On a semilattice of numberings

We study some properties of a $ \mathfrak{c} $ \mathfrak{c} -universal semilattice $ \mathfrak{A} $ \mathfrak{A} with the cardinality of the continuum, i.e., of an upper semilattice of m-degrees. In particular, it is shown that the quotient semilattice of such a semilattice modulo any countable ideal will be also $ \mathfrak{c} $ \mathfrak{c} -universal. In addition, there exists an isomorphism $ \mathfrak{A} $ \mathfrak{A} such that $ {\mathfrak{A} \mathord{\left/ {\vphantom {\mathfrak{A} {\iota \left( \mathfrak{A} \right)}}} \right. \kern-\nulldelimiterspace} {\iota \left( \mathfrak{A} \right)}} $ {\mathfrak{A} \mathord{\left/ {\vphantom {\mathfrak{A} {\iota \left( \mathfrak{A} \right)}}} \right. \kern-\nulldelimiterspace} {\iota \left( \mathfrak{A} \right)}} will be also $ \mathfrak{c} $ \mathfrak{c} -universal. Furthermore, a property of the group of its automorphisms is obtained. To study properties of this semilattice, the technique and methods of admissible sets are used. More exactly, it is shown that the semilattice of mΣ-degrees $ L_{m\Sigma }^{\mathbb{H}\mathbb{F}\left( S \right)} $ L_{m\Sigma }^{\mathbb{H}\mathbb{F}\left( S \right)} on the hereditarily finite superstructure $ \mathbb{H}\mathbb{F} $ \mathbb{H}\mathbb{F} (S) over a countable set S will be a $ \mathfrak{c} $ \mathfrak{c} -universal semilattice with the cardinality of the continuum.     

Finite-to-finite universal quasivarieties are Q-universal

Let K be a quasivariety of algebraic systems of finite type. K is said to be universal if the category G of all directed graphs is isomorphic to a full subcategory of K. If an embedding of G may be effected by a functor F:G K which assigns a finite algebraic system to each finite graph, then K is said to be finite-to-finite universal. K is said to be Q-universal if, for any quasivariety M of finite type, L(M) is a homomorphic image of a sublattice of L(K), where L(M) and L(K) are the lattices of quasivarieties contained in M and K, respectively.?We establish a connection between these two, apparently unrelated, notions by showing that if K is finite-to-finite universal, then K is Q-universal. Using this connection a number of quasivarieties are shown to be Q-universal. Received February 8, 2000; accepted in final form December 23, 2000.     

Relative universality and universality obtained by adding constants

A variety ${\mathbb{V}}${\mathbb{V}} is var-relatively universal if it contains a subvariety \mathbbW{\mathbb{W}} such that the class of all homomorphisms that do not factorize through any algebra in \mathbbW{\mathbb{W}} is algebraically universal. And \mathbbV{\mathbb{V}} has an algebraically universal α-expansion a\mathbbV{\alpha\mathbb{V}} if adding α nullary operations to all algebras in \mathbbV{\mathbb{V}} gives rise to a class a\mathbbV{\alpha\mathbb{V}} of algebras that is algebraically universal. The first two authors have conjectured that any varrelative universal variety \mathbbV{\mathbb{V}} has an algebraically universal α-expansion a\mathbbV{\alpha\mathbb{V}} . This note contains a more general result that proves this conjecture.     

\mathcal{Q}-Universal Quasivarieties of Graphs

It is proved that a quasivariety K of undirected graphs without loops is -universal if and only if K contains some non-bipartite graph.     

A semilattice of numberings. II

\mathfrakc \mathfrak{c} -Universal semilattices \mathfrakA \mathfrak{A} of the power of the continuum (of an upper semilattice of m-degrees ) on admissible sets are studied. Moreover, it is shown that a semilattice of \mathbbH\mathbbF( \mathfrakM ) \mathbb{H}\mathbb{F}\left( \mathfrak{M} \right) -numberings of a finite set is \mathfrakc \mathfrak{c} -universal if \mathfrakM \mathfrak{M} is a countable model of a c-simple theory.     

Mass equidistribution of Hilbert modular eigenforms

Let \mathbbF\mathbb{F} be a totally real number field, and let f traverse a sequence of non-dihedral holomorphic eigencuspforms on \operatornameGL₂/\mathbbF\operatorname{GL}_{2}/\mathbb{F} of weight (k₁,?,k_{[\mathbbF:\mathbbQ]})(k_{1},\ldots,k_{[\mathbb{F}:\mathbb{Q}]}), trivial central character and full level. We show that the mass of f equidistributes on the Hilbert modular variety as max(k₁,?,k_{[\mathbbF:\mathbbQ]}) ? ￥\max(k_{1},\ldots,k_{[\mathbb{F}:\mathbb{Q}]}) \rightarrow \infty.     

Maximal Chains in Positive Subfamilies of P(ω)

A family P ì [w]^w{\mathcal P} \subset [\omega]^\omega is called positive iff it is the union of some infinite upper set in the Boolean algebra P(ω)/Fin. For example, if I ì P(w){\mathcal I} \subset P(\omega) is an ideal containing the ideal Fin of finite subsets of ω, then P(w) \IP(\omega) \setminus {\mathcal I} is a positive family and the set Dense(\mathbb Q)\mbox{Dense}({\mathbb Q}) of dense subsets of the rational line is a positive family which is not the complement of some ideal on P(\mathbb Q)P({\mathbb Q}). We prove that, for a positive family P{\mathcal P}, the order types of maximal chains in the complete lattice áP è{?}, ì ?\langle {\mathcal P} \cup \{\emptyset\}, \subset \rangle are exactly the order types of compact nowhere dense subsets of the real line having the minimum non-isolated. Also we compare this result with the corresponding results concerning maximal chains in the Boolean algebras P(ω) and Intalg[0,1)_{\mathbb R}\mbox{Intalg}[0,1)_{{\mathbb R}} and the poset E(\mathbb Q)E({\mathbb Q}), where E(\mathbb Q)E({\mathbb Q}) is the set of elementary submodels of the rational line.     

Minimal subsystems of affine dynamics on local fields

We describe the dynamics of an arbitrary affine dynamical system on a local field by exhibiting all its minimal subsystems. In the special case of the field \mathbbQ_p{\mathbb{Q}_p} of p-adic numbers, for any non-trivial affine dynamical system, we prove that the field \mathbbQ_p{\mathbb{Q}_p} is decomposed into a countable number of invariant balls or spheres each of which consists of a finite number of minimal subsets. Consequently, we give a complete classification of topological conjugacy for non-trivial affine dynamics on \mathbbQ_p{\mathbb{Q}_p} . For each given prime p, there is a finite number of conjugacy classes.     

Groups whose non-linear irreducible characters are rational valued

A finite group G all of whose nonlinear irreducible characters are rational is called a \mathbbQ₁{\mathbb{Q}_1}-group. In this paper, we obtain some results concerning the structure of \mathbbQ₁{\mathbb{Q}_1}-groups.     

Metric groups attached to skew-symmetric biextensions

Let G be a commutative, unipotent, perfect, connected group scheme over an algebraically closed field of characteristic p > 0 and let E be a biextension of G × G by the discrete group \mathbbQ_p/\mathbbZ_p\mathbb{Q}_{p}/\mathbb{Z}_{p}. When E is skew-symmetric, V. Drinfeld defined a certain metric group A associated to E (when G is the perfectization of the additive group \mathbbG_a\mathbb{G}_{a}, it is easy to compute this metric group, cf. Appendix A). In this paper we prove a conjecture due to Drinfeld about the class of the metric group A in the Witt group (cf. Appendix B).     

Unifying two results of Orlov on singularity categories

Let \mathbbX\mathbb{X} be a separated Noetherian scheme of finite Krull dimension which has enough locally free sheaves of finite rank and let U í \mathbbXU\subseteq \mathbb{X} be an open subscheme. We prove that the singularity category of U is triangle equivalent to the Verdier quotient triangulated category of the singularity category of \mathbbX\mathbb{X} with respect to the thick triangulated subcategory generated by sheaves supported in the complement of U. The result unifies two results of Orlov. We also prove a noncommutative version of this result.     

On the spectral norm of algebraic numbers

In this paper we continue to study the spectral norms and their completions ([4]) in the case of the algebraic closure $ \overline {\mathbb Q} $ of ? in ?. Let $ \widetilde{\overline{\mathbb{Q}}} $ be the completion of $ \overline {\mathbb Q} $ relative to the spectral norm. We prove that $ \widetilde{\overline{\mathbb{Q}}} $ can be identified with the R‐subalgebra of all symmetric functions of C(G), where C(G) denotes the ?‐Banach algebra of all continuous functions defined on the absolute Galois group G = Gal$ {\overline {\mathbb Q}} / {\mathbb Q} $. We prove that any compact, closed to conjugation subset of ? is the pseudo‐orbit of a suitable element of $ \widetilde{\overline{\mathbb{Q}}} $. We also prove that the topological closure of any algebraic number field in $ \widetilde{\overline{\mathbb{Q}}} $ is of the form $\widetilde{\mathbb{Q}[x]}$ with x in $ \widetilde{\overline{\mathbb{Q}}} $.     

Complex cycles on algebraic models of smooth manifolds

Every compact smooth manifold M is diffeomorphic to the set X(\mathbbR){X(\mathbb{R})} of real points of a nonsingular projective real algebraic variety X, which is called an algebraic model of M. Each algebraic cycle of codimension k on the complex variety X_\mathbbC=X×_\mathbbR\mathbbC{X_{\mathbb{C}}=X\times_{\mathbb{R}}\mathbb{C}} determines a cohomology class in H^2k(X(\mathbbR);\mathbbD){H^{2k}(X(\mathbb{R});\mathbb{D})} , where \mathbbD{\mathbb{D}} denotes \mathbbZ{\mathbb{Z}} or \mathbbQ{\mathbb{Q}} . We investigate the behavior of such cohomology classes as X runs through the class of algebraic models of M.     

换位子群是不可分Abel群的有限秩可除幂零群

完整地确定了换位子群是不可分Abel群的有限秩可除幂零群的结构,证明了下面的定理.设G是有限秩的可除幂零群,则G的换位子群是不可分Abel群当且仅当G'=Q或Q_p/Z且G可以分解为G=S×D,其中当G'=Q时,■当G'=Q_p/Z时,S有中心积分解S=S_1*S_2*…*S_r,并且可以将S形式化地写成■其中■,式中s,t都是非负整数,Q是有理数加群,π_κ(k=1,2,…,t)是某些素数的集合,满足π_1■Cπ_2■…■π_t,Q_π_k={m/n|(m,n)=1,m∈Z,n为正的π_k-数}.进一步地,当G'=Q时,(r;s;π_1,π_2,…,π_t)是群G的同构不变量;当G'=Q_p/Z时,(p,r;s;π_1,π_2,…,πt)是群G的同构不变量.即若群H也是有限秩的可除幂零群,它的换位子群是不可分Abel群,那么G同构于H的充分必要条件是它们有相同的不变量.     

A geometric construction for permutation equivariant categories from modular functors

Let G be a finite group. Given a finite G-set X\cal{X} and a modular tensor category C\cal{C}, we construct a weak G-equivariant fusion category C^X\cal{C}^{\cal{X}}, called the permutation equivariant tensor category. The construction is geometric and uses the formalism of modular functors. As an application, we concretely work out a complete set of structure morphisms for \mathbbZ/2\mathbb{Z}/2-permutation equivariant categories, finishing thereby a program we initiated in an earlier paper.     

Two-dimensional complex tori with multiplication by $\sqrt {d}$

We give an elementary argument for the well known fact that the endomorphism algebra End(A)?\Bbb Q {\rm {End}}(A)\otimes {\Bbb Q } of a simple complex abelian surface A can neither be an imaginary quadratic field nor a definite quaternion algebra. Another consequence of our argument is that a two-dimensional complex torus T with \Bbb Q (?d)\hookrightarrow End_{\Bbb Q} (T){\Bbb Q }(\sqrt {d})\hookrightarrow {\rm{End_{{\Bbb Q }}}}(T) where \Bbb Q (?d){\Bbb Q }(\sqrt {d}) is real quadratic, is algebraic.     

Stable systolic category of manifolds and the cup-length

It follows from a theorem of Gromov that the stable systolic category cat_stsys M{\rm cat}_{\rm stsys} M of a closed manifold M is bounded from below by cl_\mathbbQ M{\rm cl}_{\mathbb{Q}} M, the rational cup-length of M [Ka07]. We study the inequality in the opposite direction. In particular, combining our results with Gromov’s theorem, we prove the equality cat_stsys M = cl_\mathbbQ M{\rm cat}_{\rm stsys} M = {\rm cl}_{\mathbb{Q}} M for simply connected manifolds of dimension ≤ 7.     

Initial Enlargement of Filtrations and Entropy of Poisson Compensators

Let μ be a Poisson random measure, let \mathbbF\mathbb{F} be the smallest filtration satisfying the usual conditions and containing the one generated by μ, and let \mathbbG\mathbb{G} be the initial enlargement of \mathbbF\mathbb{F} with the σ-field generated by a random variable G. In this paper, we first show that the mutual information between the enlarging random variable G and the σ-algebra generated by the Poisson random measure μ is equal to the expected relative entropy of the \mathbbG\mathbb{G}-compensator relative to the \mathbbF\mathbb{F}-compensator of the random measure μ. We then use this link to gain some insight into the changes of Doob–Meyer decompositions of stochastic processes when the filtration is enlarged from  \mathbbF\mathbb{F} to  \mathbbG\mathbb{G}. In particular, we show that if the mutual information between G and the σ-algebra generated by the Poisson random measure μ is finite, then every square-integrable \mathbbF\mathbb{F}-martingale is a \mathbbG\mathbb{G}-semimartingale that belongs to the normed space S¹\mathcal{S}^{1} relative to  \mathbbG\mathbb{G}.     

Finite lattices as relative congruence lattices of finite algebras

Let A be a finite algebra and a quasivariety. By A is meant the lattice of congruences θ on A with . For any positive integer n, we give conditions on a finite algebra A under which for any n-element lattice L there is a quasivariety such that . The author was supported by INTAS grant 03-51-4110.