over finite and local fields, Applications to coding theory and cryptography of arithmetic geometry, Algebraic functions and function fields in algebraic geometry Asymptotically good towers of function fields with small \(p\)-rank | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 present a numerical comparison between cohomology groups of a differential equation with irregular singularities on a Riemann surface and those of associated spectral curve. The author shows a coincidence of index of rigidity of differential equations with irregular singularities on a compact Riemann surface and Euler characteristic of the associated spectral curves which are recently called irregular spectral curves. Also he presents a comparison of local invariants, so called Milnor formula which links the Komatsu-Malgrange irregularity of differential equations and Milnor number of the spectral curves. This paper is organized as follows: Section 1, deals with spectral curves of differential equations. Section 2, deals with Local formal theory on differential equations. In this section, the author recalls the Hukuhara-Turrittin-Levelt theory on local structure of differential equations and the notion of irregularity introduced by \textit{H. Komatsu} [J. Fac. Sci., Univ. Tokyo, Sect. I A 18, 379--398 (1971; Zbl 0232.34026)] and \textit{B. Malgrange} [Lect. Notes Math. 712, 77--86 (1979; Zbl 0423.32014)]. Section 3, deals with Local comparison: Milnor formula. Here the author deals with a local differential module and define its characteristic polynomial with respect to a fixed basis. The zero locus of this characteristic polynomial may have a singularity at infinity which corresponds to the irregular singularity of the differential module. He compares these singularities and obtain a comparison formula between the irregularity of differential module and the Milnor number of the characteristic polynomial. Section 4, deals with global comparison: Euler characteristics. The author shows the matching of the index of rigidity of a differential equation and the Euler characteristic of the corresponding spectral curve. irregular singularity; spectral curve; index of rigidity; Euler characteristic; Komatsu-Malgrange irregularity; Milnor number Relationships between algebraic curves and integrable systems, Singularities of curves, local rings, Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms Index of rigidity of differential equations and Euler characteristic of their spectral 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 [\textit{A. Blasco} and \textit{S. Pérez-Díaz}, Comput. Aided Geom. Des. 31, No. 2, 81--96 (2014; Zbl 1301.14033)], a method for computing \textit{generalized asymptotes} of a real algebraic plane curve implicitly defined is presented. Generalized asymptotes are curves that describe the status of a branch at points with sufficiently large coordinates and thus, it is an important tool to analyze the behavior at infinity of an algebraic curve. This motivates that in this paper, we analyze and compute the generalized asymptotes of a real algebraic space curve which could be parametrically or implicitly defined. We present an algorithm that is based on the computation of the \textit{infinity branches} (this concept was already introduced for plane curves in [\textit{A. Blasco} and \textit{S. Pérez-Díaz}, ``Asymptotic behavior of an implicit algebraic plane curve'', \url{arXiv:1302.2522}]). In particular, we show that the computation of infinity branches in the space can be reduced to the computation of infinity branches in the plane and thus, the methods in [\url{arXiv:1302.2522}] can be applied. algebraic space curve; convergent branches; infinity branches; asymptotes; perfect curves Blasco, A.; Pérez-Díaz, S.: Asymptotes of space curves, (2014) Computational aspects of algebraic curves, Computer-aided design (modeling of curves and surfaces), Curves in Euclidean and related spaces Asymptotes of space curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(P_1,\dots,P_s\) be \(s\) distinct points in the projective space \({\mathbb P}^n:={\mathbb P}_k^n\) with \(k\) an algebraically closed field of arbitrary characteristic and let \(m_1,\dots,m_s\) be positive integers. The zero-dimensional scheme \(Z:=m_1P_1+\dots+m_sP_s\) defined by all forms in the polynomial ring \(R:=k[x_0,\dots,x_n]\) vanishing at \(P_i\) with multiplicity at least \(m_i\) is called a set of fat points in \({\mathbb P}^n\) and \(I:=I_Z\) is its defining ideal.
Let consider the graded ring \(A := R/I\), which is the homogeneous coordinate ring of \(Z\). It is well known that \(A =\sum_{t\geq 0}A_ t\) is a one-dimensional Cohen-Macaulay graded ring whose multiplicity is \(e(A) = \sum_{i=1}^{s}{{m_i + n-1}\choose{ n}}\).
Furthermore, the Hilbert function \(H_A(t) := \dim_k A_t\) strictly increases until it reaches the multiplicity, at which it stabilizes. The regularity index of \(A\) (or of the fat points \(Z\)) is defined to be the least integer \(t\) such that \(H_A(t) = e(A)\), and we will denote it by \(\mathrm{reg}(Z)\) (or by \(\mathrm{reg}(A)\)).
In this paper, the author shows a formula to compute the regularity index of \(s+2\) fat points not lying on a linear \((s-1)\)-space in \({\mathbb P}^n\), \(s\leq n\) (see Theorem 3.4). This result shows that the Segre bound is attained in the case of \(s+2\) fat points not lying on a linear \((s-1)\)-space. Since \(s+2\) fat points in general position never lie on a linear \((s-1)\)-space, the result also generalizes a formula to compute the regularity index of fat points in general position in \({\mathbb P}^n\) given by \textit{M. V. Catalisano, N. V. Trung} and \textit{G. Valla} [Proc. Am. Math. Soc. 118, No. 3, 717--724 (1993; Zbl 0787.14030)]. fat point; regularity index; zero-scheme DOI: 10.1080/00927872.2011.593385 Divisors, linear systems, invertible sheaves, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Regularity index of \(s + 2\) fat points not on a linear \((s - 1)\)-space | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a field of characteristic 0 and let \(K=k((t))\) be the field of Laurent series over it. Let \(X\) be the analytification, in the sense of Berkovich theory, of a smooth projective variety over \(K\). Let \(L\) be an ample line bundle on \(X\).
In [J. Algebr. Geom. 4, No. 2, 281--300 (1995; Zbl 0861.14019)], \textit{S.-W. Zhang} introduced a notion of positivity for metrics on \(L\). By a classical construction, to any model \(\mathcal{X}\) of \(X\) over \(R=k[[T]]\) and any model \(\mathcal{L}\) of \(L\) over \(\mathcal{X}\), one can associate a continuous metric \(\| \cdot\|_{\mathcal{L}}\) on \(X\). When \(\mathcal{L}\) is nef on the special fiber of \(\mathcal{X}\), this metric is called a semipositive model metric. In general, a continuous metric on \(L\) is called semipositive when it is a uniform limit of semipositive model metrics.
The aim of the present article is to extend the notion of positivity to non-necessarily continuous metrics and to prove that it satisfies the expected basic properties. To do so, the authors use fine topological properties of \(X\): it is homeomorphic to the projective limit \(\varprojlim_{\mathcal{X}} \Delta_{\mathcal{X}}\), where \(\mathcal{X}\) runs through the set of SNC models of \(X\) (regular models whose special fiber is simple normal crossing) and \(\Delta_{\mathcal{X}}\) is a compact simplicial complex. More precisely, \(\Delta_{\mathcal{X}}\) may be realized as the dual complex of the special fiber of \(\mathcal{X}\) (encoding the multiple intersections between its irreducible components), it embeds canonically into \(X\) and there is a retraction \(p_{\mathcal{X}}\) on \(X\) onto its image (still denoted by \(\Delta_{\mathcal{X}}\)). The fact that only SNC models need to be considered follows from desingularization results by \textit{M. Temkin} (see [Adv. Math. 219, No. 2, 488--522 (2008; Zbl 1146.14009)]) and use the fact that the residue field \(k\) of \(K\) has characteristic 0.
From now on we fix a reference model metric \(\|\cdot\|\) with curvature form \(\theta\). The authors say that a function \(\varphi\) on \(X\) is a \(\theta\)-plurisubharmonic (\(\theta\)-psh) model function when \(\|\cdot\| e^{-\varphi}\) is a semipositive model metric. A general \(\theta\)-psh function on \(X\) is then defined to be an upper semicontinuous function such that, for each SNC model \(\mathcal{X}\) of \(X\), we have \(\varphi \leq \varphi \circ p_{\mathcal{X}}\) and the restriction of \(\varphi\) to \(X\) is a uniform limit of restrictions of \(\theta\)-psh model functions.
The main results of the paper are analogues of well-known results in the complex case: A) the set of \(\theta\)-psh functions on \(X\) moduling scaling is compact; B) every \(\theta\)-psh function is the pointwise limit of a decreasing net of \(\theta\)-psh model functions. The proofs use computations of intersection numbers on the special fibers of the models, toroidal techniques to construct appropriate models and a certain cohomological vanishing property of multiplier ideals (that also requires residue characteristic 0). semipositive metrics; plurisubharmonic functions; Berkovich spaces S. Boucksom, C. Favre and M. Jonsson, Singular semipositive metrics in non-archimedean geometry, preprint (2011), . Rigid analytic geometry, Plurisubharmonic functions and generalizations Singular semipositive metrics in non-Archimedean geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \( G \) be an \( n\times n \) matrix with coefficients in \( {\mathbb Q}(x) \) and consider the differential module \( {\mathcal M }\) associated to the system of linear differential equations \( X' =GX\). For each prime \( p\), one can define its ``\(p\)-adic generic radius of convergence'' \( R_{p}\). Following André's terminology, the differential module \( {\mathcal M }\) is said to be a \( G\)-module if its ``inverse global radius'', namely \( \rho({\mathcal M})=\sum_{p}\max(-\log(R_{p}),0)\), is finite. It is known that all differential modules ``coming from geometry'' are \( G\)-modules and the converse is conjectured. One can also define the size \( \sigma({\mathcal M}) \) of \( {\mathcal M}\). Roughly speaking, it is the mean of the (logarithmic) size of the Taylor coefficients of its generic solutions. The Bombieri-André theorem asserts that \( 0\leq\rho(G)-\sigma(G)\leq n-1 \) and implies that \( G\)-modules are also characterized by the finiteness of their size. The aim of this paper is to improve the knowledge of \( \rho(G)-\sigma(G) \) in connection with the orders of nilpotence \( m(p) \) of the mod \( p \) reduction of \( {\mathcal M }\) (up to finitely many primes \( p \) this is well defined). Especially that difference is computed when \( {\mathcal M }\) is an hypergeometric differential module.
This paper is Dwork's last one. The way it gives a deep insight in the \( G\)-functions theory is a fine example of the cleverness of this great mathematician. It has been generalized to the several variables case by \textit{L. Di Vizio} [On the arithmetic size of linear differential equations. J. Algebra 242, No. 1, 31-59 (2001; Zbl 0998.12008)]. \(G\)-modules; \(G\)-functions; \(p\)-adic generic radius of convergence; inverse global radius; Bombieri-André theorem; hypergeometric differential module Dwork, B.: On the size of differential modules. Duke math. J. 96, 225-239 (1999) \(p\)-adic differential equations, Arithmetic ground fields (finite, local, global) and families or fibrations, Other hypergeometric functions and integrals in several variables, Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions), Transcendence theory of other special functions On the size of differential modules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article under review expands upon the existing theory of limit linear series, for nodal curves which are not of compact type, which has been developed in [\textit{B. Osserman}, J. Reine Angew. Math. 753, 57--88 (2019; Zbl 1419.14037); Math. Z. 284, No. 1--2, 69--93 (2016; Zbl 1378.14032)] and [\textit{O. Amini} and \textit{M. Baker}, Math. Ann. 362, No. 1--2, 55--106 (2015; Zbl 1355.14007)].
The author's main result is a combinatorial construction which allows him to establish a precise comparison between his theory and that of Amini and Baker. It translates the concept of limit linear series for nodal curves of non-compact type into statements about limit linear series for metric graphs. In doing so, he obtains a version of Clifford's theorem for limit linear series on nodal curves.
The author also makes comparisons between his theory of multidegrees, for dual graphs, and the concept of multidegrees for divisors on non-metric graphs in the sense of \textit{M. Baker} and \textit{S. Norine} [Adv. Math. 215, No. 2, 766--788 (2007; Zbl 1124.05049)]. As one result, he proves that if \(r \geq g\) or \(d > 2g - 2\), then there is no limit \(\mathfrak{g}^r_d\) with \(r > d - g\) on any nodal genus \(g\) curve.
Finally, the author establishes a characterization of limit linear series for curves of pseudocompact type. He then concludes the article by presenting examples before discussing a concept of Brill-Noether generality for metric graphs. limit linear series; nodal curves; metric graphs Divisors, linear systems, invertible sheaves, Families, moduli of curves (algebraic), Special divisors on curves (gonality, Brill-Noether theory) Limit linear series and the Amini-Baker construction | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main motivation of this paper is the study of bounded \(t\)-structures, after the key contributions of Antieau, Gepner and Heller who proved that obstructions to the existence of bounded \(t\)-structures on a stable \(\infty\)-category \(\mathscr{C}\) are controlled by the first negative \(K\)-group \(K_{-1}(\mathscr{C})\) [\textit{B. Antieau} et al., Invent. Math. 216, No. 1, 241--300 (2019; Zbl 1430.18009)]. This work tackles the problem assuming that \(\mathscr{C}\) is endowed with a \textit{weight structure}, a structure introduced in [\textit{M. V. Bondarko}, J. \(K\)-Theory 6, No. 3, 387--504 (2010; Zbl 1303.18019)], similar (but not dual) to a \(t\)-structure, which axiomatizes the properties of naive truncations of chain complexes. This notion is closely related to the concept of \textit{regularity} of \(\mathbb{E}_1\)-ring spectra, introduced in [\textit{C. Barwick} and \textit{T. Lawson}, ``Regularity of structured ring spectra and localization in \(K\)-theory'', Preprint, \url{arXiv:1402.6038}].
In the first part, the author contributes to the topic of regular \(\mathbb{E}_1\)-ring spectra by proving the stability of regularity spectra under localizations (Proposition \(2.8\)) and the discreteness of bounded regular \(\mathbb{E}_1\)-ring spectra which are \textit{quasicommutative} (Theorem \(2.11\)). The paper also provides a counterexample to the latter statement in the non-quasicommutative case (Construction \(2.12\)). The bulk of this section, however, is the (twofold) generalization of the concept of regularity to spectral stacks (Definition \(2.15\)): a spectral stack \(X\) is \textit{regular} if there exists a regular atlas \(\operatorname{Spec}(R) \to X\), while it is \textit{homological regular} if the standard \(t\)-structure on \(\operatorname{QCoh}(X)\) restricts to the subcategory of compact objects. While in general not equivalent, the two definitions agree if \(X\) is an affine spectral scheme or, with some additional hypothesis, if \(X\) is a quotient of a Noetherian connective \(\mathbb{E}_{\infty}\)-\(k\)-algebra under the action of a smooth affine group scheme (Theorem \(2.16\)).
In the second part, the author recalls the main definitions, properties and examples of weight structures on stable \(\infty\)-categories, and introduces the notion of \textit{adjacent structures} (Definition \(3.9\)) - i.e., a weight structure \(\left(\mathscr{C}_{w\geqslant 0},\mathscr{C}_{w\leqslant 0}\right)\) and a \(t\)-structure \(\left(\mathscr{C}_{t\geqslant 0},\mathscr{C}_{t\leqslant 0}\right)\) such that \(\mathscr{C}_{w\geqslant 0}=\mathscr{C}_{t\geqslant 0}\). Arguing that the standard \(t\)-structure and the standard weight structure on \(\operatorname{Mod}^{\operatorname{perf}}_R\) are adjacent if and only if \(R\) is regular, the author conjectures that the existence of adjacent structures on a stable \(\infty\)-category \(\mathscr{C}\) should be a noncommutative analogue of regularity. Hence, it is proposed that if all the mapping spaces in the heart of the weight structure \(\operatorname{Hw}(\mathscr{C})\) are \(N\)-truncated for some fixed \(N\), the \(\infty\)-category \(\operatorname{Hw}(\mathscr{C})\) is actually discrete (Conjecture \(3.12\)). Finally, the author proves that if \(X\) is a quotient spectral stack satisfying the assumptions of Theorem \(2.16\), then Conjecture \(3.12\) holds for \(\operatorname{QCoh}(X)\). regularity; ring spectra; spectral stacks; weight structure; \(t\)-structure; quotient stack \((\infty,1)\)-categories (quasi-categories, Segal spaces, etc.); \(\infty\)-topoi, stable \(\infty\)-categories, Generalizations (algebraic spaces, stacks), Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Regular local rings, Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) Regularity of spectral stacks and discreteness of weight-hearts | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review is about moduli spaces of ``decorated swamps'' on a fixed smooth projective curve \(X\) over the complex numbers.
``Swamps'' are vector bundles with an additional structure, namely a nontrivial homomorphism \(E_\rho\to L\) to some line bundle \(L\), where \(E_\rho\) is the vector bundle obtained from \(E\) via a (fixed) linear representation \(\rho: \mathrm{GL}(r)\to \mathrm{GL}(V)\). Stability and moduli spaces of swamps where studied in the work of Schmitt. This framework unifies the treatment of stability and construction of moduli spaces for versions of vector bundles with additional structure.
The author generalizes this study to what he calls ``decorated swamps'': on top of \(E\), the line bundle \(L\) and the homomorphism \(E_\rho\to L\), he introduces the local datum of a point \(s\) of the fiber of \((E_\sigma^\vee)_{x_0}\), where \(x_0\) is a fixed point of \(X\) and \(\sigma\) is a second linear representation of \(\mathrm{GL}(r)\). He defines a stability condition for these and, via a careful GIT computation, constructs (Theorem 3.9) a projective coarse moduli space of semi-stable decorated swamps up to \(S\)-equivalence (which is a generalization of the usual notion for vector bundles).
In the last section it is shown that if this constructions is applied to parabolic vector bundles and to vector bundles with level structure, one recovers the know notions of stability for those cases. moduli spaces; parabolic vector bundles; level structure N. Beck, \textit{Moduli of decorated swamps on a smooth projective curve}, Internat. J. Math. \textbf{26} (2015), no. 10, 1550086, 29 pp. Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles Moduli of decorated swamps on a smooth projective 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 authors of this interesting article are concerned with the study of coarse moduli spaces parameterizing families of stable coherent systems over nodal reducible curves, that is, pairs \((E,V)\) where \(E\) is a locally free sheaf and \(V\) is a subspace of global sections of \(E\). For fixed degree \(d=\text{deg}(E)>0\) and in the case when \(0<k=\text{dim}(V)<r=\text{rk}(E)\), the moduli spaces parameterize BGN extensions, and the authors generalize the notion of a BGN extension of type \((r,d,k)\) to nodal reducible curves in order to study the moduli spaces of stable coherent systems.
The analysis is depending on the choice of polarization \(\underline{w}\), and the authors use a notion of \textit{good polarization}, which in particular when the curve \(C\) is of compact type, good polarizations are exactly those for which the trivial bundle \(\mathcal{O}_C\) is \(\underline{w}\)-stable. For \((C, \underline{w})\) a polarized nodal curve with \(\underline{w}\) good and for \(0<k<r\), the irreducible components of the moduli space parameterizing coherent systems \((E,V)\) with \(E\) locally free are characterized. In particular, the main result of the article proves that any irreducible component of the moduli space is birational to a Grassmannian fibration over irreducible components of the moduli space of \(\underline{w}\)-semistable depth one sheaves whose \(\underline{w}\)-rank and \(\underline{w}\)-degree are \((r-k)\) and \(d\), respectively; these components are the only ones which contain coherent systems \((E,V)\) with \(E\) locally free. coherent system; nodal curve; polarization; stability; moduli space; BGN extension Vector bundles on curves and their moduli, Sheaves in algebraic geometry, Algebraic moduli problems, moduli of vector bundles Coherent systems and BGN extensions on nodal reducible curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study the problem of finding the bifurcation set of polynomial mappings \(\mathbb{C}^m \rightarrow \mathbb{C}^k\). They give characterization in the case of \(m = k+1 \geq 3\). The conditions involve the Betti numbers of the fibers and uses a vanishing result from real setting.
In the case of \(\mathbb{C}^{n+1} \rightarrow \mathbb{C}^n, n \geq 2\), it turns out that even if the Betti numbers of the fibers around a regular value \(\lambda\) are constant, \(\lambda\) still might be in the bifurcation set, as demonstrated by an example \(f: \mathbb{C}^3 \rightarrow \mathbb{C}^2\) where \(f(x,y,z) = (x, ((x - 1)(xz + y^2) + 1)(x(xz + y^2) - 1))\), and \(\lambda = (0,0)\).
The first main result is the following complete characterization. \(\lambda \in \mathrm{Im} f \setminus \overline{f(\mathrm{Sing}f)}\) is not in the bifurcation set if and only if the Euler characteristic of the fibers \(f^{-1}(t)\) is constant for all \(t\) in a neighborhood and no connected components of the fibers vanish at infinity as \(t\) approaches \(\lambda\). This is true in general for \(f : X \rightarrow Y\), \(X,Y\) nonsingular connected affine varieties of dimensions \(n+1\) and \(n \geq 2\), respectively. In fact, the analogous result holds for connected complex manifolds as well (the first one being Stein), provided the first two Betti numbers of the fibers are finite. The proof relies on a deep result from [\textit{Y. S. Ilyashenko}, Topol. Methods Nonlinear Anal. 11, No. 2, 361--373 (1998; Zbl 0927.32020)] and [\textit{G. Meigniez}, Trans. Am. Math. Soc. 354, No. 9, 3771--3787 (2002; Zbl 1001.55016)].
As applications of the main theorem, several nice corollaries are mentioned about characterizing the bifurcation set and establishing local triviality of mappings around specific values under certain conditions. bifurcation set; polynomial mapping; locally trivial fibration Fibrations, degenerations in algebraic geometry, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Critical points of functions and mappings on manifolds, Topological properties of mappings on manifolds, Singularities of differentiable mappings in differential topology Bifurcation set of multi-parameter families of complex curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A virtually smooth scheme of expected dimension \(d\) is a pair \((X,E^{\bullet})\) where \(X\) is a scheme that can be embedded in a smooth scheme and \(E^{\bullet}\) is a 1-perfect obstruction theory for \(X\) with rk \( E^{\bullet} = d\). Virtually smooth schemes were shown in \textit{J. Li} and \textit{G. Tian} [First Int. Press Lect. Ser. 1, 47--83 (1998; Zbl 0978.53136)] and \textit{K. Behrend} and \textit{B. Fantechi} [Invent. Math. 128, No. 1, 45--88 (1997; Zbl 0909.14006)] to admit a virtual fundamental class suited for doing intersection theory.
In the present paper, the authors define and study virtual versions of the holomorphic Euler characteristic for elements in \(K^0(X)\) and of two refinements of this: the \(\chi_{-y}\)-genus and the elliptic genus. These are shown to be deformation invariant and to reduce to the usual definitions in the case \(X\) is smooth and \(E^{\bullet}\) is the cotangent bundle. The virtual \(\chi_{-y}\)-genus is shown to be a polynomial of degree \(d\) and, as a consequence, it is also possible to define a virtual version of the topological Euler characteristic. If \(X\) has lci singularities and \(E^{\bullet}\) is the cotangent complex \(L_X^{\bullet}\), then the topological Euler characteristic is shown to coincide with Fulton's Chern class, which is then invariant under deformations for proper lci schemes.
The main results of the paper are virtual versions of the Theorems of Hirzebruch-Riemann-Roch and of Grothendieck-Riemann-Roch (in the later the target of the morphism is required to be a smooth scheme).
Similar results holding for \([0,1]\)-manifolds were proven by \textit{I. Ciocan-Fontanine} and \textit{M. Kapranov} in [Geom. Topol. 13, No. 3, 1779--1804 (2009; Zbl 1159.14002)].
Using the virtual version of Grothendieck-Riemann-Roch and Graber-Pandharipande virtual localization [see \textit{T. Graber} and \textit{R. Pandharipande} [Invent. Math. 135, No. 2, 487--518 (1999; Zbl 0953.14035)], the authors establish a virtual localization formula for the virtual Euler characteristic in the presence of an equivariant action of a torus.
The authors finish by showing how their results can be applied in the study of \(K\)-theoretic Donaldson invariants in moduli spaces of stable sheaves. Riemann-Roch theorems; virtual fundamental class; virtual Euler characteristic; genus; localization Fantechi, B.; Göttsche, L., Riemann-Roch theorems and elliptic genus for virtually smooth schemes, Geom. Topol., 14, 83-115, (2010) Riemann-Roch theorems, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Characteristic classes and numbers in differential topology Riemann-Roch theorems and elliptic genus for virtually smooth schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems An interesting new development of microlocal analysis related with real algebraic geometry is presented in this paper, especially on semi-algebraic manifolds (called Nash manifolds). The choice of Nash manifolds is naturally motivated in view of algebraic character of the adopted definitions.
More detailed, on any Nash manifold are defined analogous of the classical Schwartz spaces \({\mathcal E}_c^\infty\), \({\mathcal E}^\infty(M)\) and \({\mathcal E}^{-\infty}(M)\) (recalled in the paper) and the analogous spaces are denoted respectively by \({\mathcal S}(M)\), \({\mathcal T}(M)\) and \({\mathcal G}(M)\). It is showed for \(M=\mathbb{R}^n\), \({\mathcal S}(M)\) is put the space of classical Schwartz function and \({\mathcal G}(M)\) is the space of classical generalized Schwartz functions. When \(M\) is a compact Nash manifold the mentioned spaces coincide, i.e. \({\mathcal S}(M)={\mathcal T}(M)={\mathcal E}^\infty(M)\). The obtained results are summarized as follows:
1. If \(Z\) is a closed Nash submanifold of an affine Nash manifold \(M\), the restriction maps
\[
{\mathcal T}(M)\to{\mathcal T}(Z)\quad \text{and}\quad{\mathcal S}(M)\to{\mathcal S}(Z)
\]
are surjective maps, i.e. the restrictions are continuous and onto.
2. If \(U\) is a semi-algebraic open subset of the Nash manifold \(M\), the Schwartz functions on \(U\) admit extension by zero \({\mathcal S}(U)\to{\mathcal S}(M)\) on \(M\), i.e. Schwartz functions on \(U\) coincide with Schwartz function on \(M\) which vanish with all its derivatives on \(M\setminus U\).
It is to remark that the classical generalized functions do not have this property of extension by zero. By the way a nice exposition of a part of semi-algebraic geometry is presented and many geometric notions related with Nash manifolds are developed, like Nash vector bundles, sheaves, etc. On this base the Nash differential operators are defined. The exposition of all material is well organized, very elegant and very clear. This paper can be recommended as a model of clearness (``clarté'' in French). Nash manifolds; Schwartz spaces; entension by zero Möllers, J.: Symmetry breaking operators for strongly spherical reductive pairs and the Gross-Prasad conjecture for complex orthogonal groups. arXiv:1705.06109 Real-analytic and Nash manifolds, Nash functions and manifolds Schwartz functions on Nash manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a matrix \(A\) in \(\mathbb{R}^{m \times k}\), a singular value decomposition (SVD) of \(A\) is \(A = U \cdot \Sigma \cdot V^t\) where \(U \in \mathbb{R}^{m \times m}\) and \(V \in \mathbb{R}^{k \times k}\) are orthogonal and \(\Sigma \in \mathbb{R}^{m \times k}\) with non-negative real numbers on the diagonal. It's very well known that SVD has many applications in numerical linear algebra. In addition, the fact that the matrix \(A\) defines a linear map \(A\) from \(\mathbb{R}^k\) to \(\mathbb{R}^m\) allows authors to generalize SVD to complexes. Now, let \(C_{\bullet}\) be a finite complex of finite-dimensional \(\mathbb{R}\)-vector spaces consisting of vector spaces \(C_i \cong \mathbb{R}^{c_i}\) and differentials from \(C_i\) to \(C_{i-1}\) given by matrices \(A_i\), \(i=1,\dots,n\). In this paper, the authors extend the notion of SVD to finite complexes of vector spaces proving that, for the matrices \(A_i\), \(i=1,\dots,n\), of the differentials which define the complex \(C_{\bullet}\), there exist sequences \(U_0,\dots,U_n\) and \(\Sigma_1,\dots,\Sigma_n\) of orthogonal and diagonal matrices, respectively, verifying similar relations to the singular value decompositions. The authors also define the concept of pseudoinverse complex and they present two algorithms for computing the SVD of a finite complex of vector spaces. Many examples illustrate the algorithms. Finally, an application to syzygies, concerning the computation of Betti numbers in free resolutions of graded modules, is given. singular value decomposition; homology; complex; algorithm Geometric aspects of numerical algebraic geometry, Syzygies, resolutions, complexes and commutative rings, Eigenvalues, singular values, and eigenvectors, Numerical linear algebra, Numerical algebraic geometry Singular value decomposition of complexes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0728.00008.]
Let \(X_ 0(N)\) be the modular curve for \(\Gamma_ 0(N)\), and let \(\phi : X_ 0(N)\rightarrow E\) be a non-constant holomorphic map to an elliptic curve \(E\): \(y^ 2=4x^ 3-g_ 2x-g_ 3\); the interesting case is when \(E\) is defined over the rationals. Such a map \(\phi\) can be written as \(\phi(z)=(\alpha(z),\beta(z))\), with \(\alpha\) and \(\beta\) meromorphic automorphic functions on \(X_ 0(N)\), that satisfy the equation defining \(E\). In examples it is often possible to identify \(\alpha\) and \(\beta\) explicitly. After all, these functions are determined by having well specified singularities at the points of \(X_ 0(M)\) sent by \(\phi\) to the point at infinity of \(E\).
This paper gives a uniform description of \(\alpha\) and \(\beta\) in terms of the points sent to infinity, the ramification index of \(\phi\) at these points, and the holomorphic cusp form of weight 2 for \(\Gamma_ 0(N)\) that corresponds to the pullback of the canonical differential form \(y^{-1}dx\) on \(E\). Furthermore, it indicates how \(g_ 2\) and \(g_ 3\) may be expressed in quantities related to automorphic forms for \(\Gamma_ 0(N)\).
A remarkable aspect of the proof is the wider point of view that \(\alpha\) and \(\beta\) are even determined as harmonic automorphic forms of weight 0 with prescribed singularities. As such they can be expressed in meromorphically extended Poincaré series, taken at the value of the spectral parameter at which they are harmonic. These Poincaré series have been studied by \textit{H. Neunhöffer} [Über die analytische Fortsetzung von Poincaréreihen, S.ber. Heidelb. Akad. Wiss., Math.-Nat. Kl. 1973, 2. Abh., 33-90 (1973; Zbl 0272.10015)]. Individually, the resulting harmonic automorphic forms are not necessarily meromorphic on \(X_ 0(N)\). But linear combinations of them are found with the right singularities. Hence these combinations are equal to the meromorphic automorphic functions \(\alpha\) and \(\beta\). modular curve; holomorphic map; elliptic curve; meromorphic automorphic functions; singularities; harmonic automorphic forms; Poincaré series Holomorphic modular forms of integral weight, Forms of half-integer weight; nonholomorphic modular forms, Elliptic curves, Elliptic curves over global fields, Fourier coefficients of automorphic forms Parametrization of modular elliptic curves by Poincaré 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 In [\textit{K. Thas}, Proc. Japan Acad., Ser. A 90, No. 1, 21--26 (2014; Zbl 1329.14009)] it was explained how one can naturally associate a Deitmar scheme (which is a scheme defined over the field with one element, \(\mathbb{F}_1\)) to a so-called ``loose graph'' (which is a generalization of a graph). Several properties of the Deitmar scheme can be proven easily from the combinatorics of the (loose) graph, and known realizations of objects over \(\mathbb{F}_1\) such as combinatorial \(\mathbb{F}_1\)-projective and \(\mathbb{F}_1\)-affine spaces exactly depict the loose graph which corresponds to the associated Deitmar scheme. In this paper, we first modify the construction of [loc. cit.], and show that Deitmar schemes which are defined by finite trees (with possible end points) are ``defined over \(\mathbb{F}_1\)'' in Kurokawa's sense; we then derive a precise formula for the Kurokawa zeta function for such schemes (and so also for the counting polynomial of all associated \(\mathbb{F}_q\)-schemes). As a corollary, we find a zeta function for all such trees which contains information such as the number of inner points and the spectrum of degrees, and which is thus very different than Ihara's zeta function (which is trivial in this case). Using a process called ``surgery,'' we show that one can determine the zeta function of a general loose graph and its associated {Deitmar, Grothendieck}-schemes in a number of steps, eventually reducing the calculation essentially to trees. We study a number of classes of examples of loose graphs, and introduce the \textit{Grothendieck ring of}\(\mathbb{F}_1\)\textit{-schemes} along the way in order to perform the calculations. Finally, we include a computer program for performing more tedious calculations, and compare the new zeta function to Ihara's zeta function for graphs in a number of examples. field with one element; Deitmar scheme; loose graph; zeta function; Ihara zeta function Mérida-Angulo, M.; Thas, K., Deitmar schemes, graphs and zeta functions, J. Geom. Phys., 117, 234-266, (2017) Schemes and morphisms, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Deitmar schemes, graphs and zeta functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\alpha_1,\dots,\alpha_m\) be linear functions on \(\mathbb C^n\) and \(X=\mathbb C^n\setminus V(\alpha )\), where \(\alpha =\prod _{i=1}^m\alpha _i\) and \(V(\alpha )=\{p\in \mathbb C^n:\alpha (p)=0\}\). The coordinate ring \(\mathcal{O}_X=\mathbb C[x]_\alpha\) of \(X\) is a holonomic \(A_n\)-module, where \(A_n\) is the \(n\)-th Weyl algebra, and since holonomic \(A_n\)-modules have finite length, \(\mathcal{O}_X\) has finite length. We consider a ``twisted'' variant of this \(A_n\)-module which is also holonomic. Define \(M_\alpha^\beta\) to be the free rank 1 \(\mathbb C[x]_\alpha\)-module on the generator \(\alpha^\beta\) (thought of as a multivalued function), where \(\alpha ^\beta=\alpha_1^{\beta_1}\dots\alpha _m^{\beta_m}\) and the multi-index \(\beta =(\beta_1,\dots,\beta_m)\in\mathbb C^m\). It is straightforward to describe the decomposition factors of \(M_\alpha^\beta\), when the linear functions \(\alpha _1,\dots,\alpha _m\) define a normal crossing hyperplane configuration, and we use this to give a sufficient criterion on \(\beta\) for the irreducibility of \(M_\alpha^\beta\), in terms of numerical data for a resolution of the singularities of \(V(\alpha )\). hyperplane arrangements; D-module theory Sheaves of differential operators and their modules, \(D\)-modules, Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Relations with arrangements of hyperplanes Decomposition factors of D-modules on hyperplane configurations in general position | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 considers periodic continued fractions of the form
\[
\phi(\lambda)=b_ 0+{\lambda-\alpha_ 1\over{b_ 1 + {\lambda-\alpha_ 2\over {\ddots b_{N-1}+{\lambda-\alpha_N\over b_N-b_0+\phi}}}}}
\]
which occur in the study of integrable systems and associated periodic dressing chains. Here \(\{\alpha_i\}\) is a fixed \(N\)-periodic sequence \((\alpha_{N+i}=\alpha_i,\quad i \geq 1)\) which is the fundamental datum, \(\{b_i\}\) is a second sequence of complex numbers, \(N\)-periodic from \(i=1\), and \(\lambda\) is a formal parameter. The authors call this a ``periodic \(\alpha\)-fraction''. Convergence is not discussed but, quite formally, periodicity of the fraction implies that \(\phi\) satisfies a quadratic equation \(A(\lambda)\phi^2+2B(\lambda)\phi+ C(\lambda)=0\), where \(A,B,C\) are polynomials in \(\lambda\) with coefficients depending polynomially on the \(b_i,\alpha_i\). Thus \(\phi = -B+\sqrt{R(\lambda}/A\) where \(R=B^2-AC\), so that \(\phi\) is an algebraic function on the hyperelliptic curve \(y^2=R(\lambda)\).
The following questions are addressed:
(1) Which algebraic functions of the form \(\phi\) admit \(N\)-periodic \(\alpha\)-fraction expansions?
(2) How many such expansions are there for given \(\phi\) and how does one find them?
(3) What is the geometry of the set of functions \(\phi\) from a given hyperelliptic extension (i.e. with \(R\) fixed) which admit periodic \(\alpha\)-fraction expansions?
Answers are given assuming odd \(N\), and all \(\alpha_i\) distinct. Briefly, let \(\mathcal{A}=\prod (\lambda-\alpha_i) \). A polynomial \(R(\lambda)\) of degree \(N=2g+1\) is called \(\mathcal A\)-admissible if there exists a polynomial \(S\) of degree \(\leq g\) such that \(R=S^2+\mathcal A\). Then, if \(\phi\) has an \(N\)-periodic \(\alpha\)-expansion, the polynomials \(A,B,C\) introduced above satisfy: \(\deg B\leq g\), \(A\) is monic of degree \(g\), \(-C\) is monic of degree \(g+1\) and \(R=B^2-AC\) is \(\mathcal A\)-admissible.
Conversely, for an open dense subset of such triples the corresponding function has exactly two \(\alpha\)-periodic expansions, which can be found by an effective matrix factorization procedure.(The pure periodic case (\(b_N=b_0)\) is exceptional in that there is a unique expansion). This answers questions 1 and 2. An action of \({\mathbb Z}_2\times S_n\) on the fractions is also discussed. Question 3 is answered by relating the set of triples \((A,B,C)\) with \(R\) fixed and \(\mathcal A\)-admissible to the Jacobian of the hyperelliptic curve \(y^2=R\).
Reviewer's remark. Three types of periodic continued fraction have made appearances in connection with periodicity phenomena in integrable systems: Stieltjes fractions, in [\textit{P. van Moerbeke}, Invent. Math. 37, 45--81 (1976; Zbl 0361.15010)], the ``usual'' type (i.e. defined by analogy with \(\mathbb R\) for finite-tailed Laurent series with uniformising parameter \(1/\lambda\)) in [\textit{S. A. Andrea} and \textit{T. G. Berry}, Linear Algebra Appl. 161, 117--134 (1992; Zbl 0760.65036)] and finally the fractions of the present paper. Is there some unifying principle to be found? continued fraction; hyperelliptic curve; dressing chain Algebraic functions and function fields in algebraic geometry, Jacobians, Prym varieties, Continued fractions; complex-analytic aspects Periodic continued fractions and hyperelliptic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this work, we consider the collection of necessary homological conditions previously obtained via Conley index theory for a Lyapunov semi-graph to be associated to a Gutierrez-Sotomayor flow on an isolating block and address their sufficiency. These singular flows include regular \(\mathcal{R} \), cone \(\mathcal{C} \), Whitney \(\mathcal{W} \), double \(\mathcal{D}\) and triple \(\mathcal{T}\) crossing singularities. Local sufficiency of these conditions are proved in the case of Lyapunov semi-graphs along with a complete characterization of the branched \(1\)-manifolds that make up the boundary of the block. As a consequence, global sufficient conditions are determined for Lyapunov graphs labelled with \(\mathcal{R}, \mathcal{C}, \mathcal{W}, \mathcal{D}\) and \(\mathcal{T}\) and with minimal weights to be associated to Gutierrez-Sotomayor flows on closed singular \(2\)-manifolds. By removing the minimality condition, we prove other global realizability results by requiring that the Lyapunov graph be labelled with \(\mathcal{R}, \mathcal{C}\) and \(\mathcal{W}\) singularities or that it be linear. Conley index; isolating blocks; Lyapunov graph; Poincaré-Hopf inequalities; cone; cross caps; double; triple singularities Index theory for dynamical systems, Morse-Conley indices, Flows on surfaces, Singularities of surfaces or higher-dimensional varieties, Singularities of vector fields, topological aspects, Stratifications in topological manifolds Gutierrez-Sotomayor flows on singular 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 \(f:\mathbb C^n\to\mathbb C^m\) be a polynomial mapping such that \(\#f^{-1}(0)<\infty,\) and \(m,n\) are arbitrary. The Łojasiewicz exponent at infinity \(\mathcal L _\infty(f)\) of \(f\) is defined by
\[
\mathcal L_\infty(f)=\sup\{\nu\in\mathbb R:|f(x)|\geq C|x|^{\nu}
\]
for a constant \(C>0\) and sufficiently large \(|x|\}\). The main result of the paper is that for any linear mapping \(L:\mathbb C^m\to\mathbb C^n\) (provided \(\#(L\circ f)^{-1}(0)<\infty\)) we have
\[
\mathcal L_\infty(f)\geq\mathcal L_\infty(L\circ f)
\]
and that for generic \(L\) we have equality in the above formula. As a corollary the author obtains some inequalities for \(\mathcal L_\infty(f)\) for arbitrary \(m,n\), known in the case \(m=n\). Łojasiewicz exponent at infinity; polynomial mapping [31]S. Spodzieja, The Łojasiewicz exponent at infinity for overdetermined polynomial mappings, Ann. Polon. Math. 78 (2002), 1--10. Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Real polynomials: location of zeros, Holomorphic mappings and correspondences, Polynomials in real and complex fields: location of zeros (algebraic theorems) The Łojasiewicz exponent at infinity for overdetermined polynomial mappings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper gives a considerable extension of Arnold's lists of singularities, though it does not present a set of normal forms. It opens with a general discussion of classification problems, which decides on listing \(\mu\)-constant strata rather than normal forms (modalities are tabulated in {\S}8) and suggests an inductive approach, listing singularities of functions with given restriction to a generic hyperplane (if this has type \(A_ 3\), for example, the list is that of quadruple points of plane curves): the cases considered are sections of type \(A_ n\), \(D_ n\) and \(\tilde D_ n\). Plane curve singularities are discussed in {\S}2, where Arnold's approach is related to Puiseux exponents, and {\S}3 completes Arnold's partial enumeration of the U-series, and gives hints as to how the complete V-series would appear. Here, and in the next few sections on complete intersection singularities, the method used for classification is to reduce (using projections) to another classification problem which has already been solved. The proof that the method works for \(\mu\)-constant strata involves calculations of the Milnor fomula. Several such calculations are given, but one of the key formulae is not proved in high dimensions. There are extensive tables of triple points of functions on \({\mathbb{C}}^ 3\) and of complete intersection curve and surface singularities. constant Milnor number; classification of singularities; singularities of functions; Plane curve singularities; Puiseux exponents; complete intersection singularities Wall C.T.C.: Notes on the classification of singularities. Proc. Lond. Math. Soc. 48, 461--513 (1984) Singularities in algebraic geometry, Singularities of differentiable mappings in differential topology, Local complex singularities, Singularities of surfaces or higher-dimensional varieties, Singularities of curves, local rings, Complex singularities Notes on the classification of singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper gives a clear outline of the authors' calculation of the cohomology groups
\[
\text{Ext}^1_{\Gamma(m+1)}(BP_*,v_n^{-1}BP_*/I_n)
\]
for all \(m, n>0\), where \(\Gamma(m+1) = BP_*(BP)/(t_1,\dots,t_m)\). A change of rings isomorphism re-interprets this \(\text{Ext}\)-group as \(\text{Ext}^1_{\Sigma(n,m+1)}(K(n)_*,K(n)_*)\), where the generalized Morava stabilizer algebra \(\Sigma(n,m+1)\) is the quotient \(\Sigma(n)/(t_1,\dots,t_m)\) of the standard Morava stabilizer algebra \(\Sigma(n)\). The authors determine all these \(\text{Ext}^1\)-groups, which vary in rank from \(n+1\) to \(n^2\) according to the value of \(m\). They sketch the method of calculation, which relies heavily upon their previous and forthcoming publications.
The results are of interest for their applicability to the chromatic approach to calculating the Adams-Novikov \(E_2\)-term for Ravenel's \(p\)-local spectrum \(T(m)\), which has \(BP_*(T(m)) = BP_*[t_1,\dots,t_m] .\)
The interest of the work to a wider readership is enhanced by a lucid introduction which explains the algebraic and homotopy-theoretic significance of the Morava stabilizer groups \(S_n\) and their generalized counterparts \(S_{n,m}\). Morava~ stabilizer algebra; Adams-Novikov spectral sequence; chromatic homotopy theory; Morava~ stabilizer group Stable homotopy theory, spectra, Formal groups, \(p\)-divisible groups, Adams spectral sequences The first cohomology group of the generalized Morava stabilizer algebra | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The book under review presents the elements of the singularity theory of analytic spaces with applications; it consists of a preface, two main parts, three appendices and a bibliography including 158 items among which are 18 references on works written by the authors with collaborators.
The first part deals with complex spaces and germs. It contains basic notions and results of the general theory such as the Weierstraß preparation theorem, the finite coherence theorem with applications, finite and flat morphisms, normalization, singular locus and relations with differential calculus. In addition, the cases of isolated hypersurface and plane curve singularities are treated. Thus the authors describe some well-known invariants of hypersurface singularities including the Milnor and Tjurina numbers and methods of their computation. They also discuss the concept of finite determinacy, the property of quasihomogeneity, algebraic group actions, the classification of simple singularities, the parameterization and resolution of plane curve singularities, the intersection multiplicity and the semigroup of values associated with a plane curve singularity, the conductor and other classical topological and analytic invariants.
The second part is concerned with local deformation theory of complex space germs. First the authors describe the general deformation theory of isolated singularities of complex spaces. Then the notions of versality, infinitesimal deformations and obstructions are considered in detail. The final section contains a new treatment of equisingular deformations of plane curve singularities including a proof for the smoothness of the \(\mu\)-constant stratum which is based on properties of deformations of the parametrization. This result is obtained, in fact, as a further development of ideas by \textit{J. M. Wahl} [Trans. Am. Math. Soc. 193, 143--170 (1974; Zbl 0294.14007)]. Three appendices include a detail description of basic notions and results from sheaf theory, commutative algebra and formal deformation theory.
The book is written in a clear style, almost all key topics are followed by carefully chosen computational examples together with algorithms implemented in the computer algebra system \textsl{Singular} [see \textit{G.-M. Greuel, G. Pfister} and \textit{H. Schönemann}, Singular 3. A computer algebra system for polynomial computations. Centre for Computer Algebra, Univ. Kaiserslautern (2005), \url{http://www.singular.uni-kl.de}]. Moreover, the exposition contains many non-formal comments, remarks and good exercises illustrated by nice pictures. Without a doubt this book is comprehensible, interesting and useful for graduate students; it is also very valuable for advanced researchers, lecturers, and practicians working in singularity theory, algebraic geometry, complex analysis, commutative algebra, topology, and in other fields of pure mathematics. complex spaces and germs; isolated hypersurface singularities; equisingular deformations; \(\mu\)-constant stratum; embedded deformations; plane curve singularities; parametrization; resolution; normalization; versal deformations; obstructions; cotangent complex G.-M. Greuel, C. Lossen, E. Shustin, \(Introduction to Singularities and Deformations\) (Springer, Berlin, 2007) Deformations of complex singularities; vanishing cycles, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Invariants of analytic local rings, Equisingularity (topological and analytic), Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects), Deformations of singularities, Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Research exposition (monographs, survey articles) pertaining to algebraic geometry Introduction to singularities and deformations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper is a study of a special class of polynomial endomorphisms of \(\mathbb{C}^N\) that are called ``Locally Finite'' (LF) and have nice dynamical properties.
It is proven that such an endomorphism \(F\) may be equivalently defined in three ways:
(1) there is a polynomial in one variable \(p(T)\) such that \(p(F)=0,\) where \(F^n\) is the \(n\)th iteration of \(F;\) (2) \(\sup_{n\geq 0} \deg F^n<\infty;\) (3) \(\dim \text{Span}_{n\geq 0} \;r\circ F^n<\infty\) for each \(r\in\mathbb{C}[x_1,\dots,x_N].\) The special type of the vanishing polynomial \(p(T)\) is found.
Then the unipotent and semisimple LF polynomial automorphisms are defined and it is shown that any LF polynomial automorphism is a composition of uniquely defined commuting LF polynomial automorphisms \(F_s\) and \(F_u,\) where \(F_s\) is semisimple and \(F_u\) is unipotent. The authors prove also that the exponential defines a bijective map from the set of locally nilpotent derivations on the ring \(\mathbb{C}[x_1,\dots,x_N]\) to the set of unipotent polynomial automorphisms of \(\mathbb{C}^N.\)
For \(N=2\) the vanishing polynomial for a LF polynomial automorphism \(F\) of degree \(d\) is found explicitly and the degree of minimal vanishing polynomial is shown to be less than \(d+1.\) polynomial endomorphism; derivations Furter J.-P., Maubach S.: Locally finite polynomial endomorphisms. J. Pure Appl. Algebra 211(2), 445--458 (2007) Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Automorphisms, derivations, other operators for Lie algebras and super algebras Locally finite polynomial endomorphisms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f\) be a real analytic function defined in a neighbourhood of the origin in \(\mathbb{R}^{n}.\) The main theorem of the paper is the resolution of singularities of \(f\) in the following sense: there is a ``partition'' of a neighbourhood of the origin such that for each piece \(N\) of this partition the function \(f,\) after appropriate composition, is a monomial i.e.
\[
f\circ \Psi (x)=c(x)m(x),
\]
where \(c(x)\) is nonvanishing, \(m(x)\) is a monomial and \(\Psi \) is a composition of reflections, translations, invertible monomial maps and quasi-translations (the last maps are:
\[
(x_{1},\ldots ,x_{n})\mapsto (x_{1},\ldots ,x_{j-1},x_{j}-r(x_{1},\ldots ,x_{j-1},x_{j+1},\ldots ,x_{n}),x_{j+1},\ldots ,x_{n}),
\]
where \(r\) is analytic).
As applications the author gives:
1. the criteria for the exponents \(\epsilon _{i}\) for the integrability of \( \int_{O}\prod_{i}f_{i}^{-\epsilon _{i}},\) where \(f_{i}\) are real analytic functions and \(O\) is a neighbourhood of the origin in \(\mathbb{R} ^{n}.\)
2. a proof of the Łojasiewicz inequality\(\;\left| f_{2}\right| \geq C\left| f_{1}\right| ^{\mu },\) \(C,\mu >0,\) on compact sets for any two analytic functions \(f_{1},\) \(f_{2}\) for which \(V(f_{2})\subset V(f_{1}).\) singularity; real analytic function; resolution of singularity; Lojasiewicz inequality Greenblatt M.: A coordinate-dependent local resolution of singularities and applications. J. Funct. Anal. 255(8), 1957--1994 (2008) Local complex singularities, Real algebraic and real-analytic geometry, Global theory and resolution of singularities (algebro-geometric aspects) An elementary coordinate-dependent local resolution of singularities and applications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let us consider a reflexive sheaf \(\mathcal{E}\) on the unit ball of \(\mathbb{C}^n\) with isolated singularity at 0, and an admissible Hermitian-Yang-Mills connection \(A\) on \(\mathcal{E}\) with respect to a Kähler metric \(\omega\). The rescaled sequence of connections \(A_\lambda:=\lambda^*A\) (obtained by the dilatations \(\lambda\in \mathbb{C}^n\)), that are Hermitian-Yang-Mills with respect to a rescaled Kähler metric \(\omega_\lambda\) induced by \(\omega\), converge to a smooth Hermitian-Yang-Mills connection \(A_\infty\) outside a closed complex-analytic subvariety \(\Sigma\) of \(\mathbb{C}^n \setminus \{0\}\) by well-known results in the literature. The set \(\Sigma\) is called the analytic bubbling set. This connection \(A_\infty\) extends to an admissible Hermitian-Yang-Mills connection on \(\mathbb{C}^n\) and defines a reflexive sheaf \(\mathcal{E}_\infty\). The sequence of Yang-Mills energy measures \(\mu_\lambda= \vert F_{A_\lambda}\vert^2 \omega_\lambda^n\) converges weakly (after taking subsequences) to a Radon measure \(\mu\) on \(\mathbb{C}^n\). The triple \((A_\infty,\Sigma, \mu)\) is called by the authors an analytic tangent cone of \(A\) at \(0\). In a nutshell, the goal of this very rich paper is to investigate this analytic tangent cone and to see that this object has a canonical algebraic interpretation. The leitmotiv of the study is to show the uniqueness of the objects obtained at the analytic limit.
Under the above setting and the assumption that \(\mathcal{E}\) is isomorphic to the pull-back of some holomorphic vector bundle \(\underline{\mathcal{E}}\) over \(\mathbb{CP}^{n-1}\), it is proved that \(\mathcal{E}_\infty\) is isomorphic to the double dual of the graded object associated to the Harder-Narasimhan-Seshadri filtration \(Gr(\underline{\mathcal{E}})\) and \(A_\infty\) is gauge equivalent to the natural Hermitian-Yang-Mills connection living on it. Futhermore, the analytic bubbling set \(\Sigma\) is also independent of the choice of subsequences. It agrees with the singular set where the sheaf \(Gr(\underline{\mathcal{E}})\) fails to be locally free, and the measure \(\mu\) is completely determined by \(\underline{\mathcal{E}}\).
The paper generalizes earlier results of the authors [Duke Math. J. 169, No. 14, 2629--2695 (2020; Zbl 1457.14079)] where \(Gr(\underline{\mathcal{E}})\) was supposed to be reflexive. A nice consequence of the paper is that there exists an admissible Hermitian-Yang-Mills connection on a rank-2 reflexive sheaf over \(\mathbb{CP}^3\) such that at all of its singular points, the analytic tangent cones have flat connections but non empty bubbling sets.
Very recently, a more general result (the technical assumptions have been removed) has been proved by the authors in [Invent. Math. 225, No. 1, 73--129 (2021; Zbl 1471.32031)]. Hermitian Yang-Mills connections; tangent cone; reflexive sheaf; Harder-Narasimhan-Seshadri filtrations; singularities Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), Vector bundles on curves and their moduli Analytic tangent cones of admissible Hermitian Yang-Mills connections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 compact non-singular real algebraic variety and \(Y\) a compact non-singular rational real algebraic surface. In this paper the author gives the complete characterization of a \(C^\infty\) mapping \(f:X\to Y\) that is approximated, for the \(C^\infty\) topology, by regular mappings, using the data on the algebraic cycles. This result explains clearly several preceding results on approximation of \(C^\infty\) mappings by regular mappings. approximation of \(C^\infty\) mappings; real algebraic variety; algebraic cycles; regular mappings W. Kucharz, Algebraic morphisms into rational real algebraic surfaces, J. Algebraic Geom. 8 (1999), 569-579. Zbl0973.14030 MR1689358 Real algebraic sets, Realizing cycles by submanifolds, Rational and unirational varieties, Birational geometry, Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) Algebraic morphisms into rational real algebraic 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 Linear systems \({\mathcal L} = {\mathcal L}_{n,d}(m_1,\dots,m_s)\) of hypersurfaces of degree \(d\) in \({\mathbb P}^n\) with assigned multiple points \(P_1,\dots, P_s\) of multiplicities \(m_1,\dots,m_s\), are studied. For such a system, its expected dimension is edim \({\mathcal L} = \)max\(\{{n+d\choose n} - \sum ^s_{i=1}{n+m_i -1 \choose n}, -1\}\); when its actual dimension is greater, we say that the linear system is special. Let \(Z \subset {\mathbb P}^n\) be the scheme (of ``fat points'') associated to \({\mathcal L} \); we will use \({\mathcal L} \), by abuse of notation, also for the ideal sheaf \({\mathcal I}_Z \otimes {\mathcal O}_{{\mathbb P}^n}(d)\); to say that the linear system is special is equivalent to the fact that \(h^0({\mathcal L},{\mathbb P}^n)h^1({\mathcal L},{\mathbb P}^n) > 0\).
In the paper, the definition of a new expected dimension of a linear system is given, namely its expected linear dimension, ldim \({\mathcal L}\) , which takes into account the contribution of the linear base locus, and thus the notion of linear speciality is introduced and studied. We always have dim \({\mathcal L}\geq\) ldim \({\mathcal L} \geq \) edim \({\mathcal L}\). Sufficient conditions for a linear system to be linearly non-special for an arbitrary number of points and necessary conditions for a small number of points are given; the main tool for those results is the study of blow-ups of \({\mathbb P}^n\) at linear subspaces which are in the base locus of the system. linear systems; fat points; base locus; linear speciality; effective cone Brambilla, MC; Dumitrescu, O; Postinghel, E, On a notion of speciality of linear systems in \(\mathbb{P}^n\), Trans. Am. Math. Soc., 367, 5447-5473, (2015) Divisors, linear systems, invertible sheaves, Hypersurfaces and algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry On a notion of speciality of linear systems in \(\mathbb{P}^{n}\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper the authors develop \(\mathbb{A}^1\)-homotopy theory of schemes -- a homotopy theory of algebraic varieties where the affine line plays the role of the unit interval.
In the three chapters and nine paragraphs the authors present:
A homotopy category of a site with interval;
the \(\mathbb{A}^1\)-homotopy category of schemes over a base,
classifying spaces of algebraic groups.
First, they give a number of general results about simplicial sheaves on sites which are latter applied to the study of the homotopy category of schemes. Then, the authors study the basic properties of the \(\mathbb{A}^1\)-homotopy category \({\mathcal H}(S)\) of smooth schemes over a base scheme \(S\) with interval \(((Sm/S)_{Nis},\mathbb{A}^1)\) where \(Sm/S\) is the category of smooth schemes (of finite type) over \(S\) and Nis refers to the Nisnevich topology. They discuss the properties of the homotopy category of simplicial sheaves on \((Sm/S)_{Nis}\), then they prove three theorems with a major role in further applications of their constructions. Finally the authors consider some examples of topological realization functors.
The last chapter is dedicated to applications of the general technique developed above. The main results are: A geometrical construction of a space which represents in \({\mathcal H}(S)\) the functor \(H^1_{et}(-,G)\) for étale group schemes \(G\) of order prime to \(\text{char} (S)\), the second result shows that algebraic \(K\)-theory of a regular scheme \(S\) can be described in terms of morphisms in \({\mathcal H}(S)\) with values in the infinite Grassmannian and the third result shows how one can use \(\mathbb{A}^1\)-homotopy theory together with basic functoriality for simplicial sheaves on smooth sites to give a definition of Quillen-Thomason \(K\)-theory for all Noetherian schemes. Quillen-Thomason \(K\)-theory; homotopy category of schemes; Nisnevich topology; homotopy category of simplicial sheaves Morel, F.; Voevodsky, V., \(\mathbf{A}^1\)-homotopy theory of schemes, Publ. Math. Inst. Hautes Études Sci., 90, 45-143, (1999), 2001 Homotopy theory and fundamental groups in algebraic geometry, Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.), Nonabelian homotopical algebra \(\mathbb{A}^1\)-homotopy theory of schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathbb {P}^1=\mathbb {P}^1_{k}\) with \(k\) an algebraically closed field and let \(\mathcal{Q}=\mathbb {P}^1\times \mathbb {P}^1\) be a smooth quadric. Let \(S=k[u,u',v,v']\) be the bigraded ring, \(X\) be a zero-dimensional scheme and \(I=I(X)\) its saturated bigraded ideal in \(S\). A zero-dimensional scheme \(X\subseteq \mathcal{Q}\) is said \textit{scheme-theoretically generated} by \(r\) forms \(f_1,\dots,f_r\in S\) with \(\deg f_i=(a_i,b_i)\) if there exists a sheaves surjection \(\bigoplus_{i=1}^r \mathcal{O}_{\mathcal{Q}}(a_i,b_i)\overset{\phi}\rightarrow \mathcal{I}_X\rightarrow 0\) with \(\phi=(f_1,\dots,f_r)\). If \(r=2\), \(X\) is called \textit{scheme-theoretically complete intersection}.
Since in general in \(\mathbb {P}^1\times \mathbb {P}^1\) a bigraded ideal \(I(X)\) generated by a regular sequence is not saturated, it is interesting to study zero-dimensional schemes \(X\) defined by ideals that are the saturation of bigraded ideals generated by regular sequence.
In this paper, the authors study the case \(r=2.\)
In particular, Section 3 is devoted to the minimal case, i.e., when \(X\), arising by the ideal \((f,g),\) with \(\deg f=(a,b)\) and \(\deg g=(c,d)\), is contained in two curves of type \((a+c,0)\) and \((0,b+d).\) They found that \(X\) is the union of two \(0\)-grids and describe a minimal free bigraded resolution, showing that \(X\) has only four minimal generators (in the minimal case).
Section 4 is devoted to the general case, i.e., the curves \(C\) and \(D\) are two general curves of bidegree \((a,b)\geq (1,1)\) and \((c,d)\geq (1,1)\), respectively. They compute the Hilbert function of \(X\) and prove that the saturated ideal of \(X\) can be obtained by saturating the row ideals or the column ideals. bigraded rings; bigraded modules; complete intersection; Hilbert function; minimal free resolution; scheme-theoretic complete intersection Divisors, linear systems, invertible sheaves, Syzygies, resolutions, complexes and commutative rings, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Scheme-theoretic complete intersections in \(\mathbb P^1 \times\mathbb P^1\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper the modified Sawada-Kotera (SK) hierarchy associated with a \(3 \times 3\) matrix spectral problem is derived. The discussion relies on solving the Lenard recursion equation and the zero-curvature equation. Using the characteristic polynomial of the Lax matrix for the modified SK hierarchy, the authors introduce a trigonal curve \(K_{m-1}\) and they also give the corresponding Baker-Akhiezer function and a meromorphic function on it. With the help of the property of the Baker-Akhiezer function and the asymptotic expansions of the meromorphic function, their explicit Riemann theta function is derived. Algebro-geometric solutions of the entire modified Sawada-Kotera hierarchy are obtained using an asymptotic expansion of the meromorphic function and its Riemann theta function representation. As an application, some simple examples are given. Sawada-Kotera hierarchy; Riemann theta function; Baker-Akhiezer function; Dubrovin-type equations Wu, L.H.; He, G.L.; Geng, X.G., Algebro-geometric solutions to the modified Sawada-Kotera hierarchy, J. Math. Phys., 53, 123513, (2012) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Soliton equations, Asymptotic expansions of solutions to PDEs, Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets, Theta functions and curves; Schottky problem, Special algebraic curves and curves of low genus Algebro-geometric solutions to the modified Sawada-Kotera 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 Suppose that \(\Gamma\) is a finitely generated group. Denote by \(r_n(\Gamma)\) the number of irreducible \(n\)-dimensional complex representations of \(\Gamma\), up to equivalence. If the sequence \(r_n(\Gamma)\) is bounded by a polynomial in \(n\), then the representation zeta function \(\zeta_\Gamma(s)=\sum_n r_n(\Gamma)n^{-s}\) converges in a suitable half complex plane. \textit{A. Lubotzky} and \textit{B. Martin} [Isr. J. Math. 144, 293-316 (2004; Zbl 1134.20056)] proved that an arithmetic lattice in characteristic zero has the congruence subgroup property if and only if the sequence \(r_n(\Gamma)\) grows polynomially, or equivalently, if and only if the abscissa of convergence of \(\zeta_\Gamma(s)\) is finite.
The main result in this paper is the following: if an arithmetic lattice \(\Gamma\) in characteristic zero satisfies the congruence subgroup property, then the abscissa of convergence of \(\zeta_\Gamma(s)\) is a rational number. The proof follows a general strategy of Igusa and Denef. \textit{M. Larsen} and \textit{A. Lubotzky} [J. Eur. Math. Soc. (JEMS) 10, No. 2, 351-390 (2008; Zbl 1142.22006)] proved that \(\Gamma\) contains a finite index subgroup \(\Delta\) such that the representation zeta function of \(\Delta\) has an Euler-like factorization. The author proves that the abscissa of convergence of \(\zeta_\Gamma(s)\) is unchanged when passing to a finite-index subgroup and previous results ensure that the abscissa of convergence is rational for every local factor of \(\zeta_\Delta(s)\). Therefore, in order to prove that the abscissa of convergence of the global zeta function is rational, it remains to understand the relation between the local zeta functions for different primes. Indeed, the paper is mainly an attempt to give an approximate formula to the local zeta functions, which is uniform in the prime \(p\). motivic integration; orbit method; arithmetic lattices; irreducible complex representations; algebraic groups; congruence subgroup property; polynomial representation growth; irreducible complex characters; representation zeta functions; subgroups of finite index; local zeta functions N., Avni., Arithmetic groups have rational representation growth, Annals of Mathematics, 174, 1009-1056, (2011) Representation theory for linear algebraic groups, Other Dirichlet series and zeta functions, Discrete subgroups of Lie groups, Arcs and motivic integration, Analysis on \(p\)-adic Lie groups, Subgroup theorems; subgroup growth Arithmetic groups have rational representation growth. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 article, we study the following problem. Let \(\mathbf{G}\) be a linear algebraic group over \(\mathbb{Q}\), let \(\Gamma\) be an arithmetic lattice, and let \(\mathbf{H}\) be an observable \(\mathbb{Q}\)-subgroup. There is a \(H\)-invariant measure \(\mu_H\) supported on the closed submanifold \(H\Gamma/\Gamma\). Given a sequence \((g_n)\) in \(G\), we study the limiting behavior of \((g_n)_*\mu_H\) under the weak-\(*\) topology. In the non-divergent case, we give a rather complete classification. We further supplement this by giving a criterion of non-divergence and prove non-divergence for arbitrary sequence \((g_n)\) for certain large \(\mathbf{H}\). We also discuss some examples and applications of our result. This work can be viewed as a natural extension of the work of \textit{A. Eskin} et al. [Ann. Math. (2) 143, No. 2, 253--299 (1996; Zbl 0852.11054); Geom. Funct. Anal. 7, No. 1, 48--80 (1997; Zbl 0872.22009)] and \textit{U. Shapira} and \textit{C. Zheng} [Compos. Math. 155, No. 9, 1747--1793 (2019; Zbl 1422.37001)]. homogeneous dynamics; integral points; equidistribution; linear algebraic group; arithmetic lattice Homogeneous flows, Counting solutions of Diophantine equations, Lattice points in specified regions, Homogeneous spaces and generalizations, Discrete subgroups of Lie groups, Homogeneous spaces Translates of homogeneous measures associated with observable subgroups on some homogeneous spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A smooth map \(f:N\to P\) between manifolds \(N,P\) without boundary is said to be stable if there exists an open neighborhood \(\mathcal{U}\subset C^\infty(N,P)\) of \(f\) in the Whitney \(C^\infty\) topology, and maps \(\Theta:\mathcal{U}\to\mathrm{Diff}(N)\), \(\theta:\mathcal{U}\to\mathrm{Diff}(P)\) such that \(f=\theta(g)\circ g\circ\Theta(g)\) for every \(g\in\mathcal{U}\). It is called strongly stable if \(\Theta\) and \(\theta\) can be chosen to be continuous. \textit{J. N. Mather} [Ann. Math. (2) 89, 254--291 (1969; Zbl 0177.26002); Adv. Math. 4, 301--336 (1970; Zbl 0207.54303)] proved that proper maps are stable provided they are infinitesimally stable. The main theorem of the present paper yields a criterion for a Morse function \(f:N\to\mathbb{R}\) to be stable. In addition it states that a Morse function is strongly stable iff there exists a neighborhood \(V\subset\mathbb{R}\) of the set of critical values of \(f\) such that the restriction \(f|_{f^{-1}(V)}:f^{-1}(V)\to V\) is proper.
As a consequence it is proved that the map \(\mathbb{R}\to\mathbb{R}\), \(x\mapsto\exp(-x^2)\sin(x)\), is strongly stable but not infinitesimally stable. The paper contains also a criterion for the stability of Nash functions \(f:\mathbb{R}^n\to\mathbb{R}\), and it is proved that Nash functions become stable after a generic linear perturbation. stability of smooth maps; strong stability of smooth maps; Morse functions; Nash functions Differentiable mappings in differential topology, Nash functions and manifolds, Functions of several variables, Critical points and critical submanifolds in differential topology Stability of non-proper functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper deals with trajectories of sub-Riemannian (also called horizontal) gradient of polynomials. Given \(p\) analytic vector fields \(X_1, \dots, X_{p}\) on \({\mathbb R}^n\), the horizontal gradient of a function \(f \in C^\infty ({\mathbb R}^n, {\mathbb R})\) is defined as
\[
\nabla^h f = \sum_{i=1}^{p} (X_i f) X_i.
\]
A well-known consequence of the gradient Łojasiewicz inequality is that the length of the bounded trajectories \({\mathbf x}(t)\) of the (Riemannian) gradient \(\nabla f\) of an analytic function \(f:U \subset {\mathbb R}^n \to {\mathbb R}\) have a limit point and are uniformly bounded by a constant. In the sub-Riemannian case this is no longer true. In fact, the authors show an example where the trajectories of the horizontal gradient are not uniformly bounded and another one with trajectories accumulating on a closed curve.
However, for the class of sub-Riemannian metrics coming from splitting distributions (which contains those of Heisenberg and Martinet), the trajectories of the horizontal gradient of a generic polynomial \(f\) (that is, for \(f\) in a Zariski open and dense subset of the space of polynomials of degree at most \(d\) in \(n\) variables) have the properties already mentioned for the Riemannian case. For this class of sub-Riemannian metrics the distribution is generated by \(p=n-1\) polynomial vector fields of the form:
\[
X_i = \frac{\partial}{\partial x_i} + P_i \frac{\partial}{\partial x_n}, \; i= 1, \dots, n-1
\]
where the \(P_i\) are polynomials in the variables \((x_1, \dots, x_n)\). sub-Riemannian metric; semi-algebraic; Lojasiewicz inequality Goresky M, MacPherson R. Stratified Morse theory: Springer; 1988. Semialgebraic sets and related spaces, Sub-Riemannian geometry, Vector distributions (subbundles of the tangent bundles) Horizontal gradient of polynomials | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $X:=\mathrm{Spec}(R)$ be an affine Noetherian scheme, and $\mathcal{M} \subset \mathcal{N}$ be a pair of finitely generated $R$-modules. Denote their Rees algebras by $\mathcal{R}(\mathcal{M})$ and $\mathcal{R}(\mathcal{N})$. Let $\mathcal{N}^{n}$ be the $n$-th homogeneous component of $\mathcal{R}(\mathcal{N})$ and let $\mathcal{M}^{n}$ be the image of the $n$th homogeneous component of $\mathcal{R}(\mathcal{M})$ in $\mathcal{N}^n$. Denote by $\overline{\mathcal{M}^{n}}$ the integral closure of $\mathcal{M}^{n}$ in $\mathcal{N}^{n}$. One of the main goals of this paper is to describe the sets $\mathrm{Ass}_{X}(\mathcal{N}^{n}/\overline{\mathcal{M}^{n}})$ and $\mathrm{Ass}_{X}(\mathcal{N}^{n}/\mathcal{M}^{n}).$ \par The author obtains a complete classification of the points $x$ of $X$ that appear in the set $\bigcup_{n=1}^\infty \mathrm{Ass}_{X}(\mathcal{N}^{n}/\overline{\mathcal{M}^{n}})$ when $X$ is universally catenary. In particular, he proves that this set is finite. More generally, without assuming $X$ to be universally catenary and any additional hypothesis on the pair $\mathcal{M} \subset \mathcal{N},$ the author proves that $\mathrm{Ass}_{X}(\mathcal{N}^{n}/\overline{\mathcal{M}^{n}})$ and $\mathrm{Ass}_{X}(\mathcal{N}^{n}/\mathcal{M}^{n})$ are asymptotically stable, generalizing known results for the case where $\mathcal{M}$ is an ideal or where $\mathcal{N}$ is a free module. \par Suppose that either $\mathcal{M}$ and $\mathcal{N}$ are free at the generic point of each irreducible component of $X$ or $\mathcal{N}$ is contained in a free $R$-module. If $X$ is universally catenary, the author proves a generalization of a classical result due to \textit{S. McAdam} [Proc. Am. Math. Soc. 80, 555--559 (1980; Zbl 0445.13002)] and obtain a geometric classification of the points appearing in $\mathrm{Ass}_{X}(\mathcal{N}^{n}/\overline{\mathcal{M}^{n}})$. More precisely, he shows that if $x \in \mathrm{Ass}_{X}(\mathcal{N}^{n}/\overline{\mathcal{M}^{n}})$ for some $n$, then $x$ is the generic point of a codimension-one component of the nonfree locus of $\mathcal{N}/\mathcal{M}$ or $x$ is a generic point of an irreducible set in $X$ where the fiber dimension $\mathrm{Proj}(\mathcal{R}(\mathcal{M})) \rightarrow X$ jumps. He proves a converse of this result without requiring $X$ to be universally catenary. \par Finally, he recovers, strengthens, and proves a sort of converse of an important result of Kleiman and Thorup about integral dependence of modules. Rees algebra of a module; associated points; integral closure of modules; Bertini's theorem for extreme morphisms Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Integral dependence in commutative rings; going up, going down, Integral closure of commutative rings and ideals, Schemes and morphisms Associated points and integral closure of modules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\Delta\) be a complete \(n\)-dimensional fan in \(\mathbb R^n\) and let \(B(\Delta)\) denote the first barycentric subdivision of \(\Delta\). Let \(A\) denote the ring of polynomial function on \(\mathbb R^n\). Denote by \(\mathcal A(B(\Delta))\) the algebra of continuous functions on \(\mathbb R^n\) which restrict to polynomial functions on each cone of \(B(\Delta)\). One has a natural \(\mathbb N^n\)-grading on \(\mathcal A(B(\Delta))\), and an \(A\)-module structure \(\mathcal A(B(\Delta))\) so that the \(\mathbb N^n\) grading preserves the module structure and passes to the quotient \(\mathcal A(B(\Delta)) \otimes_A\mathbb R\). One has the Poincaré polynomial in the variables \(t_1,\cdots,t_n\) \(P_n(\Delta)=\sum_Sh_St^S\) where \(t^S=t_1^{S_1}\cdots t_n^{S_n}\) for \(S=(S_1,\cdots,S_n)\) and \(h_S\) is the dimension of the degree \(S\) part of \(\mathcal A(B(\Delta))\). It turns out that \(h_S=0\) unless \(0\leq S_i\leq 1\) for all \(i\leq n\).
The Poincaré polynomial \(P_n(\Delta)\) can be expressed as a polynomial with integer coefficients in non-commuting variables \(c,d\) where a monomial \(w(c,d)\) in \(c,d\) corresponds to the polynomial in the \(t_i\) obtained as follows: Replace the \(i\)-th letter in \(w(c,d)\) by \(t_j+1\) or \(t_j+t_{j+1}\) according as whether the letter is \(c\) or \(d\) respectively; the replacement is made from left to right and using each of the variables \(t_1,\cdots, t_n\) exactly once in order. Bayer and Klapper have shown that the resulting polynomial is homogeneous in \(c,d\) where degrees of \(c\) and \(d\) are \(1\) and \(2\) respectively. This homogeneous polynomial is called the \(cd\)-index of \(\Delta\). The \(cd\)-index can be defined for Eulerian posets. It has been conjectured by J. Fine [see \textit{M. M. Bayer} and \textit{A. Klapper}, Discrete Comput. Geom. 6, 33--47 (1991; Zbl 0761.52009)] that the \(cd\)-indices of Eulerian posets have positive coefficients.
A rank \(n\) graded poset \(\Lambda\) is called Gorenstein\(^*\) if the simplicial complex \(B(\Lambda)\) of chains of \(\Lambda\setminus\{0,1\}\) is a homology sphere of dimension \((n-1)\). Examples of Gorenstein\(^*\) posets are complete fans and regular \(CW\)-decompositions of spheres. \textit{R. Stanley} [Math. Z. 216, No. 3, 483--499 (1994; Zbl 0805.06003)] has conjectured that the \(cd\)-index of a Gorenstein\(^*\) poset has positive coefficients. The main result of the paper establishes this conjecture. Special cases of this conjecture have been settled by Stanley (for shellable \(CW\) spheres) and \textit{M. Purtill} [Trans. Am. Math. Soc. 338, No. 1, 77--104 (1993; Zbl 0780.05002)] (for quasi-simplicial polytopes and their duals as well as for all polytopes of dimensions at most five). The proof involves a description of the \(cd\)-index using the restriction \(C:I(\Delta^{\leq m}){\text{lr}} I(\Delta^{\leq m-1})\) and a certain operator \( ~D:I(\Delta^{\leq m}){\text{lr}} I(\Delta^{\leq m-2})\) where \(I(\Delta^{\leq m})\) denotes integer valued functions on the \(m\)-skeleton \(\Delta^{\leq m}\) of \(\Delta\). The result of evaluating a monomial \(w(c,d)\) of degree \(n\) on the constant function \(1_\Delta\in I(\Delta)\) is a function on \(\Delta^0=0\), i.e., an integer. It is proved that this integer is nothing but the coefficient of \(w(c,d)\) in the \(cd\)-index. The poset \(\Delta\) can be viewed as a topological space. One has the derived category of bounded complexes of sheaves of \(\mathcal A\)-modules.
The author introduces the notion of semi-Gorenstein sheaves on \(\Delta^{\leq m}\). There are operators \(\mathcal C\) and \(\mathcal D\) on semi-Gorenstein sheaves on \(\Delta\). The sheaf \(\mathcal C(F)\) is the restriction to \(\Delta^{\leq m-1}\) the definition of \(\mathcal D\) involves duality theory developed by \textit{P. Bressler} and \textit{V. A. Lunts} [Compos. Math. 135, No. 3, 245--278 (2003; Zbl 1024.52005)]. The coefficient of a degree \(n\) monomial \(w(c,d)\) in the \(cd\)-index is then the dimension of the vector space (= sheaf over \(\Delta^0\), the zero cone), got by applying \(w(\mathcal C,\mathcal D)\) to the constant sheaf \({\mathbb R}_\Delta\). Kalle Karu, The \?\?-index of fans and posets, Compos. Math. 142 (2006), no. 3, 701 -- 718. Toric varieties, Newton polyhedra, Okounkov bodies, Combinatorics of partially ordered sets The \(cd\)-index of fans and posets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 topological recursion is a framework which associates certain invariants to a spectral curve via a recursive definition. These invariants have very interesting properties and appear in a variety of contexts in mathematics and physics. Many known and well studied invariants can be recovered via certain specializations; the topological recursion gives a broader pictures which connects different areas of mathematics. The importance of the topic is well recognized in the literature, and techniques based on the topological recursion are widely applied, starting from the seminal works by Eynard and Orantin.
This paper concerns the study of the intersection numbers on the Deligne-Mumford moduli spaces of curves, and in particular the properties of a generating function of higher Weil-Petersson volumes. Higher Weil-Petersson volumes are a generalization of the ordinary Weil-Petersson volumes, which are the hyperbolic volumes of the moduli spaces of bordered Riemann surfaces with fixed genus \(g\) and \(n\) geodesic boundaries of fixed length. In this ordinary Weil-Petersson case the existence of a recursion formula has been established by \textit{M. Mirzakhani} [J. Am. Math. Soc. 20, No. 1, 1--23 (2007; Zbl 1120.32008)]. This result has been generalized to higher Weil-Petersson volumes by \textit{K. Liu} and \textit{H. Xu} [Int. Math. Res. Not. 2009, No. 5, 835--859 (2009; Zbl 1186.14059)].
These invariants also obey the Virasoro constraints, in the sense that an appropriate generating function is annihilated by operators which obey a Virasoro algebra. The main result of the paper is the proof that such a Virasoro constraint is equivalent to the Eynard-Orantin topological recursion for a certain spectral curve. A different proof using matrix models was provided earlier by \textit{B. Eynard} [Ann. Henri Poincaré 12, No. 8, 1431--1447 (2011; Zbl 1245.14013)]. The approach taken in the paper is more geometric. The paper is very technical, probably not suited for someone not familiar with the subject. On the other hand the proofs of the results are explained carefully and are easy to follow; the author has made an effort to make a rather technical result readable. intersection numbers; moduli spaces of curves; Eynard-Orantin topological recursion Zhou, J.: Topological recursions of eynard-orantin type for intersection numbers on moduli spaces of curves. Lett. math. Phys. 103, No. 11, 1191-1206 (2013) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Topological recursions of Eynard-Orantin type for intersection numbers on moduli spaces of curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper concerns the problem of approximating a uniformly continuous semialgebraic map \(f: S \to T\) from a compact semialgebraic set \(S\) to an arbitrary semialgebraic set \(T\) by a semialgebraic map \(g: S \to T\) that is differentiable of class \(C^\nu\), where \(\nu\) is a positive integer or \(\infty\). It is known that if \(T\) is an \(C^\nu\) semialgebraic manifold, then arbitrarily good (in the \(C^\nu\)-norm) \(C^\nu\) semialgebraic approximations exist. The authors show that for \emph{any semialgebraic \(T\)}, arbitrarily good \(\nu = 1\) approximations are possible. For \(\nu \geq 2\), they obtain density results when: (1) \(T\) is compact and locally \(C^\nu\) semialgebraically equivalent to a polyhedron, or (2) \(T\) is an open semialgebraic subset of a Nash set. The paper includes a useful review of approximation results in semialgebraic geometry, including discussion of key references. approximation of semialgebraic maps; approximation of maps between polyhedra Semialgebraic sets and related spaces, Approximations in PL-topology, Real algebraic sets, Nash functions and manifolds Differentiable approximation of continuous semialgebraic 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 Let \(s_1, \ldots, s_r, t_1, \ldots, t_r\) be positive integers and consider two sets of variables
\[
\{x_{ij}\}_{1\leq i\leq r, 1\leq j\leq s_i} = :x,\;\{y_{ij}\}_{1\leq i\leq r, 1\leq j\leq t_i} = :y.
\]
Define a multigraduation by \(\deg(x_{ij})=\deg (y_{ik})=a^{(i)}\in\mathbb Z^d\) and assume there exist \(\omega\in\mathbb Q^d\) such that \({}^t\omega\cdot a^{(i)}=1\) for all \(i\). Let \(\mathcal A=\{a^{(1)}, \ldots, a^{(r)}\}\) and \(I, J\) be homogeneous ideals in \(K[x]\) and \(K[y]\) respectively. Let \(R=K[x]/I\) and \(S=K[y]/J\) and \(\Phi_{I,J}:K[z]\to R\otimes_K S\) be defined by \(\Phi_{I,J}(z_{ijk})=x_{ij}\otimes y_{ik}\) with \(z=(z_{ijk})_{1\leq i\leq r, 1\leq j\leq s_i, 1\leq k\leq t_i}\). The kernel of \(\Phi_{I,J}\) is called the toric fibre product \(I\times_\mathcal A J\) of \(I\) and \(J\). A Gröbner basis of \(I\times_\mathcal A J\) is explicitly constructed in terms of a Gröbner basis of \(I\) and a Gröbner basis of \(J\) under the assumption that \(\mathcal A\) is a linearly independent set. The result is applied to algebraic statistics. Gröbner basis; algebraic statistics; hierarchical model; phylogenetic invariants S. Sullivant, \textit{Toric fiber products}, J. Algebra, 316 (2007), pp. 560--577. Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Toric varieties, Newton polyhedra, Okounkov bodies, Computational aspects in algebraic geometry Toric fiber products | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 of the article continue their study of representation growth and rational singularities of moduli spaces of local systems
[\textit{A. Aizenbud} and \textit{N. Avni}, Invent. Math. 204, No. 1, 245--316 (2016; Zbl 1401.14057)], now from a more general point of view.
They relate algebraic geometry over finite rings with representation theory, and obtain two main results concerning estimates (bounds) for the number of points of schemes over finite rings (Theorem A) and estimates (bounds) for the number of irreducible representations of arithmetic lattices of algebraic group schemes (Theorem B).
Let $X$ be a scheme of finite type over ${\mathbb Z}$ with the reduced, absolutely irreducible generic fiber
$X_{\mathbb Q} := X \times_{\operatorname{Spec}{\mathbb Z}} \operatorname{Spec}{\mathbb Q}$ of $X$ which is a local complete intersection.
Let $\#X(R)$ (the authors use the notation $|X(R)|$) be the number of points of the scheme $X$ over the finite ring $R$.
Theorem A states, in particular, that the next conditions are equivalent:
For any $m$, $\lim_{p \to \infty} \frac{\#X({\mathbb Z}/{p^m})}{p^{m \cdot \dim X_{\mathbb Q}}} = 1$; $X_{\mathbb Q}$ has rational singularities.
Theorem B states that for any algebraic group scheme $G$, whose generic fiber $G_{\mathbb Q}$ is simple, connected, simply connected, and of ${\mathbb Q}$-rank at least 2, and for every $C > 40$, the number of irreducible representations of $G({\mathbb Z})$ of dimension $n$ is equal to $ o(n^C)$.
The main tools in proving these theorems and their generalizations are Poincare series by Borevich-Shafarevich, Igusa zeta functions and other $p$-adic integrals, Lang-Weil bounds, deformation schemes and the theorem of Frobenius.
``For a topological group $\Gamma$, let $r_n(\Gamma)$ be the number of isomorphism classes of irreducible, $n$-dimensional, complex, continuous representations of $\Gamma$'', and let $\zeta_{\Gamma}(s)$ be the representation zeta function of $\Gamma$.
Let $k$ be a global field and let $T$ be a finite set of places of $k$ containing all Archimedean places.
By ${\mathcal O}_{k,T}$ authors denote the ring of $T$-integers of $k$ and by $\widehat{{\mathcal O}_{k,T}}$ the profinite completion of ${\mathcal O}_{k,T}$.
Let $\alpha(\Gamma)$ be the abscissa of convergence of $\zeta_{\Gamma}(s)$.
The second section of the article deals with preliminaries, which include (along with the above) elements of singularities and theorems by \textit{J. Denef} [Am. J. Math. 109, 991--1008 (1987; Zbl 0659.14017)], and by \textit{M. Mustaţă} [Invent. Math. 145, No. 3, 397--424 (2001; Zbl 1091.14004)].
In the next sections authors of the article under review ``study the number of points of schemes over finite rings'' and prove
(a generalization) of Theorem A.
The article closes with results on representation zeta functions of compact $p$-adic groups, of adelic groups and of arithmetic groups.
In the forth section the authors prove Theorem II on the abscissa of convergence $\alpha(G({\mathcal O}_{k,T}))$, Theorem III on abscissa of convergence $\alpha(G({\widehat{{\mathcal O}_{k,T}}}))$ and Theorem V on estimates of representation zeta function values at integer points $2n - 2, \; n \ge 2$.
Reviewer's remark: It is interesting to relay results of this article with results of the paper by \textit{B. Frankel} [J. Algebra 510, 393--412 (2018; Zbl 1436.14041)]. representation growth; Igusa zeta function; points of schemes over finite ring; complete intersection; rational singularities; representation zeta function Rational points, Singularities in algebraic geometry, Asymptotic properties of groups, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Representation theory for linear algebraic groups, Linear algebraic groups over global fields and their integers Counting points of schemes over finite rings and counting representations of arithmetic lattices | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems An affine Nash manifold in \(\mathbb{R}^n\) is a semialgebraic analytic submanifold of \(\mathbb{R}^n\). A Nash mapping between affine Nash manifolds is an analytic mapping with semialgebraic graph. Several important global results (solutions to separation, global equations, extension and factorization problems) concerning Nash functions on affine Nash manifolds were obtained by \textit{M. Coste}, \textit{J. M. Ruiz} and \textit{M. Shiota} [Am. J. Math. 117, 905-927 (1995; Zbl 0873.32007) and Compos. Math. 103, 31-62 (1996; Zbl 0885.14029)] and by \textit{M. Coste} and \textit{M. Shiota} [Ann. Sci. Éc. Norm. Super., IV. Ser. 33, 139-149 (2000; see the preceding review Zbl 0981.14027)]. In the present article the authors prove uniform bounds on the complexity of Nash functions and (using the Tarski-Seidenberg principle) obtain a solution to extension and global equations problems over arbitrary real closed fields. This allows to prove that the Artin-Mazur description holds for abstract Nash functions on the real spectrum of any commutative ring, and solve extension and global equations problems in this setting. The authors also prove the idempotency of the real spectrum and an abstract version of the separation problem. Conditions for the rings of abstract Nash functions to be noetherian are discussed. affine Nash manifold; complexity of Nash functions; Tarski-Seidenberg principle; real spectrum; separation problem Coste, M; Ruiz, JM; Shiota, M, Uniform bounds on complexity and transfer of global properties of Nash functions, J. Reine Angew. Math., 536, 209-235, (2001) Nash functions and manifolds, Real algebra, Real-analytic sets, complex Nash functions, Real-analytic and Nash manifolds Uniform bounds on complexity and transfer of global properties of Nash functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(B_n(F)\) be the number of places of degree \(n\) of \(F/\mathbb{F}_q\) and \(g(F)\) be the genus of \(F\). Let \(F=(F_0,F_1,F_2,\cdots)\) be a tower of a function fields over \(\mathbb{F}_q\) and
\[
\lambda(F)=\lambda(F/\mathbb{F}_q)=\lim _{i \rightarrow \infty} B_1(F_i)/g(F_i).
\]
The author defines \(\Delta_n(F)=\Delta_n(F/\mathbb{F}_q)=\lim _{i \rightarrow \infty} B_n(F_i)/g(F_i)\) and studies the exact values of \(\Delta_n(F)\), and upper and lower bounds for \(\Delta_n(F)\). Using the results on \(\Delta_n(F)\), it is proved that for all \(n \geq 2\), \(\Delta_n(F/\mathbb{F}_q)=0\) and \(\lambda_n(F\mathbb{F}_{q^n}/\mathbb{F}_{q^n})=\sqrt{q}-1\) if \(q\) is a square and \(F\) is an optimal tower function fields over \(\mathbb{F}_q\) where \(F\mathbb{F}_{q^n}=(F_0\mathbb{F}_{q^n},F_1\mathbb{F}_{q^n},F_2\mathbb{F}_{q^n},\cdots)\) is defined to be composite tower of \(F\) over \(\mathbb{F}_{q^n}\). In particular, it is shown that the composite tower \(F\mathbb{F}_{q^n}\)is not optimal. Moreover, simple criterion whether a tower is optimal or not and many new recursive towers of finite ramification type are given. finite fields; towers of function fields; congruence zeta functions DOI: 10.3836/tjm/1202136690 Arithmetic theory of algebraic function fields, Finite ground fields in algebraic geometry, Algebraic functions and function fields in algebraic geometry A note on optimal towers over finite fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper deals with finiteness problems concerning semialgebraic and Nash sets, nad Nash functions. A~subset \(X\subset \mathbb{R}^n\) is \textit{semialgebraic} when it has a description by finite boolean combination of polynomial equations and inequalities. An \textit{affine Nash manifold} is a pure dimensional semialgebraic subset \(M\subset \mathbb{R}^n\) that is a smooth submanifold of an open subset of \(\mathbb{R}^n\). A \textit{Nash function} on an open semialgebraic set \(U\subset M\) is a semialgebraic smooth function on \(U\). A \textit{Nash subset} of \(U\) is the zero set of a Nash function on \(U\). Nash functions are also considered in more general case. A \textit{Nash function} on a semialgebraic set \(X\subset M\) is a cross-section over \(X\) of the sheaf of germs of Nash functions on Nash manifold \(M\).
The results of the paper are within the range of research of comparison the Euclidean and semialgebraic topology initiated by \textit{G. W. Brumfiel} [Partially ordered rings and semi-algebraic geometry. Cambridge etc.: Cambridge University Press (1979; Zbl 0415.13015)]. Finiteness problems arise in connection with the fact that the semialgebraic topology is not a true topology. For instance, any Nash function \(f:X\to\mathbb{R}\) on a semialgebraic set \(X\subset M\) can be extended to an analytic function defined on an open set \(U\subset M\) in the Euclidean topology. The set \(U\) as infinite union of open semialgebraic sets is not necessary semialgebraic neighbourhood of \(X\). Therefore, a problem arises, if we can find such an extension, which is defined on an open semialgebraic set. The authors obtain an affirmative answer to this question in Theorem 1.3.
The remaining results of the article are related to possibility of finite description for properties of local nature in Euclidean topology. Among other results, it is shown that: a Nash set \(X\) that has only normal crossings in \(M\) can be covered by finitely many open semialgebraic sets \(U\) equipped with Nash diffeomorphisms \((u_1,\dots,u_m):U\to \mathbb{R}^m\) such that \(U\cap X=\{u_1\cdots u_r=0\}\) for some \(r\) (Theorem 1.6, Theorem 1.7 has a similar nature). Every affine Nash manifold with corners \(N\) is a closed subset of an affine Nash manifold \(M\) where the Nash closure of the boundary \(\partial N\) of \(N\) has only normal crossings and \(N\) can be covered with finitely many open semialgebraic sets \(U\) such that each intersection \(N\cap U=\{u_1\geq 0,\dots u_r\geq 0\}\) for a Nash diffeomorphism \((u_1,\dots,u_m):U\to \mathbb{R}^m\) (Theorem 1.11). Finiteness; Nash functions and Nash sets; semialgebraic sets; Nash manifolds with corners; extension; normal crossings at a point; normal crossing divisor. Fernando, José F.; Gamboa, J. M.; Ruiz, Jesús M., Finiteness problems on Nash manifolds and Nash sets, J. Eur. Math. Soc. (JEMS), 16, 3, 537-570, (2014) Nash functions and manifolds, Real-analytic and Nash manifolds, Real-analytic manifolds, real-analytic spaces Finiteness problems on Nash manifolds and Nash sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a field of characteristic zero. For \(1\leq j\leq p\) \quad let \(X_j\) be a smooth variety over \(k\) and \( f_j: X_j\rightarrow \mathbb A_k^{1}\) be a function. Denote by \(X_0( f)\) the set of common zeroes on \(X={\prod}_{j}X_j\) of the compositions of the appropriate projections with the functions \(f_j\). Let \(P\in k[y_1,\dots,y_p]\) be a polynomial which is nondegenerate with respect to its Newton polyhedron and \(P(f)\) be the corresponding composition. The authors show that the motivic nearby cycles \({\mathcal S}_{P(f)}\) on \(X_0( f)\) of the function \(P(f)\) on \(X\) can be expressed as a sum over the set of compact faces \(\delta\) of the Newton polyhedron of \(P\). Let \(P_{\delta}\) denote the quasihomogoneous polynomial corresponding to \(\delta\). The authors define a convolution operator \({\Psi}_{P_{\delta}}\). For a compact face \(\delta\) they define generalized nearby cycles \({\mathcal S}_{f}^{{\sigma}(\delta)}\). The main result of the paper is the following formula:
\[
i^{*}{\mathcal S}_{P(f)}={\Sigma}_{J\subset\{1,\dots,p\}}{\Sigma}_{\delta\in {\Gamma}^J} {\Psi}_{P_{\delta}}({\mathcal S} _{f_J}^{{\sigma}(\delta), l_{\Gamma}}).
\]
G. Guibert, F. Loeser and M. Merle, Nearby cycles and composition with a non-degenerate polynomial, Int. Math. Res. Not. IMRN 31 (2005), 1873-1888. Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Algebraic cycles, Singularities in algebraic geometry, Zeta functions and \(L\)-functions, Toric varieties, Newton polyhedra, Okounkov bodies Nearby cycles and composition with a nondegenerate polynomial | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author deals with a general version of a theorem of Lojasiewicz on semi-analytic sets.
Let \(X\) be a topological space and \(A \subset C(X)\) a subalgebra. A subset \(E \subset X\) is said to be described by \(A\) iff \(E\) is of the form
\[
E = \bigcup^ n_{i=1} \bigcap^ m_{j=1} \{a_{i,j} = 0, b_{i,j} > 0\} \quad \text{with} \quad a_{i,j}, b_{i,j} \in A.
\]
Under a condition on \(A\) -- which is not obvious in the semi-analytic case -- he shows: Let \(E \subset X \times \mathbb{R}\) be described by \(A[t]\), then there exist
-- a finite decomposition \(J\) of \(X\), elements of which are components of sets described by \(A\),
-- a decomposition of \(X \times \mathbb{R}\) compatible with \(E\) into sets
\[
\bigl\{ (x,t) : \;x \in Y,\;\xi^ Y_ \nu (x) < t < \xi^ Y_{\nu + 1} (x) \bigr\}, \qquad \bigl\{ (x,t) : \;x \in Y,\;t = \xi^ Y_ \nu (x) \bigr\}, \tag{\(*\)}
\]
\(Y \in J\) and \(\nu = 0, \dots, k_ Y - 1\) resp. \(1, \dots, k_ Y - 1\), where \(-\infty = \xi^ Y_ 0 < \cdots < \xi^ Y_{k_ Y} = \infty\) are continuous functions on \(A\).
The sets \((*)\) are of the form \(F \cap (Y \times \mathbb{R})\) with some \(F \subset X \times \mathbb{R}\) described by \(A[t]\).
The author gets the result by a constructive method. theorem of Lojasiewicz; semi-analytic sets Semi-analytic sets, subanalytic sets, and generalizations, Real-analytic and semi-analytic sets A real analytic constructive proof of the Lojasiewicz 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 [For the entire collection see Zbl 0722.00006.]
A scheme consisting of \(e\) points in a projective space is called a Cayley-Bacharach scheme if the subsets of \(e-1\) points have all the same postulation (i.e. Hilbert function), this condition being a weaker one than that arising in Harris' uniform position lemma [cf. \textit{J. Harris}, Math. Ann. 249, 191-204 (1980; Zbl 0449.14006)]. The main technical result of the paper is the determination of the relation between the Cayley-Bacharach property of \(X\) and the structure of the canonical module of the coordinate ring of \(X\). A characterization of CB-schemes in terms of liason is given, the so called ``weak inequalities'' are proved for Hilbert functions of CB-schemes, some ``strong inequalities'' are conjectured and relations with some new results of R. Stanley about the Hilbert function of graded Cohen-Macaulay domains are explained. Cayley-Bacharach scheme; postulation; Hilbert function; liason Geramita, A., Kreuzer, M., Robbiano, L.: Cayley--Bacharach schemes and Hilbert Functions, Preprint (1990) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Linkage, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Algebraic moduli problems, moduli of vector bundles Cayley-Bacharach schemes and their canonical modules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The subject of this work are linear systems in \({\mathbb P}^n\) of type \({\mathcal L} = {\mathcal L}_{n,d}(-\sum _{i=1}^h m_iP_i)\), \(P_i\in {\mathbb P}^n\), \(m_i\in {\mathbb N}\); which are given by hypersurfaces of degree \(d\) passing through \(h\) generic points \(P_i\) with multiplicities \(\geq m_i\).
Even for \(n=2\), the dimension of \({\mathcal L}\) is not known in general, but several equivalent conjectures (the first due to B. Segre) state that the dimension is the expected one except in the cases when the linear system contains a multiple fixed component which is a \((-1)\)-curve. In a previous paper, the author proposed yet other two equivalent forms for such conjectures, stating that \({\mathcal L}_{2,d}(-\sum _{i=1}^h m_iP_i)\) is special (i.e. it does not have the expected dimension), if and only if it is ``numerically special'' or ``cohomologically special'', where the main interest for those two concepts is that they could be generalized to \(n\geq 3\), where no general conjecture for the dimension of \({\mathcal L}\) is known.
We say that \({\mathcal L}\) is numerically special if it exists an \(\alpha\)-special effect variety \(Y\) for \({\mathcal L}\), i.e. an irreducible variety \(Y\) such that \({\mathcal L}-\alpha Y\) has positive virtual dimension greater than \({\mathcal L}\) (a few more conditions are required if \(\dim Y = n-1\)). We say that \({\mathcal L}\) is cohomologically special, instead, if it exists a \(h^1\)-special effect variety \(Y\), i.e. an irreducible \(Y\) such that \({\mathcal L}- Y\) has positive dimension, \(h^0({\mathcal L}|_Y)=0\) and \(h^1({\mathcal L}|_Y)>h^2({\mathcal L}-Y)\) (an example showing that the two definitions do not coincide is given).
The author conjectures that for \({\mathcal L}\) to be special is equivalent to being numerically special and also to being cohomologically special. In the case \(n=2\), \(\alpha\)-special effect varieties and \(h^1\)-special effect varieties are actually \((-1)\)-curves, while for \(n\geq 3\) examples are given using rational normal curves, hypersurfaces or linear spaces, and those covers most known examples of special \({\mathcal L}\)'s.
Moreover, the case of linear systems in a product \({\mathbb P}^{n_1}\times {\mathbb P}^{n_2}\times\dots\times {\mathbb P}^{n_t}\) is studied and examples (known and new) are given of systems whose speciality is due to special effect varieties. linear systems; multiple points; fat poins; Hilbert function Bocci, C.: Special effect varieties in higher dimension. Collect. Math. \textbf{56}(3), 299-326 (2005). ISSN: 0010-0757 Divisors, linear systems, invertible sheaves, Projective techniques in algebraic geometry, Singularities of curves, local rings Special effect varieties in higher dimension | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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' goal in this paper is to develop a decomposition theory for numerical classes of cycles of any codimension, which generalize the Zariski decomposition of pseudo-effective divisor on a smooth surface. Let \(X\) be a projective variety over an algebraically closed field. Let \(N_k(X)\) denote the \(\mathbb{R}\)-vector space of \(k\)-cycles with \(\mathbb{R}\)-coefficients modulo numerical equivalence. The pseudo-effective cone \(\overline{\text{Eff}}_k(X)\subset N_k(X)\) is the closure of the cone generated by classes of effective cycles. The classes in the interior of \(\overline{\text{Eff}}_k(X)\subset N_k(X)\) is called big classes. The movable cone \(\overline{\text{Mov}}_k(X)\) is the closure in \(N_k(X)\) of the cone generated by members of irreducible families of \(k\)-cycles which dominate \(X\). In [\textit{B. Lehmann}, ``Geometric characterizations of big cycles'', Preprint, \url{arXiv:1309.0880}] a homogeneous continuous function \(m(\alpha)\) on \(N_k(X)\), called the mobility, is introduced and is shown to be positive precisely for big classes. The main theorem of this paper shows that an arbitrary pseudo-effective class \(\alpha\) can be decomposed as a sum \(\alpha=P(\alpha)+N(\alpha)\) where \(P(\alpha)\) is movable, \(N(\alpha)\) is pseudo-effective, and \(\text{mob}(P(\alpha))=\text{mob}(\alpha)\). algebraic cycles; numerical equivalence; Zariski decomposition M. Fulger and B. Lehmann. \textit{Zariski decompositions of numerical cycle classes. Journal of Algebraic Geometry}. arXiv:1310.0538 (2013). Algebraic cycles, Divisors, linear systems, invertible sheaves Zariski decompositions of numerical cycle 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 Let \(S = k[x_1, \dots , x_n]\) be a polynomial ring graded by \(\deg(x_i) = 1\) for all \(i\) over an algebraically closed field \(k\) of characteristic zero. Let \(P = (x^{a_1}_ 1 ,\dots , x^{a^n}_n )\), with \(a_1\leq a_2\leq \dots\leq a_n\leq \infty\) (where \(x_i^{\infty} = 0\)) and set \(W = S/P\). Then \(W\) is called a \textit{Clements-Lindström} ring. We refer to \(x^{a_1}_ 1 ,\dots , x^{a^n}_n\) as the \(P\)-powers. The \textit{Clements-Lindström} Theorem states that Macaulay's Theorem holds over \(W\), that is, for every graded ideal in \(W\) there exists a lex ideal with the same Hilbert function.
Let Hilbert scheme \(H_{W}(h)\) be the scheme that parametrizes all graded ideals in \(W\) with a fixed Hilbert function \(h\). Equivalently, this Hilbert scheme parametrizes all graded ideals in \(S\) containing the \(P\)-powers and with a fixed Hilbert function. Let us define a \(P\)-deformation, that is, a deformation that connects ideals containing the \(P\)-powers. With this notation in Section 3, \(\S 3.7\), the authors prove Theorem 1.3, that the Hilbert scheme \(H_{W}(h)\) is connected; and that every graded ideal in the polynomial ring \(S\) that contains the \(P\)-powers, is connected by a sequence of \(P\)-deformations to the lex-plus- powers ideal with the same Hilbert function.
Macaulay's Theorem was generalized to Betti numbers by Bigatti, Hulett, and Pardue as follows: (see Theorem 1.4.) Every lex ideal in \(S\) attains maximal Betti numbers among all graded ideals with the same Hilbert function.
Aramova, Herzog, and Hibi proved that every lex ideal in an exterior algebra attains maximal Betti numbers among all graded ideals with the same Hilbert function. It was conjectured by Gasharov, Hibi, and Peeva that Theorem 1.4 holds over the \textit{Clements-Lindström} ring \(W\). In Section 4, the main theorem is Theorem 4.6.5, i.e., the proof of the conjecture: (see Theorem 1.5.) Every lex ideal in \(W\) attains maximal Betti numbers among all graded ideals with the same Hilbert function.
Note that Theorem 1.4 is about finite resolutions, while Theorem 1.5 is about infinite ones. In order to prove Theorem 1.5 the authors construct special changes of coordinates and use them to build a construction that starting with a monomial ideal yields a lex-closer ideal with bigger Betti numbers. Hilbert scheme; deformations; Betti numbers S. Murai and I. Peeva, Hilbert schemes and Betti numbers over a Clements-Lindström ring, submitted. Formal methods and deformations in algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Syzygies, resolutions, complexes and commutative rings Hilbert schemes and Betti numbers over Clements-Lindström rings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems If \(P\) is a lattice polyhedron in a rational vector space \(V\) and \(f\) is a polynomial function defined on \(P\), then one is interested in nice expressions for the sum \(s_f(P)\) of all \(f(x)\) where \(x\) runs through all lattice points of \(P\). In particular, such a formula is called ``local Euler-Maclaurin'', if it expresses \(s_f(P)\) as a sum over all faces \(F\) of \(P\) of integrals of the form \(\int_F D(P,F)\cdot f\). Here \(D(P,F)\) is a differential operator of infinite order, but with constant coefficients that acts as a kind of a weight working on \(f\). The striking point is the locality, i.e.\ \(D(P,F)\) is supposed to depend only on the (conical) shape of \(P\) in a neighborhood of a general point of \(F\).
Those formulae are known to exist (McMullen), but the operators are not uniquely determined. The choice can become canonical if one uses some additional structure like a scalar product on \(V\) or the presence of a complete flag in \(V^*\). In the present paper, the authors obtain a solution of a similar problem involving exponential sums arising as the kernels of the Laplace transformation. Here, the weight becomes a true function \(\mu(P,F)\) called interpolator, and knowing \(\mu\) does even imply solutions of the above Euler-Maclaurin problem and its inverse.
Moreover, the interpolator \(\mu\) is effectively computable, and it becomes canonical if understood as being parametrized by the flag variety on \(V^*\). lattice polytopes; Euler-MacLaurin formula; flag varieties; exponential sums; interpolators Garoufaldis, S.; Pommersheim, J.: Sum-integral interpolators and the Euler-Maclaurin formula for polytopes, Trans. amer. Math. soc. 364, 2933-2958 (2012) Toric varieties, Newton polyhedra, Okounkov bodies, Topology of general 3-manifolds, Knots and links in the 3-sphere Sum-integral interpolators and the Euler-Maclaurin formula for 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 Let \(\mathcal{Q}\subset\mathbb{C}\langle Z_1, \dots, Z_n\rangle\) be an arbitrary set of polynomials in noncommutative indeterminates such that \(q(0)=0\) for all \(q\in \mathcal{Q}\). The noncommutative variety
\[
\mathcal{V}_{f,\mathcal{Q}}^m(\mathcal{H}):=\left\{ X=(X_1, \dots, X_n)\in \mathbf{D}_f^m(\mathcal{H}): q(X)=0 \text{ for all } q\in \mathcal{Q}\right\},
\]
where \(\mathbf{D}_f^m(\mathcal{H})\) is a \textit{noncommutative regular domain} in \(B(\mathcal{H})^n\) and \(B(\mathcal{H})\) is the algebra of bounded linear operators on a Hilbert space \(\mathcal{H}\), admits a \textit{universal model} \[B^{(m)}=(B_1^{(m)}, \dots, B_n^{(m)})\] such that \(q(B^{(m)})=0\), \(q\in \mathcal{Q}\), acting on a \textit{model space} which is a subspace of the full Fock space with \(n\) generators. In this paper, we obtain a Beurling type characterization of the joint invariant subspaces under the operators \(B_1^{(m)}, \dots, B_n^{(m)}\), in terms of partially isometric multi-analytic operators acting on model spaces. More generally, a Beurling-Lax-Halmos type representation is obtained and used to parametrize the wandering subspaces of the joint invariant subspaces under \(B_1^{(m)}\otimes I_{\mathcal{E}}, \dots, B_n^{(m)}\otimes I_{\mathcal{E}}\), and to characterize when they are generating for the corresponding invariant subspaces. Similar results are obtained for any \textit{pure} \(n\)-tuple \((X_1, \dots, X_n)\) in the noncommutative variety \(\mathcal{V}_{f,\mathcal{Q}}^m(\mathcal{H})\). We characterize the elements in the noncommutative variety \(\mathcal{V}_{f,\mathcal{Q}}^m(\mathcal{H})\) which admit \textit{characteristic functions}, develop an operator model theory for the \textit{completely non-coisometric} elements, and prove that the characteristic function is a complete unitary invariant for this class of elements. This extends the classical Sz.-Nagy-Foiaş functional model for completely non-unitary contractions, based on characteristic functions. Our results apply, in particular, when \(\mathcal{Q}\) consists of the noncommutative polynomials \(Z_iZ_j-Z_jZ_i\), \(i,j=1, \dots, n\). In this case, the model space is a symmetric weighted Fock space, which is identified with a reproducing kernel Hilbert space of holomorphic functions on a Reinhardt domain in \(\mathbb{C}^n\), and the universal model is the \(n\)-tuple \((M_{\lambda_1}, \dots, M_{\lambda_n})\) of multipliers by the coordinate functions. noncommutative regular domain; joint invariant subspaces; Beurling-Lax-Halmos-type representation; wandering subspaces; characteristic functions; universal model Invariant subspaces of linear operators, Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones), Canonical models for contractions and nonselfadjoint linear operators, Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc., Noncommutative algebraic geometry Invariant subspaces and operator model theory on noncommutative 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 \(\mathbb{K}\) be an infinite field and \(P\) either \(\mathbb{K}[x_1,\dots,x_n]\) or \(\mathbb{K}[[x_1,\dots,x_n]]\) (formal power series). Let \(\hat{P} = \Hom_{P_0}(P,E)\), where \(E\) is the injective hull of \(\mathbb{K}\) over \(P_0\); we get a correspondence between graded ideals \(I\) of \(P\) and \(P\)-submodules \(I^\perp\) of \(\hat{P}\), under which the ideals \(I\) for which \(P/I\) is Artinian correspond to finitely generated submodules \(I^\perp\); \(I^\perp\) is known as the inverse system of \(I\) (by Macaulay).
This correspondence as been extended to the case of positive dimension and such generalization is studied here, by using the varius socles in the inverse system. The main result is an explicit description of inverse limits of Macaulay's inverse systems, obtained by dividing out powers of a linear regular sequence. Macaulay inverse system; Matlis duality; Rees isomorphism Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Graded rings, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Power series rings, Secant varieties, tensor rank, varieties of sums of powers Inverse limits of Macaulay's inverse systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For \(n \geq 3\), let \(\mathbb H_n \subset \text{Hilb} (\mathbb P^n)\) be the component of the Hilbert scheme whose general member is a union \(A \cup B\) with \(A \subset \mathbb P^n\) a quadric of codimension \(2\) and \(B \subset \mathbb P^n\) a quadric of codimension \(3\). Given linear subspaces \(H,L \subset \mathbb P^n\) of codimensions \(1\) and \(2\) and a general quadric hypersurface \(Q\), the map \((H,L,Q) \mapsto (H \cap Q) \cup (L \cap Q)\) induces a birational map \(\sigma: \mathbb X_n \to \mathbb H_n\), where \(\mathbb X_n \to \mathbb G(n-1,n) \times \mathbb G(n-2,n)\) is the projective bundle whose fiber over \((H,L)\) is the family of quadric hypersurfaces on \(\mathbb P^n\) modulo the quadrics in \(I_H \cdot I_L\). The map \(\sigma\) fails to be regular along the locus where \(H \subset Q\) or \(L \subset Q\). The authors assert an explicit sequence of three blow-ups \(\mathbb X_n^3 \to \mathbb X_n^2 \to \mathbb X_n^1 \to \mathbb X_n\) along smooth centers for which the rational map \(\mathbb X_n^3 \to \mathbb H\) induced by \(\sigma\) is a morphism for \(n \geq 3\), however they only give a proof for \(n=3\). Their proof analyzes orbits under the projective general linear group and uses some computations by the software SINGULAR. Similar results are known for the Hilbert scheme component whose general member is a union of a pair of codimension two linear subspaces [\textit{D. Chen}, \textit{I. Coskun} and the reviewer, Commun. Algebra 39, No. 8, 3021--3043 (2011; Zbl 1238.14012)] and more generally for the Hilbert scheme component whose general member is a union of two linear subspaces of any codimension [\textit{R. Ramkumar}, Math. Z. 300, No. 1, 493--540 (2022; Zbl 1481.14008)]. Inspired by computations of \textit{A. L. Meireles Araújo} et al. [Int. J. Algebra Comput. 29, No. 1, 9--21 (2019; Zbl 1411.14013)] and \textit{J. A. D. Maia} et al. [J. Pure Appl. Algebra 217, No. 8, 1379--1394 (2013; Zbl 1268.14047)], the authors then give an enumerative formula for the degree of the variety of degree \(d\) surfaces \(S \subset \mathbb P^3\) containing a conic and two points varying on a fixed line. schematic unions; components of Hilbert scheme Parametrization (Chow and Hilbert schemes) Schematic union of \((n-3)\) and \((n-2)\) dimensional quadrics in \(\mathbb{P}^n\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper generalizes \textit{M. M. Kapranov}'s [Adv. Sov. Math. 16(2), 29--110 (1993; Zbl 0811.14043)] well-known construction of \(\overline{\mathcal{M}}_{0,n}\) as the Chow quotient \((\mathbb{P}^1)^d /\!/_{\text{Ch}} \text{SL}_2\) by considering the set
\[
U_{d,n} := \left\{ (p_1, \dots, p_n) \in (\mathbb{P}^d)^n \,\mid\, p_i \text{ are distinct points lying on a rational normal curve} \right\}
\]
and its closure \(V_{d,n} := \overline{U_{d,n}}\) in \((\mathbb{P}^d)^n\). This closure is shown to consist of all configurations of (possibly coinciding) points that lie on a degeneration of a rational normal curve. The author shows that for \(d \leq n-3\) the Chow quotient \(V_{d,n} /\!/_{\text{Ch}} \text{SL}_{d+1}\) is isomorphic to \(\overline{\mathcal{M}}_{0,n}\), and that for any effective linearization \(L\) there exists a morphism to the GIT quotient \(\varphi: \overline{\mathcal{M}}_{0,n} \to V_{d,n} /\!/_L \text{SL}_{d+1}\), extending the isomorphism \(\mathcal{M}_{0,n} \widetilde{\to}\; U_{d,n} / \text{SL}_{d+1}\). Every effective linearization can be encoded by a tuple \(L = (x_1, \dots, x_n)\) of rational numbers with the conditions that \(0 \leq x_i \leq 1\) and \(\sum_{i=1}^n x_i = d+1\), and the morphism \(\varphi\) is shown to factor through the corresponding moduli space \(\overline{\mathcal{M}}_{0,L}\) of \(L\)-weighted pointed stable curves that was constructed in [\textit{B. Hassett}, Adv. Math. 173, No. 2, 316--352 (2003; Zbl 1072.14014)].
Moreover it is shown that if \(L = (\frac{d+1}{n}, \dots, \frac{d+1}{n})\) is the unique \(S_n\)-invariant linearization and \(\mathcal{L}_d\) denotes the GIT polarization on \(V_{d,n} /\!/_L \text{SL}_{d+1}\), then \(\varphi^* \mathcal{L}_d\) spans the same ray in \(N^1(\overline{\mathcal{M}}_{0,n})\) as a certain conformal blocks divisor studied previously in [\textit{M. Arap} et al., Int. Math. Res. Not. 2012, No. 7, 1634--1680 (2012; Zbl 1271.14034)]. Results from that paper imply that for \(d = 1, \dots, \lfloor \frac{n}{2} \rfloor - 1\) the line bundles \(\varphi^* \mathcal{L}_d\) span distinct extremal rays of the symmetric nef cone of \(\overline{\mathcal{M}}_{0,n}\). On the other hand, using the classical Gale transform the author shows that \(\varphi^* \mathcal{L}_d\) and \(\varphi^* \mathcal{L}_{n - d - 2}\) span the same ray in \(N^1(\overline{\mathcal{M}}_{0,n})\), so the spaces \(V_{d,n} /\!/_L \text{SL}_{d+1}\) and \(V_{n-d-2,n} /\!/_L \text{SL}_{n-d-1}\), with \(L\) the respective symmetric linearizations, have isomorphic normalizations. conformal blocks; moduli space of stable pointed rational curves; Chow quotient; GIT quotient; weighted pointed stable curves; Gale duality Giansiracusa, N., Conformal blocks and rational normal curves, J. Algebraic Geom., 22, 4, 773-793, (2013) Families, moduli of curves (algebraic), Geometric invariant theory, Parametrization (Chow and Hilbert schemes) Conformal blocks and rational normal curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper is arranged in the following way. In \S1 we introduce notations for Schubert cells and we will show what cells and their neighbourhoods give rise to a uniquely constructed Schubert stratification.
In \S2 we show how a flattening corresponds to a Schubert cell.
In \S3 we introduce the concept of stable equivalence of flattenings, which allows us to compare cascades that generally consist of a different number of curves lying in spaces of different dimensions.
The relation of equivalence of flattenings is constructed in \S4.
In \S5 we give a classification of the flattenings occurring in generic three-parameter families of cascades; relative to this equivalence, we study the singularities of their bifurcation diagrams, and give results of V. I. Arnol'd and O. P. Shcherbak on the connection of these singularities with the geometry of the swallowtail, tangential singularities, and the singularities of projections.
In \S6 we give lists of the simple flattenings of curves, and also of cascades, corresponding to complete flags. The methods used in the proof of the classification theorems of \S\S5 and 6 are validated in \S\S7-9. In \S7 we prove a generalization of the Frobenius theorem on integrable distributions to the case of distributions with singularities. In \S8 this result is carried over to the case of the infinite-dimensional space of germs of cascades. Using these results, we prove the finite determinacy and versality theorems in \S9.
Section 10 and 11 are devoted to applications of the theory of flattenings to the study of oscillatory properties of linear differential equations and to the decomposition of Weierstrass points of algebraic curves, respectively. Grassmannians; flag manifolds; Schubert cells; Schubert stratification; flattening; equivalence; singularities; bifurcation; Weierstrass points; algebraic curves DOI: 10.1070/RM1991v046n05ABEH002844 Deformations of complex singularities; vanishing cycles, Formal methods and deformations in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Differentiable maps on manifolds, Theory of singularities and catastrophe theory Flattenings of projective curves, singularities of Schubert stratifications of Grassmannians and flag varieties, and bifurcations of Weierstrass points of algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper gives a detailed description of the effective cone of the Gieseker moduli space \(M(\xi)\) of stable coherent shaves on \(\mathbb{P}^2\) with Chern character \(\xi\).
Determination of the effective cone is reduced to the higher rank interpolation problem (see Problem 1.2), which asks to determine the minimal slope \(\mu^+(\mathbb{Q})\) satisfying some cohomological condition for a given sheaf \(U \in M(\xi)\). This interpolation problem has an algorithmic solution indicated in the main Theorem 1.3. The algorithm involves some graphs in the slope-discriminant plane \(\{(\mu,\Delta)\}\), in particular the parabola \(Q_{\xi}\) encoding sheaves cohomologically orthogonal to \(U \in M(\xi)\). Theorem 1.3 is proved by a resolution of \(U\) in terms of a triplet of exceptional bundles, which comes from Beilinson spectral sequence.
The \((\mu,\Delta)\)-plane and the graphs appearing in the interpolation problem enjoys remarkable arithmetic properties as discussed in \S4. There is a curve \(\delta(mu)\) giving the positive dimension criteria of the moduli space of vector bundles with slope \(\mu\). \(\delta(\mu)\) is fractal-like in the sense that it intersects with the line \(\Delta=1/2\) in a Cantor set \(C\). It is shown that the parabola \(Q_\xi\) does not intersect the line \(\Delta=1/2\) along \(C\).
The effective cone has two extremal edges. One edge is the pullback of the ample generator of Kronecker modules as discussed in \S6. The other edge is the following description. For \(\xi\) with rank greater than \(2\), the Serre duality gives an isomorphism of Picard groups of moduli spaces preserving effective cones. Lower rank cases have more explicit description. For example, if the rank is \(2\), then the morphism [\textit{J. Li}, J. Differ. Geom. 37, No. 2, 417--466 (1993; Zbl 0809.14006)] \(M(\xi) \to M^{\text{DUY}}(\xi)\) to the Donaldson-Uhlenbeck-Yau moduli space determines the nef divisor giving the other edge. moduli spaces of sheaves; Bridgeland stability; effective cone; Brill-Noether divisors I. Coskun, J. Huizenga and M. Woolf, The effective cone of the moduli space of sheaves on the plane, preprint (2014), . Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Minimal model program (Mori theory, extremal rays), Algebraic moduli problems, moduli of vector bundles, Syzygies, resolutions, complexes and commutative rings The effective cone of the moduli space of sheaves on the plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is concerned with approximation of differentiable mappings into spheres by polynomial or rational regular functions (in the \(C^{\infty}\) topology). Let \(X\subseteq R^ n\) and \(Y\subseteq R^ p\) be real algebraic sets. Denote by \({\mathcal R}(X,Y)\) the set of rational maps \((f_ 1/g_ 1,...,f_ p/g_ p)\) between X and Y such that \(f_ i^{-1}(0)\cap X=\emptyset\) for \(i=1,...,p\). \({\mathcal E}(X,Y)\) is the set of smooth maps with \(C^{\infty}\) topology.
Theorem 1: For each \(n\in {\mathbb{N}}\), and for \(k=1,2,4\), \({\mathcal R}(S^ n,S^ k)\) is dense in \({\mathcal E}(S^ n,S^ k).\)
Theorem 2: Let X be a compact non singular real algebraic set and let f be a smooth map from X to \(S^ k\). If \(k=1,2,4\) the following conditions are equivalent: (1) f can be approximated in \({\mathcal E}(X,S^ k)\) by elements of \({\mathcal R}(X,S^ k)\); (2) f is homotopic to an element of \({\mathcal R}(X,S^ k).\)
Theorem 3: (as corollary of a more general theorem on maps between X and \(S^ 1\) and relations between the homology and algebraic homology groups of X): Let X be a non singular compact real algebraic curve; then \({\mathcal R}(X,S^ 1)\) is dense in \({\mathcal E}(X,S^ 1).\)
Theorem 4: Let X be a non orientable compact connected real algebraic surface. Then \({\mathcal R}(X,S^ 2)\) is dense in \({\mathcal E}(X,S^ 2)\) if one of the following conditions holds: (1) there exist a non singular algebraic curve \(C\subset X\) such that {\#}\({}_ 2(C,C;X)=2\) \((=\) the modulo 2 self-intersection number of C in X); (2) \(H_ 1^{alg}(X,Z_ 2)=H_ 1(X,Z_ 2);\) (3) the genus of X (as a smooth surface) is odd.
The authors give also many examples. approximation of differentiable mappings by real rational maps Fichou, G., Huisman, J., Mangolte, F., Monnier, J.-Ph.: Fonctions régulues, arXiv:1112.3800v3 [math.AG], to appear in J. Reine Angew. Math Real algebraic and real-analytic geometry, Differentiable mappings in differential topology, Approximation by rational functions Algebraic approximation of mappings into spheres | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The classical notion of normalization is not sufficient to address many natural questions one encounters in the theory of complex spaces and in algebraic geometry. To remedy this, two notions related to normalization were introduced about 50 years ago, namely, \textit{weak normalization} and \textit{seminormalization}. However, these by now standard concepts, are not adequate for problems in real algebraic geometry. The main difficulty is to capture the behavior of algebraic varieties defined over \(\mathbb{R}\) near their real loci. Therefore it is natural to focus on affine real algebraic sets. The set of central points of a real algebraic set X, denoted by \( \operatorname{Cent}X\), is the closure in the Euclidean topology of the set of non-singular points of \(X\). In general, \(\operatorname{Cent}X\) is different from \( X\). The authors of the paper under review construct the weak normalization and seminormalization of \(X\) relative to \(\operatorname{Cent}X\). The idea is as follows. The ring \(\mathcal{P}(X)\) of polynomial functions on \(X\) is a subring of the ring \(\mathcal{K}^{0}(\operatorname{Cent}X)\) of rational functions on \(X\) admitting continuous extensions to \(\operatorname{Cent}X\). The integral closure of \(\mathcal{P}(X)\) in \(\mathcal{K}^{0}(\operatorname{Cent}X)\) is a finite module over \(\mathcal{P}(X)\), and therefore it coincides with the polynomial ring of a real algebraic set, denoted \(X^{w_{c}}\). Moreover, there is a natural finite birational polynomial morphism \(\pi ^{w_{c}}:X^{w_{c}}\rightarrow X\) which induces a homeomorphism for the Euclidean topology between the central loci. This morphism is called the \textit{weak normalization of} X \textit{relative to} \(\operatorname{Cent}X\). One of the main results is the following universal property (see Theorem 4.8): Let \(X\) be a real algebraic set. Let \(Y\) be a real algebraic set equipped with a finite birational morphism \(\pi :Y\rightarrow X\). Then \(\pi \) induces a bijection from \(\operatorname{Cent}Y\) onto \(\operatorname{Cent}X\) if and only if \( \pi ^{w_{c}}:X^{w_{c}}\rightarrow X\) factorizes through \(\pi \). To construct the \textit{seminormalization of} \(X\) \textit{relative to} \(\operatorname{Cent}X,\) one proceeds in a similar manner, replacing \(\mathcal{K}^{0}(\operatorname{Cent} X)\) by the ring \(\mathcal{R}^{0}(\operatorname{Cent}X)\) of hereditarily rational functions on \(X\) admitting continuous extensions to \(\operatorname{Cent}X.\) The authors establish many useful algebraic and geometric properties of their constructions. real algebraic set; seminormalization; weak-normalization; continuous rational function; hereditarily rational function Real algebraic sets, Integral closure of commutative rings and ideals Weak and semi normalization in real algebraic geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper begins with two genericity results for rigid analytic morphisms (in the adic setting) analogous to classical results in algebraic geometry. Let \(K\) be a non-Archimedean complete valued field. Let \(f \colon X \to Y\) be a quasi-compact map of rigid spaces over \(K\), with \(Y\) geometrically reduced. Then \(f\) is flat over a dense open subset of~\(Y\). If \(K\) is of characteristic 0, the result holds with smooth instead of flat. Note that similar results were proved by \textit{A. Ducros} in the setting of Berkovich spaces in [Families of Berkovich spaces. Paris: Société Mathématique de France (SMF) (2018; Zbl 1460.14001)]. Even though there is no good notion of generic point in rigid geometry, the proof presented in the paper makes use of weakly Shilov points (introduced by the authors), which behave similarly in some aspects. For instance, a morphism as above is always flat (and smooth in characteristic 0) over such a point, and the property extends to a neighborhood.
The authors then apply those results to the theory of Zariski-constructible étale sheaves on rigid spaces, as defined by \textit{D. Hansen} [Compos. Math. 156, No. 2, 299--324 (2020; Zbl 1441.14085)]. Under the assumption that the base field \(K\) is of characteristic 0, they prove that the notion of Zariski-constructibility is local for the étale topology and develop a six-functor formalism in this setting.
As further applications of the theory of constructible sheaves, the authors introduce perverse sheaves and intersection cohomology on rigid spaces in characteristic 0, and prove that those notions are well-behaved. rigid analytic spaces; étale cohomology; generic smoothness; six functors; constructible sheaves; perverse sheaves; intersection cohomology Rigid analytic geometry, Étale and other Grothendieck topologies and (co)homologies, Sheaves in algebraic geometry The six functors for Zariski-constructible sheaves in rigid geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A continuous map \(f :\mathbb{K}^{n} \rightarrow \mathbb{K}^{N}\), \(\mathbb{K} =\mathbb{R}\) or \(\mathbb{C}\), is \(k\)-regular if the image of any \(k\) distinct points span a \(k\)-dimensional linear subspace of \(\mathbb{K}^{N} .\) Such mappings are related to interpolation problem by the following (known) theorem: there exists a \(k\)-regular mapping \(f :\mathbb{R}^{n} \rightarrow \mathbb{R}^{N}\) if and only if there exists \(N\)-dimensional vector subspace \(V \subset C^{0}(\mathbb{R}^{n})\) of the space of continuous function on \(\mathbb{R}^{n}\) such that for any distinct points \(P_{1} ,\ldots ,P_{k} \in \mathbb{R}^{n}\) and scalars \(\lambda _{1} ,\ldots ,\lambda _{k} \in \mathbb{R}^{}\) there exists \(f \in V\) such that \(f\left (P_{i}\right ) =\lambda _{i}\), \(i =1 ,\ldots ,k\). The authors, using the classical Veronese embedding, prove the following theorems in complex case:
1. For any \(k >3\) the mapping
\[
\mathbb{C}^{n} \rightarrow \mathbb{C}^{\binom{n +k -2}{k -2} +1}
\]
which consecutive components are: all monomials of degree at most \((k -3) ,\) all monomials of degree \((k -2)\) with exception of pure powers, \((n -1)\) polynomials \(x_{i}^{k -1} -x_{i +1}^{k -2}\) for \(i =1 ,\ldots ,n -1\) and the single monomials \(x_{1}^{k -2}\) and \(x_{n}^{k -1}\), is a \(k\)-regular map,
2. For any \(k >4\) the maping
\[
\mathbb{C}^{n} \rightarrow \mathbb{C}^{\binom{n +k -3}{n} +n +1}
\]
which consecutive components are: all monomials of degree at most \((k -3)\), \((n -1)\) polynomials \(x_{i}^{k -1} -x_{i +1}^{k -2}\) for \(i =1 ,\ldots ,n -1\) and the single monomials \(x_{1}^{k -2}\) and \(x_{n}^{k -1}\), is a \(k\)-regular map.
In particular they obtain a \(4\)-regular polynomial mapping \(\mathbb{C}^{3} \rightarrow \mathbb{C}^{11}\) and a \(5\)-regular polynomial mapping \(\text{}\mathbb{C}^{3} \rightarrow \mathbb{C}^{14} .\) interpolation space; Veronese embedding; areole; \(k\)-regular map M. Michałek and C. Miller, \textit{Examples of \(k\)-regular maps and interpolation spaces}, preprint, arXiv:1512.00609, 2015, . Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Interpolation in approximation theory, Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs Examples of \(k\)-regular maps and interpolation 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 Resorting to the Lenard recursion equations, we derive the Newell hierarchy associated with a \(3\times 3\) matrix spectral problem and establish Dubrovin-type equations in terms of the introduced trigonal curve \({\mathcal K}_{m-1}\) of arithmetic genus \(m-1\). Based on the theory of trigonal curve, we construct the corresponding Baker-Akhiezer functions and meromorphic functions on \({\mathcal K}_{m-1}\). The known zeros and poles for the Baker-Akhiezer functions and meromorphic functions allow one to find their theta function representations, from which we give algebro-geometric constructions of quasi-periodic flows of the Newell hierarchy and their explicit theta function representations. Newell hierarchy; algebro-geometric constructions; quasi-periodic solutions Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relationships between algebraic curves and integrable systems Algebro-geometric constructions of quasi-periodic flows of the Newell hierarchy and applications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k[x_1,\dots,x_n]\) denote the polynomial ring in \(n\) variables over the field \(k\) and let \(c\geq 1\) be an integer. For an integer \(t\geq1\), let \(X_t\) denote the matrix with \((i,j)\)-entry equal to \(x_{(i-1)c+j}\), for \(1\leq i\leq t\) and \(1\leq j\leq n-(t-1)c\). The matrix \(X_t\) is called the ``extended Hankel matrix''. If \(c=1\), then \(X_t\) is the classical Hankel matrix. Let \(A_t\) be the \(k\)-subalgebra of \(k[x_1,\dots,x_n]\) generated by the \(t\times t\) minors of \(X_t\). The present paper calculates the multiplicity, \(e(A_t)\), of \(A_t\). It is shown that \(e(A_t)\) is equal to the number of facets of a simplicial complex \(\Delta_t\). Ultimately, it is shown that \(e(A_t)\) is equal to a sum of the number (denoted \(f^{\lambda/\mu}\)) of standard skew tableaux of shape \(\lambda/\mu\) for certain partitions \(\lambda\) and \(\mu\). In particular, if \(t\) attains its maximum possible value, namely \(m=\lfloor\frac{n+c}{c+1}\rfloor\), then \(e(A_m)=f^{\lambda}\), where \(\lambda\) is the partition which consists of \(m\) copies \(n-(m-1)(c+1)-1\). The present paper is a continuation of ``The determinatal ideals of extended Hankel matrices'' by the same author [J. Pure Appl. Algebra 215, No. 6, 1502--1515 (2011; Zbl 1223.13006)]. The earlier paper describes a Gröbner basis for the defining ideal of \(A_t\) and this Gröbner basis is the starting point for the present work. multiplicity; determinantal ideal; extended Hankel matrix Linkage, complete intersections and determinantal ideals, Determinantal varieties Multiplicity of algebras of minors of extended Hankel matrices | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main result of the paper is that if \(f:\quad X\to S\) is a proper smooth family, then the period map \(P:\quad S\to \Gamma \setminus D\) is generically injective modulo polarized equivalence (this is generic Torelli) whenever variational Torelli holds for the family. Recall that variational Torelli for \(f:\quad X\to S\) means that the fibers \(X_ s=f^{-1}(s)\) are determined up to polarized equivalence by the real part \((H_{{\mathbb{R}}},F^.,Q,T,\delta)\) of the infinitesimal variation of Hodge structure (IVHS). In practice, \(\Gamma\) is usually either \(\Gamma_ 0\), the monodromy group of the family, or \(G_{{\mathbb{Z}}}\), the integral automorphisms of the Hodge structure. However, it turns out that \(\Gamma\) may be taken to be any discrete subgroup of \(G_{{\mathbb{R}}}\) that contains the monodromy group \(\Gamma_ 0.\)
Previous treatments of this result always needed extra hypotheses, including infinitesimal Torelli, the existence of a suitable quotient of S, and the existence of a regular value of the period map P. This paper shows that none of these are needed. - The two main tools used in the proof are the principle of prolongation and the existence of generic quotients. Both of these results are part of the mathematical ''folklore'', and are given precise formulations and proofs. - The one limitation of the proof is that ''generic'' has to be interpreted as the complement of an analytic subvariety. It would be preferable to know that the subvariety were algebraic. This is related to the question of putting an algebraic structure on the period map. period map; generic Torelli; variational Torelli; infinitesimal variation of Hodge structure; IVHS DOI: 10.1007/BF01388918 Transcendental methods, Hodge theory (algebro-geometric aspects), Transcendental methods of algebraic geometry (complex-analytic aspects) Variational Torelli implies generic Torelli | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper deals with a generalization of Hamburger's theorem on the characterization of zeta-functions. Originally, Hamburger's theorem claims that the Riemann zeta-function is characterized by its functional equation. The author of this paper proves Hamburger's theorem for Epstein's zeta-function, which is a typical example of a zeta-function associated with a prehomogeneous vector space.
Let \(V={\mathbb{R}}^ n\) be a real n-dimensional vector space equipped with the inner product \((x,y)=\sum^{n}_{i=1}x_ iy_ i\). Let L be a lattice in V and let \(L^*\) be its dual lattice. Let P(x) be a homogeneous harmonic polynomial of degree d on V. For a function \(a: L\setminus \{0\}\to {\mathbb{C}}\) satisfying \(a(m)<c\cdot (m,m)^ M\) with some positive contants c and M, we put
\[
\zeta_ P(a,L;s):=\sum_{m\in L\setminus \{0\}}a(m)\cdot P(m)/(m,m)^{s+(d/2)}.
\]
We may consider \(\zeta^*_ P(a^*,L^*;s)\) for \(a^*: L^*\setminus \{0\}\to {\mathbb{C}}\) satisfying the same condition. Then \(\zeta_ P\) and \(\zeta^*_ P\) are absolutely convergent for \(Re(s)>(n/2)+M.\)
Assume, for any harmonic polynomial P(x), (1) meromorphic extendability of \(\zeta_ P\) and \(\zeta^*_ P\), (2) the condition that \(\zeta_ P\) and \(\zeta^*_ P\) have only finite poles at the same locations, and (3) \(\zeta_ P\) and \(\zeta^*_ P\) are connected by a functional equation. Under the additional condition that \(\zeta_ P\) and \(\zeta_ P^*\) are entire functions if d is sufficiently large, the author proves that a(m) and \(a^*(m)\) are constant functions. This is nothing but a generalization of Hamburger's theorem. The author drove some of its consequences assuming the above result. generalization of Hamburger's theorem; Epstein's zeta-function; prehomogeneous vector space Analytic theory (Epstein zeta functions; relations with automorphic forms and functions), Hurwitz and Lerch zeta functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Special varieties The Hamburger theorem for the Epstein 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 Let \(S(x)\) be a real analytic function defined in a neighbourhood of the origin in \(\mathbb{R}^{n}\) and \(\phi (x)\) a smooth real-valued function such that \(\text{supp}\,\phi \subset U.\) The author studies the behaviour of the integrals
\[
\begin{aligned} I_{S,\phi }(\epsilon ) &= \underset{\{x:0<S(x)<\epsilon \}}{\int }\phi (x)dx, \\
\text{\(\phi(x)\) nonnegative and \(\phi(0)>0\),}
J_{S,\phi }(\lambda ) &= \underset{\mathbb{R}^{n}}{\int }e^{i\lambda S(x)}\phi (x)dx
\end{aligned}
\]
as \(\epsilon \rightarrow 0\) and \(\lambda \rightarrow \infty .\) Assuming \( S(0)=0\) and \(\nabla S(0)=0\) by Hironaka's resolution of singularities one has an asymptotic expansion
\[
\begin{aligned} I_{S,\phi }(\epsilon ) &\sim \sum_{j=0}^{\infty }\sum_{i=0}^{n-1}c_{ij}(\phi )\ln (\epsilon )^{i}\epsilon ^{r_{j}}, \\ J_{S,\phi }(\lambda ) &\sim \sum_{j=0}^{\infty }\sum_{i=0}^{n-1}d_{ij}(\phi )\ln (\lambda )^{i}\lambda ^{-s_{j}}, \end{aligned}
\]
where \(\{r_{j}\}\) and \(\{s_{j}\}\) are increasing arithmetic progressions of positive rational numbers independent of \(\phi .\) The problem is to find the orders of these series. \textit{A. N. Varchenko} [Funct. Anal. Appl. 10, 175-196 (1976); translation from Funkts. Anal. Prilozh. 10, No.3, 13--38 (1976; Zbl 0351.32011)] found a formula for the second series in terms of the Newton diagram of \(S\) under assumption on the Kouchnirenko nondegeneracy of \(S\). The author generalizes the Varchenko result (in the form of inequalities and equalities for \(n=2)\) to the general case (without the assumption of nondegeneracy). oscilatory integral; real analytic function; Newton polyhedron M. Greenblatt, Oscillatory integral decay, sublevel set growth, and the Newton polyhedron, Math. Ann. 346 (2010), no. 4, 857-895. Singularities in algebraic geometry, Real-analytic and semi-analytic sets, Local complex singularities Oscillatory integral decay, sublevel set growth, and the Newton polyhedron | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors investigate the regularity of \(I^n\) where \(I\) is an ideal in a polynomial ring\break \(K[X_1,\dots,X_m]\). The main result is that the Castelnuovo-Mumford regularity \(\text{reg }I^n\) satisfies
(i) \(\text{reg }I^n\leq nd(I)+e\), and
(ii) \(\text{reg }I^n\) is a linear function for \(n\gg 0\).
Here \(d(I)\) is the largest degree of a generating element of \(I\). [A similar result has independently been obtained by \textit{V. Kodiyalam}, Proc. Am. Math. Soc. 128, 407-411 (2000; Zbl 0929.13004).] The proof uses essentially that the Rees algebra \({\mathcal R}(I)\) is a finitely generated bigraded \(K\)-algebra, and therefore one obtains a similar bound if \(I\) is replaced by its integral closure. Furthermore the number \(e\) can be determined from the bigraded structure of \(R(I)\). The linearity for \(n\gg 0\) is proved by an argument of linear programming applied to a suitable initial ideal.
The upper bound on \(\text{reg }I^n\) also holds for \(\text{reg }\widetilde I^n\) where \(^\sim\) denotes saturation. However, the assertion on linearity is false in general since the ``saturated'' Rees algebra need not be finitely generated. The paper contains several interesting examples for the behaviour of \(\text{reg }\widetilde I^n\), in particular one showing that the quotient \((\text{reg }\widetilde I^n)/n\) may converge to an irrational number. polynomial ring; Castelnuovo-Mumford regularity; Rees algebra Cutkosky, S. Dale; Herzog, Jürgen; Trung, Ngô Viêt, Asymptotic behaviour of the Castelnuovo--Mumford regularity, Compositio Math., 118, 3, 243-261, (1999) Local cohomology and commutative rings, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Polynomial rings and ideals; rings of integer-valued polynomials, Vanishing theorems in algebraic geometry Asymptotic behaviour of the Castelnuovo-Mumford regularity | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author investigates the converging radius of the ``generalized zeta function''. Let \(X\) be a projective scheme defined over \(\mathbb{F}_q\), and let \(N_d(X)\) denote the number of effective zero-dimensional cycles \(Y\) defined over \(\mathbb{F}_q\) with \(\deg Y=d\). The author observes that the usual zeta function \(\zeta(X;t)\) of \(X\) can be expressed as \(\sum_{d=0}^\infty N_d(X)t^d\), and proposes the definition of a generalized zeta function in the following way. Let \((X,H)\) be a polarized quasi-projective scheme of dimension \(n\) over \(\mathbb{F}_q\) with a fixed compactification \((\overline{X},\overline{H})\). For an integer \(l\) with \(0\leq l\leq n-1\) and any positive integer \(d\), let \(N_d(X,H,l;\mathbb{F}_q)\) be the number of effective \(l\)-dimensional cycles \(Y\) defined over \(\mathbb{F}_q\) with \(\deg_{\overline{H}}Y=d\), and he sets \(Z(X,H,l;t)=\sum_{d=0}^\infty N_d(X,H,l;\mathbb{F}_q)t^{d^{l+1}}\), which is the proposed generalized zeta function. The main theorem of this paper shows that the function converges at the origin \(t=0\) with a natural converging radius. algebraic cycles; zeta function Divisors, linear systems, invertible sheaves, Algebraic cycles The number of 1-codimensional cycles on projective varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper provides an alternative method to the topological recursion for computing the one-point function \(W_1(x)\) in the case of Laguerre and generalized Laguerre ensembles, i.e. an Hermitian matrix model with potential \(V(x)=x\) on \(\mathbb{R}_+\). The main result of the paper is to provide an explicit three terms recursion, which is similar to the one of \textit{J. Harer} and \textit{D. Zagier} [Invent. Math. 85, 457--485 (1986; Zbl 0616.14017)] or \textit{N. Do} and \textit{P. Norbury} [Topological recursion for irregular spectral curves, \url{arXiv:1412.8334}] in the Gaussian case, for computing the coefficients of the one-point function. The main advantage of this method compared to the topological recursion, is that it does not require the knowledge of the other correlation functions and therefore computations are much easier and faster. Thus, as explained in the paper, it is particularly well-suited for the matching of the coefficients with integers arising in enumerative geometry. In the Laguerre case, the author presents the details of the connection with the number of unicellular two-colored maps. The proof of the three terms recursion relies first on a replica trick, then the use of the Harish-Chandra-Itzykson-Zuber integral, and finally some contour deformations and standard complex analysis techniques. Consequently, the paper is relatively short and pleasant to read. However, since the proof relies on the knowledge of some specific formulas, the method may not easily generalize to more complicated models. replica method; Laguerre ensemble; topological recursion Chekhov, L. O., The harer-Zagier recursion for an irregular spectral curve, J. Geom. Phys., 110, 30-43, (2016) Enumerative problems (combinatorial problems) in algebraic geometry, Random matrices (algebraic aspects), Exact enumeration problems, generating functions, Random matrices (probabilistic aspects) The Harer-Zagier recursion for an irregular spectral curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review is the synopsis of a course given by the author at the Centre Émile Borel, France, during the special semester on \(p\)-adic cohomology and its arithmetic applications held in 1997. Its main goal is to provide a comprehensive survey on the recent developments towards a generalization of the classical theory of algebraic \({\mathcal D}\)-modules (over \(\mathbb{C}\)) to the case of base schemes over a field of arbitrary characteristic, emphasizing the arithmetically relevant case of characteristic \(p> 0\).
The objective of such an arithmetic theory of \({\mathcal D}\)-modules is to furnish an adequate toolkit, like in the classical case of characteristic \(0\), for the effective study of the various \(p\)-adic cohomology theories once initiated by A. Grothendieck in the 1960s.
Recently, a remarkable progress in setting-up a \(p\)-adic theory of algebraic \({\mathcal D}\)-modules has been achieved by P. Berthelot himself, mainly through his series of research papers titled ``\({\mathcal D}\)-modules arithmétiques I, II, III, IV'', and published between 1996 and 2004.
Actually, the article under review basically surveys the contents of these recent, fundamental papers by explaining systematically both their underlying philosophy and their main results obtained so far.
In this vein, the present treatise is intended as the general introduction to the more detailed research papers ``\({\mathcal D}\)-modules arithmétiques I, II, III, IV'', thereby focusing more on the conceptual and methodological framework rather than on complete proofs of the principal results therein. Accordingly, the presentation begins with a sweeping introduction of both historical and strategical nature, which is already very enlightening by itself alone.
The main body of the paper consists of five rather extensive sections treating the following topics:
1. Differential calculus modulo \(p^n\);
2. Cohomology operations modulo \(p^n\);
3. Passage to formal schemes;
4. The characteristic variety and holonomy.
In the course of the entire exposition, the author explains how some fundamental classical results on sheaves of differential operators and their associated sheaves of (coherent) modules can be generalized to the case of characteristic \(p\), and how the construction of different completions can be carried out by using deeper results on formal \(p\)-adic sciemes. Also, the author describes several results and conjectures concerning the concept of holonomy on characteristic \(p> 0\) with regard to \({\mathcal D}\)-modules with Frobenius action.
Altogether, the author has set high value on comprehensive and lucid explanations of the whole matter which, in the abstract, is both conceptually and technically utmost subtle, entangled and advanced. The analogy to the classical prototype on characteristic \(0\) as far as extant, is thoroughly discussed wherever it is appropriate, thereby enforcing the strategical character of this brilliant survey. Without any doubt, this extensive treatise is a perfect introduction to the very recent developments on the arithmetical theory of \({\mathcal D}\)-modules as a whole. divided powers; isocrystal; overconvergence; perfect complex; cohomological operation; de Rham cohomology; crystalline cohomology; rigid cohomology; characteristic variety Berthelot, Pierre, Introduction à la théorie arithmétique des \(\mathcal{D}\)-modules, Cohomologies \(p\)-adiques et applications arithmétiques, II, Astérisque, 279, 1-80, (2002) Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, \(p\)-adic cohomology, crystalline cohomology, de Rham cohomology and algebraic geometry, Rigid analytic geometry, Rings of differential operators (associative algebraic aspects), Sheaves of differential operators and their modules, \(D\)-modules, Commutative rings of differential operators and their modules, Formal methods and deformations in algebraic geometry Introduction to the arithmetic theory of \(\mathcal D\)-modules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let for \(i=1,\dots, m,\) \(Q_i\) be real symmetric \(n\times n\) matrices, \(q_i(x)=\langle Q_i x,x\rangle=x^{\mathsf{t}} Q_i x\) be the associated quadratic forms and \(\alpha_i\) be real numbers. The authors establish sufficient conditions in order that the system of \(m\) real quadratic equations \(q_i(x)=\alpha_i,\) has a real solution. By solving for \(X\) the equation \(\text{trace }Q_iX=\alpha_i\) subject to \(X\succeq 0\) -- this can be done rapidly by semidefinte programming -- it is elementary to show that it is sufficient to establish sufficient conditions for solvability of systems of equations of the form \(q_i(x)=\text{trace }Q_i.\) It is not hard to see that such a criterion should not so much depend on the \(Q_i\) per se but rather on the space only which the \(Q_i\) (supposed to be linearly independent) generate; and that the sum of the squares of the elements of an orthonormal basis with respect to the Frobenius inner product of that space is independent of the choice of the basis. The main theorem is formulated accordingly:
Theorem 1.1: Assuming the above notations and supposing that for some (equivalently, for any) orthonormal basis \(A_1,\dots,A_m\) of the subspace \(\text{span}(Q_1,\dots, Q_m)\) there holds the operator norm inequality \(\|\sum_{i=1}^m A_i^2 \|_{\mathrm{op}}\leq \frac{10^{-6}}{m},\) the system of quadratic equations \(q_i(x)=\text{trace }Q_i,\) \(i=1,\dots, m\) has a solution \(x\in \mathbb{R}^n.\)
(The real \(10^{-6}\) can probably be replaced by a better value but this problem is not investigated.) The condition of Theorem 1.1 can be checked in polynomial time and is satisfied for example for random \(q_i,\) provided \(m\leq \gamma \sqrt{n},\) for some absolute constant \(\gamma.\) The authors also give sufficient conditions, similar to the above, and with similar proof for the nontrivial solvability of homogeneous systems \(q_i(x)=0,\) \(i=1,\dots,m\) of quadratic equations, so we concentrate on giving details for the proof of T1.1. Theorem 1.1 is deduced from the following theorem where \(t\) is short for \((\tau_1,\dots,\tau_m),\) \(Q(t)= I- \mathbf i\sum_{i=1}^m \tau_i Q_i,\) and \(\det^{-1/2} Q(t)\) is a branch that has value 1 at \(t=0.\) (\(\mathbf i =\sqrt{-1}\))
Theorem 2.1: Let \(\alpha_1,\dots,\alpha_m\) be real numbers and \(q_i(x)=\frac{1}{2}\langle Q_i x, x\rangle \) quadratic forms. Provided that \( \int_{\mathbb R^m} \det^{-1/2} Q(t) \exp(-\mathbf i\sum_{i=1}^m \alpha_i \tau_i) dt\) is nonzero and absolutely convergent, the system \(q_i(x)=\alpha_i\) has a solution.
The connection between the hypothesis and the conclusion of this theorem can be glimpsed at via Lemma 3.1 which says that for \(\sigma>0,\)
\(\sigma^m \int_{\mathbb R^m} \exp(-\frac{\sigma^2}{2} \sum_{i=1}^m (q_i(x)-\alpha_i)^2) e^{-\frac{\|x\|^2}{2}} dx = (2\pi)^{\frac{n-m}{2}} \int_{\mathbb R^m} \det^{-1/2} Q(t) \exp(-\mathbf i\sum_{i=1}^m \alpha_i \tau_i) e^{-\frac{\|t\|^2}{2\sigma^2}} dt.\)
The hypothesis of Theorem 2.1 tells us that as \(\sigma \rightarrow \infty\) the right hand side goes to a value \(\neq 0.\) Supposing the system \(q_i(x)=\alpha_i\) has no solution it is shown that the left hand side goes to 0. So the proof is by contradiction and thus we have the interesting case of a paper which proves that the existence of a solution can (in favorable circumstances) be rapidly established while it gives no clue of where to find the solution (rapidly).
For the deduction of Theorem 1.1 from 2.1 a translation of Theorem 2.1 to polar coordinates is made: Let \(w=(w_1,\dots,w_m)\in \mathbb S^{m-1}\) and define \(A(w)=\sum_{i=1}^m w_i A_i,\) where \(A_1,.\dots,A_m\) is an orthonormal basis of \(\text{span}(Q_1,\dots, Q_m)\) and denote by \(\lambda_j(w)\) the eigenvalues of of \(A(w)\) for \(w\in \mathbb S^{m-1}.\) For getting to Theorem 1.1 it is sufficient to show that \(\int_{\mathbb S^{m-1}} ( \int_0^\infty \tau^{m-1} \det^{-1/2}(I-\mathbf i\tau A(w)) \exp(-\frac{\mathbf i\tau}{2} \text{trace} A(w)) d\tau )dw\neq 0. \) This is done by splitting the inner integral into two parts as \(\int_{5\sqrt{m}}^\infty \dots\) and \(\int_0^{5\sqrt{m}} \dots\) and showing that the contribution of the first part is negligible, while the contribution of the second part follows from eigenvalue estimates. The possibility of such an approach comes from noting that \(\det^{-1/2}(I-\mathbf i\tau A(w)) \exp(-\frac{\mathbf i\tau}{2} \text{trace} A(w))= \exp ( \frac{1}{2}\sum_{k=2}^\infty \frac{(\tau \mathbf i )^k}{k} \sum_{j=1}^n \lambda_j^k(w)).\) For carrying out this program, in Section 4 many eigenvalue estimates are given. As an example we mention the expectation inequality \(\mathbb E(\sum_{j=1}^n \lambda_j^3(w))^2 \leq \frac{120}{(m+2)(m+4)}\|\sum_{i=1}^m A_i^2\|_{\mathrm{op}}.\) Relationships between the operator norm, the Hilbert Schmidt or Frobenius norm and the Schatten 4 norm are also presented. In Section 5 a number of integrals are estimated, and in the final Section 6 the non-zeroness of above integral is obtained by applying probabilistic language, viewing the integral above essentially as an expectation value, and using the Markov inequality and a concentration inequality adapted for the Schatten 4 norm.
In Section 2 an outline of the proof of Theorem 1.1 is given, facilitating thus following the rather involved ideas, but in Section 6 the reviewer thinks to have spotted a number of mathematically significant typos. For example in both equations (6.2) and (6.5) and after (6.7) the factor \(\tau^{m-1}\) in the inner integrals seems to be missing and the symbol \(\sum_{i=1}^m\) should be cancelled. In (6.7) \(A\) should be replaced by \(A(w).\) Well, refereeing never was nor ever will be a glamorous activity \ldots\ in particular as long as it remains anonymous. For the `well-known' facts used in Lemma 3.1 the reviewer relegates the reader e.g. to Chapter 2 of \textit{E. F. Beckenbach} and \textit{R. Bellman} [Inequalities. Berlin-Göttingen-Heidelberg: Springer-Verlag (1961; Zbl 0097.26502)] and to \textit{M. J. Lighthill} [Introduction to Fourier analysis and generalised functions. Cambridge: At the University Press (1958; Zbl 0078.11203)] or \textit{S. Lang} [Real and functional analysis. 3. ed. New York: Springer-Verlag (1993; Zbl 0831.46001)], respectively. quadratic equations; positive semidefinite relaxation; Fourier analysis; eigenvalue estimates; algorithms; expectation; concentration inequalities Real algebraic sets, Computational real algebraic geometry, Semidefinite programming When a system of real quadratic equations has a solution | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\{S_\lambda\}\) be a directed inverse system of schemes with affine transition maps, so that the limit scheme \(S := \varprojlim_\lambda S_\lambda\) exists. For each \(\lambda\), let \(G_\lambda\) be an \(S_\lambda\)-group scheme satisfying \(G_{\lambda'} = G_\lambda \times_{S_\lambda} S_{\lambda'}\) for every transition map \(S_{\lambda'} \to S_{\lambda}\). Let \(G\) denote the \(S\)-group scheme \(\varprojlim_\lambda G_\lambda\). When the \(G_\lambda\)'s are commutative, one may form the étale cohomology groups \(H^q(S_\lambda,G_\lambda)\) for each integer \(q \geq 0\), and one obtains a canonical arrow
\[
\varinjlim_\lambda H^q(S_\lambda,G_\lambda) \to H^q(S,G).
\]
Assuming a certain finiteness hypothesis on the \(G_\lambda\)'s, \textit{A. Grothendieck} [Sem. Geom. algebrique Bois-Marie 1963/64, SGA 4, No.7, Lect. Notes Math. 270, 341--365 (1972; Zbl 0255.14009)] showed that the displayed arrow is an isomorphism; and he remarked without proof that the same result holds for \(H^1\) for possibly noncommutative groups. The present paper gives a detailed proof of Grothendieck's remark under a few mild finiteness hypotheses on the \(S_\lambda\)'s. limit of schemes; non-abelian cohomology Benedictus Margaux, Passage to the limit in non-abelian Čech cohomology, J. Lie Theory 17 (2007), no. 3, 591-596. Étale and other Grothendieck topologies and (co)homologies, Group schemes, Infinite-dimensional Lie (super)algebras, Generalizations (algebraic spaces, stacks) Passage to the limit in non-abelian Čech 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 Consider the space \(M_n^{\mathrm{nor}}\) of square normal matrices \(X = (x_{i j})\) over \(\mathbb{R} \cup \{- \infty \}\), i.e., \(- \infty \leq x_{i j} \leq 0\) and \(x_{i i} = 0\). Endow \(M_n^{\mathrm{nor}}\) with the tropical sum \(\oplus\) and multiplication \(\odot\). Fix a real matrix \(A \in M_n^{\mathrm{nor}}\) and consider the set \(\Omega(A)\) of matrices in \(M_n^{\mathrm{nor}}\) which commute with \(A\). We prove that \(\Omega(A)\) is a finite union of alcoved polytopes; in particular, \(\Omega(A)\) is a finite union of convex sets. The set \(\Omega^A(A)\) of \(X\) such that \(A \odot X = X \odot A = A\) is also a finite union of alcoved polytopes. The same is true for the set \(\Omega'(A)\) of \(X\) such that \(A \odot X = X \odot A = X\).{
}A topology is given to \(M_n^{\mathrm{nor}}\). Then, the set \(\Omega^A(A)\) is a neighborhood of the identity matrix \(I\). If \(A\) is strictly normal, then \(\Omega'(A)\) is a neighborhood of the zero matrix. In one case, \(\Omega(A)\) is a neighborhood of \(A\). We give an upper bound for the dimension of \(\Omega'(A)\). We explore the relationship between the polyhedral complexes \(\operatorname{span}A\), \(\operatorname{span}X\) and \(\operatorname{span}(A X)\), when \(A\) and \(X\) commute. Two matrices, denoted \(\underline{A}\) and \(\overline{A}\), arise from \(A\), in connection with \(\Omega(A)\). The geometric meaning of them is given in detail, for one example. We produce examples of matrices which commute, in any dimension. tropical algebra; commuting matrices; normal matrix; idempotent matrix; alcoved polytope; convexity Linde, J.; de la Puente, M. J., Matrices commuting with a given normal tropical matrix, Linear Algebra Appl., 482, 101-121, (2015) Max-plus and related algebras, , Hermitian, skew-Hermitian, and related matrices Matrices commuting with a given normal tropical matrix | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(A\) be a commutative ring and \(A[x]\) the polynomial ring in one variable over \(A\). A ring homomorphism \(\sigma:A\rightarrow A[x]\) is called an exponential map on \(A\) if whenever \(a\in A\) and \(\sigma(a) = \sum_{i=0}^{m}a_{i}x^{i}\) (\(a_{i}\in A\) for \(i=0,\dots, m\)), one has \(a_{0} = a\) and \(\sum_{i=0}^{m}\sigma(a_{i})y^{i} = \sum_{i=0}^{m}a_{i}(x+y)^{i}\) in \(A[x, y]\). (Note that for every such an exponential map \(\sigma\), one can consider a collection \((\delta_{i})_{i=0}^{\infty}\) of endomorphisms of the additive group of \(A\) defined by \(\sigma(a) = \sum_{i\geq 0}\delta_{i}(a)x^{i}\) for every \(a\in A\). The family \((\delta_{i})_{i=0}^{\infty}\) is called a locally finite iterative higher derivation on \(A\); this concept is naturally equivalent to the concept of an exponential map.)
In [Osaka J. Math. 15, 655--662 (1978; Zbl 0393.13007)] \textit{Y. Nakai} proved the following structure theorem: Let \(k\) be an algebraically closed field, \(A\) a \(k\)-domain, and \(\sigma\) a nontrivial exponential map on \(A\) over \(k\). If \(A^{\sigma} = \{a\in A|\sigma(a) = a\}\) is a finitely generated PID over \(k\) and every prime element of \(A^{\sigma}\) is a prime element of \(A\), then \(A\) is the polynomial ring in one variable over \(A^{\sigma}\).
The main result of the paper under review generalizes the Nakai's theorem by removing the conditions that the ground field \(k\) is algebraically closed, that \(A\) is an integral domain, and that the \(k\)-algebra \(A^{\sigma}\) is finitely generated. As one of the consequences of this generalization of Nakai's theorem, the author obtains the following result that generalizes the cancellation theorem of \textit{A. J. Crachiola} [J. Pure Appl. Algebra 213, No. 9, 1735--1738 (2009; Zbl 1168.14041)]: Let \(k\) be a field, \(\overline{k}\) its algebraic closure, and \(A\) and \(A'\) finitely generated \(k\)-domains with \(A[x]\simeq_{k}A'[x]\). If \(A\) and \(\overline{k}\bigotimes_{k}A\) are UFDs and trans.\(\deg_{k}A = 2\), then \(A\simeq_{k}A'\). S. Kuroda, A generalization of Nakai's theorem on locally finite iterative higher derivations, Osaka J. Math. 54 (2017), 335--341. Derivations and commutative rings, Actions of groups on commutative rings; invariant theory, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) A generalization of Nakai's theorem on locally finite iterative higher derivations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main result of the paper is the construction of a sort of global version of a Néron model of the Jacobian of a family of curves. To be more precise, let \(S\) denote the spectrum of a noetherian normal domain, complete and separated with respect to an ideal, and let \(C\to S\) be a semistable curve with some additional properties which are omitted here.
Let \(V\) denote the spectrum of a discrete valuation ring \(R\) with generic point \(\eta\). If \(V\to S\) is a morphism such that \(C_V\) is a regular scheme with \(C_\eta\) smooth and some additional property on the closed fibre, it is well known that there exists a Néron model of \(\text{Pic}^0(C_\eta|k(\eta))\) over \(V\). The question is, does there exist a scheme over \(S\) whose base change via any morphism \(V\to S\) is the Néron model of \(\text{Pic}^0(C_\eta|k(\eta))\)? The main result of the paper is an affirmative answer to this question. The idea is to construct suitable compactifications of \(\text{Pic}^0(C|S)\) such that their smooth locus has the wanted property? The main technique is the so called Mumford uniformization. F. Andreatta, On Mumford's uniformization and Néron models of Jacobians of semistable curves over complete rings, in Moduli of Abelian Varieties, Progress in Mathematics, Vol. 195, 2001, Birkhäuser, Basel, pp. 11--126. Arithmetic ground fields for abelian varieties, Jacobians, Prym varieties, Formal methods and deformations in algebraic geometry, Abelian varieties of dimension \(> 1\) On Mumford's uniformization and Néron models of Jacobians of semistable curves over complete rings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\Gamma\) be a Fuchsian group acting on the hyperbolic upper half-plane \(\mathbb{H}\). A canonical method of constructing a function which is invariant under \(\Gamma\) is by taking averages of some ``nice'' function \(f\) over a quotient \(\Gamma'\backslash\Gamma\) where \(\Gamma'\) is a subgroup of \(\Gamma\) (of finite or infinite index). By modifying the average (i.e. using the so-called slash action) one can also obtain functions with more general transformation properties (i.e. non-zero weight and/or non-trivial multiplier system). These types of averages are usually called \textit{Poincaré series} and the simplest examples of such functions are the \textit{Eisenstein series} (holomorphic or non-holomorphic).
Non-holomorphic (parabolic) Eisenstein series occur naturally in the spectral theory of the hyperbolic Laplacian \(\Delta=-y^{2}\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right)\) on the quotient orbifold \(\mathcal{M}=\Gamma\backslash\mathbb{H}\) (recall that an orbifold is essentially a Riemannian manifold where ``corners'' and cusps are also allowed). If \(\Gamma\) has parabolic elements (i.e. \(\mathcal{M}\) has cusps) then it has a continuous spectrum with multiplicity equal to the number of inequivalent cusps. The eigenpacket corresponding to scattering from a cusp \(p\) in \(\Gamma\) is given by the Eisenstein series with respect to \(\Gamma_{p},\) the stabilizer of \(p\) in \(\Gamma.\) For example, if \(p\) is the cusp at \(\infty\), the corresponding Eisenstein series is
\[
E_{\infty}\left(z;s\right)=\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma}\Im\left(\gamma z\right)^{s}
\]
where \(z\in\mathbb{H}\) and \(\Re s>1\). The function \(E_{\infty}\) has a meromorphic continuation, satisfies a functional equation (in terms of the vector of Eisenstein series with respect to all the cusps of \(\Gamma\)) and satisfies \(\Delta E_{\infty}\left(z;s\right)=s\left(1-s\right)E_{\infty}\left(z;s\right)\).
Other than parabolic subgroups of \(\Gamma\) there are two other types of possible (non-trivial) subgroups generated by stabilizers of points in the upper half-plane or the real line: elliptic and hyperbolic. If \(e\in\mathbb{H}\) is the fixed point of an element of \(\Gamma\) then this is said to be an elliptic fixed-point and the stabilizer \(\Gamma_{e}\) is necessarily a finite cyclic group. Similarly, if \(x,y\in\mathbb{R}\) are two different fixed points of some \(\gamma_{h}\in\Gamma\) then the geodesic in \(\mathbb{H}\) with endpoints in \(x\) and \(y\) is fixed under the infinite cyclic subgroup \(\Gamma_{h}\) generated by \(\gamma_{h}\).
One can now construct elliptic and hyperbolic Eisenstein series in much the same manner as the parabolic Eisenstein series above. The sum over \(\Gamma_{p}\backslash\Gamma\) will instead be taken over \(\Gamma_{e}\backslash\Gamma\) and \(\Gamma_{h}\backslash\Gamma\). Similarly, the imaginary part, \(z\mapsto\Im z\), which is closely related to the hyperbolic distance function in cartesian coordinates, is replaced by the corresponding function under a suitable change of coordinates.
Let \(\left\{ \mathcal{M}_{n}=\Gamma_{n}\backslash\mathbb{H}\right\} \) be an elliptically degenerating family of orbifolds (of finite volume) with limit surface \(\mathcal{M}_{\infty}=\Gamma_{\infty}\backslash\mathbb{H}\). By elliptical degeneration we mean that a ``corner'' is degenerating into a cusp.
The main result in the paper under review is that the Eisenstein series are ``well-behaved'' under this kind of degeneration. To be precise: all parabolic, hyperbolic and non-degenerating elliptical Eisenstein series \(E_{*}\) on \(\mathcal{M}_{n}\) converge to the corresponding Eisenstein series on \(\mathcal{M}_{\infty}\). Furthermore, the elliptic Eisenstein series of a degenerating elliptic element converges to the parabolic Eisenstein series on \(\mathcal{M}_{\infty}\) associated to the newly formed parabolic point. In all cases the convergence is uniform on compact subsets bounded away from the degenerating points and half-planes \(\Re s\geq1+\delta\).
The method of proof is to first construct certain counting functions and rewrite the Eisenstein series in terms of Stieltjes integrals with respect to the corresponding counting measures. The convergence of the Eisenstein series then essentially follow from convergence of the counting functions.
Other than elliptic, one can also consider a family of hyperbolically degenerating orbifolds, i.e. ``pinching'' along a geodesic until a cusp is formed. The authors remark that the corresponding elliptical Eisenstein converge in this situation as well (the convergence of parabolic and hyperbolic Eisenstein series in this setting was proved earlier by \textit{D. Garbin, J. Jorgenson} and \textit{M. Munn} [Comment. Math. Helv. 83, No. 4, 701--721 (2008; Zbl 1154.30032)]). Eisenstein series; Fuchsian groups; degenerating Riemann surfaces Garbin, D.; Von Pippich, A. -M.: On the behavior of Eisenstein series through elliptic degeneration. Comm. math. Phys. 292, 511-528 (2009) Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas), Spectral theory; trace formulas (e.g., that of Selberg), Moduli, classification: analytic theory; relations with modular forms On the behavior of Eisenstein series through elliptic degeneration | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article is a merely technical appendix to the author's foregoing treatise in the same volume
[Ann. Inst. Fourier 54, No. 3, 499--630 (2004; Zbl 1062.14014)]. Its purpose is to give, in full detail, some results used there concerning geometric quotients, Zariski regularity, and \(S\)-stability. These results are important in their own right, as they have much wider applications, and they are discussed here for the sake of generality, completeness, coherence, and reference.
The first part of this appendix gives a simplified and detailed proof of the existence of geometric quotients of compact connected normal complex spaces modulo certain analytic equivalence relations.
This existence theorem was proved by the author himself more than 20 years ago, and that in a special case [Invent. Math. 63, 187--223 (1981; Zbl 0436.32024)]. Now he offers a general proof of the existence of such geometric quotients, which actually can be reduced to the earlier proof in the special case. This requires some subtle technical work of remarkable length, all of which is carried out in full detail and rigor. Basically, the author follows closely his original approach, but essentially simplifying it in one crucial step.
Part 2 introduces the notion of Zariski regularity for subsets of arbitrary analytic spaces. This is a weak notion of countable constructibility which is useful in the study of certain fibres of holomorphic maps between complex spaces. Also, the notion of Zariski regularity is shown to have important applications to the construction of meromorphic quotients.
Part 3 deals with a stability condition for Zariski-regular subsets of the Chow scheme of a compact connected normal complex space \(X\), again with applications to the study of fibrations. This stability condition was already used by the author in an earlier paper on connectedness properties of compact Kähler manifolds [Contemp. Math. 241, 85--96 (1999; Zbl 0965.32021)]. Kähler manifolds; fibrations; complex analytic spaces; meromorphic quotients; Zariski regularity; stability Campana, F.: Orbifolds, special varieties and classification theory: an appendix. Ann. Inst. Fourier (Grenoble) \textbf{54}(3), 631-665 (2004) Transcendental methods, Hodge theory (algebro-geometric aspects), Rational and birational maps, \(n\)-folds (\(n>4\)), Compact Kähler manifolds: generalizations, classification, Kähler manifolds, Hyperbolic and Kobayashi hyperbolic manifolds, Classification theorems for complex manifolds Orbifolds, special varieties and classification theory: appendix. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 connected separated scheme of finite type over \({\mathbb F}_{p}.\) Let \(F: X\rightarrow X\) be the absolute Frobenius map. N. Katz proved that there exists equivalence of categories between: {\parindent=6mm \begin{itemize}\item[(1)] the category \((LS_{{\mathbb F}_{p}})/{X_{\mathrm{et}}})\) of smooth étale \({\mathbb F}_{p}\)-sheaves on \(X\) of finite rank \item[(2)] the category \((UR_{{\mathbb F}_{p}})/{X})\) of pairs \(({\mathcal F}, {\phi})\) consisting of {\parindent=12mm \begin{itemize}\item[(i)] a locally free \({\mathcal O}_{X}\)-module \({\mathcal F}\) of finite rank \item[(ii)] an isomorphism \(\phi : F^{*}{\mathcal F} \rightarrow {\mathcal F}\) of locally free \({\mathcal O}_{X}\)-modules.
\end{itemize}}\end{itemize}} The full subcategory of unipotent étale \({\mathbb F}_{p}\)-local systems is denoted as \((NLS_{{\mathbb F}_{p}})/{X_{\mathrm{et}}})\) and the corresponding subcategory of \((UR_{{\mathbb F}_{p}})/{X})\) as \((NUR_{{\mathbb F}_{p}})/{X}).\) Assume that there exists an \({\mathbb F}_{p}\)- valued point \(x\) in \( X.\) Taking a geometric point over \(x\) one gets the fundamental group of the category \((NLS_{{\mathbb F}_{p}})/{X_{\mathrm{et}}})\) with respect to the base point \(\bar x\). This is called the \({\mathbb F}_{p}\)-completion \({\pi}_{1}(X, {\bar x})^{{\mathbb F}_{p}}\) if the étale fundamental group of \(X.\) The topological dual \({\mathbb F}_{p}[[{\pi}_{1}(X, {\bar x})]]^{*}\) of the dual of the completion of \({\mathbb F}_{p}[{\pi}_{1}(X, {\bar x})]\) with respect to the augmentation ideal \(I\) has a structure of a Hopf algebra. In the paper the author shows that the Hopf algebra \({\mathbb F}_{p}[[{\pi}_{1}(X, {\bar x})]]^{*}\) is isomorphic to the cohomology of the bar complex of the Artin-Schreier DGA of \(X.\) If the DGA \(A^{\bullet}\) is graded commutative then the corresponding bar complex \(B(A^{\bullet}, {\epsilon})\) has a multiplication defined by the shuffle product. The author shows that Artin-Schreier DGA is commutative up to homotopy and defines a shuffle product up to higher homotopy. Hopf algebra; Artin-Schreier DGA \(F_{p}\)-completion Finite ground fields in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies The Artin-Schreier DGA and the \(F_p\)-fundamental group of an \(F_p\) scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors of this paper concerns about the problem of computing the Lebesgue volume of compact basic semialgebraic sets, using the Moment-SOS methodology. This involves solving an infinite-dimensional linear program (LP) and obtaining the volume by taking the limit of a sequence of solutions as it converges.
However, the convergence of the sequence can be slow due to the Gibbs phenomenon, which can impede the accuracy of the approximations. This issue can be resolved by introducing additional linear moment constraints obtained from an application of Stokes' theorem for integration on the set, which greatly improves convergence.
While this approach has shown significant promise, the rationale behind its efficacy was unclear so far. The authors provide a refined version of the LP formulation, demonstrating that when the set is a smooth super-level set of a single polynomial, the dual of the refined LP has an optimal solution that is a continuous function. As a result, the dual approximates a continuous function with a polynomial, eliminating the Gibbs phenomenon and further accelerating convergence.
The authors utilize recent results on Poisson's partial differential equation (PDE) in their proof of this technique. Overall, this paper presents a valuable contribution to the field of computational mathematics, providing a deeper understanding of the effective computation of Lebesgue volumes of semialgebraic sets. numerical methods for multivariate integration; real algebraic geometry; convex optimization; Stokes' theorem; Gibbs phenomenon Semialgebraic sets and related spaces, Semidefinite programming, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Effectivity, complexity and computational aspects of algebraic geometry, Length, area, volume, other geometric measure theory, Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation, Integral geometry, Numerical integration, Approximation methods and heuristics in mathematical programming Stokes, Gibbs, and volume computation of semi-algebraic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper develops a ``localized intersection theory'' of the following form. Let \(S\) be the spectrum of an unequal-characteristic complete discrete valuation ring with algebraically closed residue field. Let \(s\) and \(\eta\) denote the closed and generic points of \(S\), respectively. In what follows all schemes are assumed to be separated and of finite type over \(S\). Let \(i\:X\to Y\) be a closed immersion of \(S\)-schemes such that the generic fiber \(X_\eta\) is nonempty and such that the conormal sheaf \(\mathcal N_X Y\) has finite homological dimension over \(X\) and is locally free of rank \(d\) over \(X_\eta\). Let \(V\) be a purely \(k\)-dimensional scheme, let \(f\:V\to Y\) be a morphism, and let \(W=X\times_Y V\). Then the author defines a ``localized intersection product'' \((X\mathbin.V)_{\text{loc}}\in A_{k-d-1}(W_s)\). This is used to define a localized Gysin map \(i^!_{\text{loc}}\:Z_k(Y')\to A_{k-d-1}(X\times_Y Y')\) for any \(Y'\to Y\). This Gysin map is compatible with proper push-forward and flat pull-back, and it satisfies an excess formula.
The special case in which the above map \(i\) is the diagonal closed immersion \(i\:\Delta_X\to X\times_S X\) for an arithmetic surface \(X\) is of special interest. If \(\sigma\) is an \(S\)-automorphism of \(X\) and \(\Gamma\) is its graph, then a Lefschetz fixed point formula is proved, giving the degree of the zero-cycle \((\Delta_X\mathbin.\Gamma)_{\text{loc}}\in A_0(X_s)\). This implies that the localized Euler characteristic, i.e., the degree of \(c_{2,X_s}^X(\Omega^1_{X/S})\cap[X]=(\Delta_X\mathbin.\Delta_X)_{\text{loc}}\), is minus the Artin conductor. This fact was conjectured by \textit{S. Bloch} [in: Algebraic geometry, Proc. Summer Res. Inst. Brunswick 1985, part 2, Proc. Symp. Pure Math. 46, 421-450 (1987; Zbl 0654.14004)]. Also, the Lefschetz fixed point formula was conjectured by \textit{K. Kato, S. Saito}, and \textit{T. Saito} [Am. J. Math. 110, 49-75 (1988; Zbl 0673.14020)], and proved by them in the geometric (equal characteristic) case.
Finally, the paper gives an application of the Lefschetz formula to a conjecture of Serre on the existence of Artin representations for two-dimensional regular local rings in the unequal characteristic case. arithmetic surface; localized intersection theory; bivariant class; Lefschetz fixed point formula; Artin representation; Swan conductor; localized Gysin map A. Abbes, Cycles on arithmetic surfaces, Compos. Math., 122 (2000), 23--111. Arithmetic varieties and schemes; Arakelov theory; heights, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Algebraic cycles, Étale and other Grothendieck topologies and (co)homologies Cycles on arithmetic 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 We show that, under suitable assumptions, the Hecke eigensystems arising from \(\pmod p\) modular forms of PEL type associated to an algebraic group \(G/\mathbb Q\) of type A or C, coincide with the Hecke eigensystems arising from \(\pmod p\) algebraic modular forms associated to an inner form \(I\) of \(G\). The form \(I\) has to be compact modulo center at infinity and satisfies: \(I_{\mathbb Q_v} = G_{\mathbb Q_v}\) for \(v\neq p,\infty\). In order to realize the Hecke correspondence, we first adapt a result of \textit{M. Rapoport} and \textit{Th. Zink} [Period spaces for \(p\)-divisible groups. Princeton, NJ: Princeton Univ. Press (1996; Zbl 0873.14039)] to obtain an explicit uniformization of the superspecial locus of PEL Shimura varieties over \(\overline{\mathbb F}_p\) (such a uniformization in the general context of PEL Shimura varieties was first described by \textit{C.-F. Yu} [Forum Math. 22, No. 3, 565--582 (2010; Zbl 1257.11059)]). We then compare (\(mod\)\ \(p\)) modular forms with superspecial modular forms by exploiting the Hermitian \(\mathbb F_{p^2}\)-structure of the cotangent spaces of the \(p\)-divisible groups of the superspecial points. We study in detail the case in which \(G\) is of type A, and we prove that the number \(\mathcal N\) of \(\pmod p\) Hecke eigensystems arising from unitary modular forms for a fixed quadratic imaginary field, having signature \((r,s)\), prime-to-\(p\) level \(N\geq 3\) and variable weight satisfies the estimate \(\mathcal N \in O(p^{g^2+g+1-rs})\) for \(p\) large, where \(g=r+s\). The paper generalizes results of \textit{J.-P. Serre} [Isr. J. Math. 95, 281--299 (1996; Zbl 0870.11030)] for GL\((2)\) and \textit{A. Ghitza} [Int. Math. Res. Not. 2004, No. 55, 2983--2987 (2004; Zbl 1084.11021); CIJ. Number Theory 106, No. 2, 345--384 (2004; Zbl 1121.11040)] for GSp\((2g)\) \((g>1)\). Hecke eigensystems; Hecke correspondence; PEL Shimura varieties; superspecial points D. A.REDUZZI,\textit{Hecke eigensystems for (mod}\textit{p}\textit{) modular forms of PEL type and algebraic modular} \textit{forms}, Int. Math. Res. Not. IMRN 9 (2013), 2133--2178. http://dx.doi.org/10.1093/imrn/rns114.MR3053416 Hecke-Petersson operators, differential operators (one variable), Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties Hecke eigensystems for \(\pmod p\) modular forms of PEL type and algebraic modular forms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Each isolated critical point of a holomorphic function has a spectrum which is a finite set of rational numbers [see \textit{J. H. M. Steenbrink}, in ''Real and compl. Singul.'', Proc. Nordic Summer Sch., Sympos. Math., Oslo 1976, 525-536 (1977; Zbl 0373.14007)]. The spectrum is closely related with most of the other characteristics of a critical point. An interesting problem is to explain how the spectrum varies under decomposition of a critical point into simpler critical points. According to a conjecture of V. I. Arnol'd the spectrum is lower semicontinuous under decomposition of a critical point [see \textit{V. I. Arnol'd} in ''Geometry and analysis'', Pap. dedic. Mem. V. K. Patodi, 1-9 (1981; Zbl 0492.58006)]. This article is a first step to a proof of the Arnol'd conjecture. In the article a new notion of a semicontinuity subset of a real line is introduced and a problem of a description of semicontinuity subsets is posed. With the help of this notion Arnol'd's conjecture is generalized and formulated in the form: any semiline of a real line is a semicontinuity set. It is proven that for any irrational number \(\alpha\) the union of intervals \(\cup_{k\in {\mathbb{Z}}}(\alpha +2k,\alpha +2k+1)\) is a semicontinuity set. As a consequence one gets the validity of Arnol'd's conjecture for decompositions of simple critical points and unimodular critical points of a series T. The proofs are based on a theory of mixed Hodge structures in vanishing cohomologies. Further development of ideas of this article see in the author's papers in Sov. Math., Dokl. 27, 735-740 (1983), Funkts. Anal. Prilozh. 17, No.4, 77-78 (1983) and Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 22, 130-166 (1983) and \textit{J. Steenbrink} [''Semicontinuity of the singularity spectrum'', Prepr. 23, Math. Inst., Univ. Leiden, 1-9 (1983)]. In particular in this preprint Steenbrink proved Arnol'd's conjecture. lower semicontinuous spectrum; spectrum of critical point; decomposition of a critical point; Arnol'd conjecture; mixed Hodge structures in vanishing cohomologies --, The spectrum and decompositions of a critical point of a function.Sov. Math. Dokl. 27(1983), 575--579. Singularities in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), Local complex singularities, Hodge theory in global analysis The spectrum and decompositions of a critical point of a function | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors define the intersection multiplicity for locally analytic subsets (of pure dimension) of a complex analytic manifold at an isolated point of the intersection. The approach is analytic; using as main tools the intersection multiplicity theory for the proper isolated intersections [\textit{R. Draper}, Math. Ann. 180, 175--204 (1969; Zbl 0157.40502)] and the diagonal construction [\textit{P. Samuel}, J. Math. Pures Appl. (9) 30, 159--274 (1951; Zbl 0044.02701) and \textit{A. Weil}, Foundations of algebraic geometry. New York: AMS (1946; Zbl 0063.08198); rev. ed. (1962; Zbl 0168.18701)] the authors extend the definition to the improper case. They also compare this definition with the algebraic approach by Samuel's multiplicity.
First they define the intersection multiplicity between a locally analytic subset \(X\) of pure dimension \(k\) and a submanifold \(N\) (both in an analytic manifold \(M\) of dimension \(m)\) at the isolated point \(a\), \(\tilde i (X \cdot N,a)\), as the infimum of the intersection multiplicities of \(X\) with locally analytic subsets \(V\) of dimension \(m - k\) containing the germ of \(N\) at \(a\) and with \(a\) isolated point of \(X \cap V\) \((V\) intersect properly \(X\) in \(a)\). Also the subsets \(V\) for which \(\tilde i (X \cdot N,a)\) is reached is characterized in terms of the relative tangent cone of \(X\) and \(N\) and the tangent spaces of the subsets involved. After that the intersection multiplicity between two locally analytic subsets of \(X\) and \(Y\) is defined as \(i(X \cdot Y,a) = \tilde i((X \times Y) \cdot \Delta, (a,a))\). Some properties like additivity on the components are proved. Also the formula \(i(X \cdot Y,a) \geq \deg_aX \deg_aY\) is given, characterizing the equality by the condition that the tangent cones have trivial intersection. At the end the comparison with the Samuel's theory of multiplicity is given, the intersection multiplicity above defined is equal to the multiplicity of the primary ideal of the diagonal \(\Delta\) in the local ring \({\mathcal O}_{x \times Y,(a,a)}\). improper intersection; intersection multiplicity; analytic subsets [1]R. Achilles, P. Tworzewski, and T. Winiarski, On improper isolated intersection in complex analytic geometry, Ann. Polon. Math. 51 (1990), 21--36. Analytic subsets and submanifolds, Complex surface and hypersurface singularities, Germs of analytic sets, local parametrization, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry On improper isolated intersection in complex analytic geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a quasi-algebraic variety defined over the reals by polynomial equalities \(f_1=\dots=f_s=0\) and inequalities. The authors describe a method to determine when a connected component of \(S\) contains smooth points of the complex variety \(V\) associated with \((f_1,\dots,f_s)\). The existence of smooth points of \(V\) in a component \(\Omega\subset S\) is relevant to decide if one can use the geometry of \(V\) in order to find some information on \(\Omega\), like its dimension, or the number of points of \(\Omega\) which lie in some model.
The method illustrated by the authors is based on the construction, from the Jacobian matrix of \(f_1,\dots,f_s\), of a polynomial \(g\) which vanishes on the singular locus of \(V\), but not on the whole \(V\). When \(V\) is irreducible, the extremal points of \(g\) provide a smooth point in every connected component of \(V\cap \mathbb R^n\). The authors adapt the method to the case where \(V\) has many components, possibly of different dimensions. The method is implemented by using homotopy continuation algorithms, and tested in several examples, compared with purely symbolic techniques. computational real algebraic geometry; real dimension; Kuramoto model Computational real algebraic geometry, Semialgebraic sets and related spaces Smooth points on semi-algebraic sets | 0 |
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