Volume preserving subgroups of $${{\mathcal A}}$$ and $${{\mathcal K}}$$ and singularities in unimodular geometry

For a germ of a smooth map f from \mathbb Kⁿ{{\mathbb K}^n} to \mathbb K^p{{\mathbb K}^p} and a subgroup G_Wq{{{G}_{\Omega _q}}} of any of the Mather groups G for which the source or target diffeomorphisms preserve some given volume form Ω _q in \mathbb K^q{{\mathbb K}^q} (q = n or p) we study the G_Wq{{{G}_{\Omega _q}}} -moduli space of f that parameterizes the G_Wq{{{G}_{\Omega _q}}} -orbits inside the G-orbit of f. We find, for example, that this moduli space vanishes for G_Wq = A_Wp{{{G}_{\Omega _q}} ={{\mathcal A}_{\Omega _p}}} and A{{\mathcal A}}-stable maps f and for G_Wq = K_Wn{{{G}_{\Omega _q}} ={{\mathcal K}_{\Omega _n}}} and K{{\mathcal K}}-simple maps f. On the other hand, there are A{{\mathcal A}}-stable maps f with infinite-dimensional A_Wn{{{\mathcal A}_{\Omega _n}}} -moduli space.     

Tiling Groupoids and Bratteli Diagrams II: Structure of the Orbit Equivalence Relation

In this second paper, we study the case of substitution tilings of \mathbb R^d{{\mathbb R}^d} . The substitution on tiles induces substitutions on the faces of the tiles of all dimensions j = 0, . . . , d − 1. We reconstruct the tiling’s equivalence relation in a purely combinatorial way using the AF-relations given by the lower dimensional substitutions. We define a Bratteli multi-diagram B{{\mathcal B}} which is made of the Bratteli diagrams B^j, j=0, ?d{{\mathcal B}^j, j=0, \ldots d} , of all those substitutions. The set of infinite paths in B^d{{\mathcal B}^d} is identified with the canonical transversal Ξ of the tiling. Any such path has a “border”, which is a set of tails in B^j{{\mathcal B}^j} for some j ≤ d, and this corresponds to a natural notion of border for its associated tiling. We define an étale equivalence relation R_B{{\mathcal R}_{\mathcal B}} on B{{\mathcal B}} by saying that two infinite paths are equivalent if they have borders which are tail equivalent in B^j{{\mathcal B}^j} for some j ≤ d. We show that R_B{{\mathcal R}_{\mathcal B}} is homeomorphic to the tiling’s equivalence relation R_X{{\mathcal R}_\Xi} .     

Theory of Completeness for Logical Spaces

A logical space is a pair (A, B){(A, {\mathcal{B}})} of a non-empty set A and a subset B{{\mathcal{B}}} of P A{{\mathcal{P}} A} . Since P A{{\mathcal{P}} A} is identified with {0, 1}^A and {0, 1} is a typical lattice, a pair (A, F){(A, {\mathcal{F}})} of a non-empty set A and a subset F{{\mathcal{F}}} of \mathbbB^A{{\mathbb{B}}^A} for a certain lattice \mathbbB{{\mathbb{B}}} is also called a \mathbbB{{\mathbb{B}}} -valued functional logical space. A deduction system on A is a pair (R, D) of a subset D of A and a relation R between A* and A. In terms of these simplest concepts, a general framework for studying the logical completeness is constructed.     

Algebraic characterization of isometries of the complex and the quaternionic hyperbolic planes

Let H²_{\mathbb F}{{\bf H}^{\bf 2}_{\mathbb F}} denote the two dimensional hyperbolic space over \mathbb F{\mathbb F} , where \mathbb F{\mathbb F} is either the complex numbers \mathbb C{\mathbb C} or the quaternions \mathbb H{\mathbb H} . It is of interest to characterize algebraically the dynamical types of isometries of H²_{\mathbb F}{{\bf H}^{\bf 2}_{\mathbb F}} . For \mathbb F=\mathbb C{\mathbb F=\mathbb C} , such a characterization is known from the work of Giraud–Goldman. In this paper, we offer an algebraic characterization of isometries of H²_{\mathbb H}{{\bf H}^{\bf 2}_{\mathbb H}} . Our result restricts to the case \mathbb F=\mathbb C{\mathbb F=\mathbb C} and provides another characterization of the isometries of H²_{\mathbb C}{{\bf H}^{\bf 2}_{\mathbb C}} , which is different from the characterization due to Giraud–Goldman. Two elements in a group G are said to be in the same z-class if their centralizers are conjugate in G. The z-classes provide a finite partition of the isometry group. In this paper, we describe the centralizers of isometries of H²_{\mathbb F}{{\bf H}^{\bf 2}_{\mathbb F}} and determine the z-classes.     

Singularity Spectrum of Generic ��-H?lder Regular Functions After Time Subordination

A question of Yves Meyer motivated the research concerning “time” subordinations of real functions. Denote by B₁^a{\mathcal {B}}_{1}^{\alpha} the metric space of functions with Lipschitz constant 1 defined on [0,1], equipped with the complete metric defined via the supremum norm. Given a function g ? B₁^ag\in {\mathcal {B}}_{1}^{\alpha} one obtains a time subordination of g simply by considering the composite function Z=g○f, where f∈ℳ:={f:f(0)=0, f(1)=1 and f is a continuous nondecreasing function on [0,1]}. The metric space E^a=M×B₁^a\mathcal {E}^{\alpha}=\mathcal {M}\times {\mathcal {B}}_{1}^{\alpha} equipped with the product supremum metric is a complete metric space. In this paper for all α∈[0,1) multifractal properties of g○f are investigated for a generic (typical) element (f,g)∈ℰ ^α . In particular we determine the generic H?lder singularity spectrum of g○f.     

On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime

Given a finite family F\mathcal{F} of linear forms with integer coefficients, and a compact abelian group G, an F\mathcal{F}-free set in G is a measurable set which does not contain solutions to any equation L(x)=0 for L in F\mathcal{F}. We denote by d_F(G)d_{\mathcal{F}}(G) the supremum of μ(A) over F\mathcal{F}-free sets A⊂G, where μ is the normalized Haar measure on G. Our main result is that, for any such collection F\mathcal{F} of forms in at least three variables, the sequence d_F(\mathbb Z_p)d_{\mathcal{F}}({\mathbb {Z}}_{p}) converges to d_F(\mathbb R/\mathbb Z)d_{\mathcal{F}}({\mathbb {R}}/{\mathbb {Z}}) as p→∞ over primes. This answers an analogue for ℤ _p of a question that Ruzsa raised about sets of integers.     

Some Weighted Energy Classes of Plurisubharmonic Functions

In this paper we study relations between the weighted energy class E_c\mathcal{E}_{\chi} introduced by S. Benelkourchi, V. Guedj and A. Zeriahi recently with Cegrell’s classes E\mathcal{E} and N\mathcal{N}. Next we establish a generalized comparison principle for the operator M _χ . As an application, we prove a version of existence of solutions of Monge–Ampère type equations in the class E_c(H,W)\mathcal{E}_{\chi}(H,\Omega).     

Free limits of Thompson’s group <Emphasis Type="Italic">F</Emphasis>

We produce a sequence of markings S _k of Thompson’s group F within the space G_n{{\mathcal G}_n} of all marked n-generator groups so that the sequence (F, S _k ) converges to the free group on n generators, for n ≥ 3. In addition, we give presentations for the limits of some other natural (convergent) sequences of markings to consider on F within G₃{{\mathcal G}_3}, including (F, {x ₀, x ₁, x _n }) and (F,{x₀,x₁,x₀ⁿ}){(F,\{x_0,x_1,x_0^n\})}.     

On thin-complete ideals of subsets of groups

Let F ì P_G \mathcal{F} \subset {\mathcal{P}_G} be a left-invariant lower family of subsets of a group G. A subset A ⊂ G is called F \mathcal{F} -thin if xA ?yA ? F xA \cap yA \in \mathcal{F} for any distinct elements x, y ∈ G. The family of all F \mathcal{F} -thin subsets of G is denoted by t( F ) \tau \left( \mathcal{F} \right) . If t( F ) = F \tau \left( \mathcal{F} \right) = \mathcal{F} , then F \mathcal{F} is called thin-complete. The thin-completion t^*( F ) {\tau^*}\left( \mathcal{F} \right) of F \mathcal{F} is the smallest thin-complete subfamily of P_G {\mathcal{P}_G} that contains F \mathcal{F} . Answering questions of Lutsenko and Protasov, we prove that a set A ⊂ G belongs to τ*(G) if and only if, for any sequence (g _n ) _n∈ω of nonzero elements of G, there is n ∈ ω such that

Orthogonal basis for spherical monogenics by step two branching

Spherical monogenics can be regarded as a basic tool for the study of harmonic analysis of the Dirac operator in Euclidean space \mathbb R^m{{\mathbb R}^m}. They play a similar role as spherical harmonics do in case of harmonic analysis of the Laplace operator on \mathbb R^m{{\mathbb R}^m}. Fix the direct sum \mathbb R^m=\mathbb R^p ?\mathbb R^q{{\mathbb R}^m={\mathbb R}^p \oplus {\mathbb R}^q}. In this article, we will study the decomposition of the space M_n(\mathbb R^m, \mathbb C_m){{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)} of spherical monogenics of order n under the action of Spin(p) × Spin(q). As a result, we obtain a Spin(p) × Spin(q)-invariant orthonormal basis for M_n(\mathbb R^m, \mathbb C_m){{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}. In particular, using the construction with p = 2 inductively, this yields a new orthonormal basis for the space M_n(\mathbb R^m, \mathbb C_m){{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}.     

Some properties for the largest component of random geometric graphs with applications in sensor networks

In this paper we consider the standard Poisson Boolean model of random geometric graphs G（Hλ,s; 1） in Rd and study the properties of the order of the largest component L1 （G（Hλ,s; 1）） . We prove that ElL1 （G（Hλ,s; 1））] is smooth with respect to A, and is derivable with respect to s. Also, we give the expression of these derivatives. These studies provide some new methods for the theory of the largest component of finite random geometric graphs （not asymptotic graphs as s - co） in the high dimensional space （d 〉 2）. Moreover, we investigate the convergence rate of E[L1（G（Hλ,s; 1））]. These results have significance for theory development of random geometric graphs and its practical application. Using our theories, we construct and solve a new optimal energy-efficient topology control model of wireless sensor networks, which has the significance of theoretical foundation and guidance for the design of network layout.     

A Hilbert theorem for vertex algebras

Given a simple vertex algebra A \mathcal{A} and a reductive group G of automorphisms of A \mathcal{A} , the invariant subalgebra A^G {\mathcal{A}^G} is strongly finitely generated in most examples where its structure is known. This phenomenon is subtle, and is generally not true of the classical limit of A^G {\mathcal{A}^G} , which often requires infinitely many generators and infinitely many relations to describe. Using tools from classical invariant theory, together with recent results on the structure of the W_{1 + ￥} {\mathcal{W}_{{1 + }\infty }} algebra, we establish the strong finite generation of a large family of invariant subalgebras of βγ-systems, bc-systems, and bcβγ-systems.     

Uniformization of {\mathcal {G}}-bundles

We show some of the conjectures of Pappas and Rapoport concerning the moduli stack Bun_G{{\rm Bun}_\mathcal {G}} of G{\mathcal {G}}-torsors on a curve C, where G{\mathcal {G}} is a semisimple Bruhat-Tits group scheme on C. In particular we prove the analog of the uniformization theorem of Drinfeld-Simpson in this setting. Furthermore we apply this to compute the connected components of these moduli stacks and to calculate the Picard group of Bun_G{{\rm Bun}_\mathcal {G}} in case G{\mathcal {G}} is simply connected.     

The {\overline{\partial}} operator along the leaves and Guichard’s theorem for a complex simple foliation

In (Ann Sc ENS Sér 3 4:361–380, 1887) Guichard proved that, for any holomorphic function g on ${{\mathbb C}}In (Ann Sc ENS Sér 3 4:361–380, 1887) Guichard proved that, for any holomorphic function g on \mathbb C{{\mathbb C}}, there exists a holomorphic function h (on \mathbb C{{\mathbb C}}) such that h - h °t = g{h - h \circ \tau = g} where τ is the translation by 1 on \mathbb C{{\mathbb C}}. In this note we prove an analogous of this theorem in a more general situation. Precisely, let (M,F){(M,{\mathcal F})} be a complex simple foliation whose leaves are simply connected non compact Riemann surfaces and γ an automorphism of F{{\mathcal F}} which fixes each leaf and acts on it freely and properly. Then, the vector space H_F(M){{\mathcal H}_{\mathcal F}(M)} of leafwise holomorphic functions is not reduced to functions constant on the leaves and for any g ? H_F(M){g \in {\mathcal H}_{\mathcal F}(M)}, there exists h ? H_F(M){h \in {\mathcal H}_{\mathcal F}(M)} such that h - h °g = g{h - h \circ \gamma = g}. From the proof of this theorem we derive a foliated version of Mittag–Leffler Theorem.     

Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres

Let X ì \mathbb Rⁿ{{\bf X} \subset {\mathbb R}^n} be a generalised annulus and consider the Dirichlet energy functional

The Deligne–Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3

In the moduli space M \mathcal{M} _g of genus-g Riemann surfaces, consider the locus RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} of Riemann surfaces whose Jacobians have real multiplication by the order O \mathcal{O} in a totally real number field F of degree g. If g = 3, we compute the closure of RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} in the Deligne–Mumford compactification of M \mathcal{M} _g and the closure of the locus of eigenforms over RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} in the Deligne–Mumford compactification of the moduli space of holomorphic 1-forms. For higher genera, we give strong necessary conditions for a stable curve to be in the boundary of RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} . Boundary strata of RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} are parameterized by configurations of elements of the field F satisfying a strong geometry of numbers type restriction.     

Goodness in the enumeration and singleton degrees

We investigate and extend the notion of a good approximation with respect to the enumeration ${({\mathcal D}_{\rm e})}We investigate and extend the notion of a good approximation with respect to the enumeration (D_e){({\mathcal D}_{\rm e})} and singleton (D_s){({\mathcal D}_{\rm s})} degrees. We refine two results by Griffith, on the inversion of the jump of sets with a good approximation, and we consider the relation between the double jump and index sets, in the context of enumeration reducibility. We study partial order embeddings i_s{\iota_s} and [^(i)]_s{\hat{\iota}_s} of, respectively, D_e{{\mathcal D}_{\rm e}} and D_T{{\mathcal D}_{\rm T}} (the Turing degrees) into D_s{{\mathcal D}_{\rm s}} , and we show that the image of D_T{{\mathcal D}_{\rm T}} under [^(i)]_s{\hat{\iota}_s} is precisely the class of retraceable singleton degrees. We define the notion of a good enumeration, or singleton, degree to be the property of containing the set of good stages of some good approximation, and we show that i_s{\iota_s} preserves the latter, as also other naturally arising properties such as that of totality or of being G⁰_n{\Gamma^0_n} , for G ? {S,P,D}{\Gamma \in \{\Sigma,\Pi,\Delta\}} and n > 0. We prove that the good enumeration and singleton degrees are immune and that the good S⁰₂{\Sigma^0_2} singleton degrees are hyperimmune. Finally we show that, for singleton degrees a _s < b _s such that b _s is good, any countable partial order can be embedded in the interval (a _s, b _s).     

Relatively thin and sparse subsets of groups

Let G be a group with identity e and let I \mathcal{I} be a left-invariant ideal in the Boolean algebra P_G {\mathcal{P}_G} of all subsets of G. A subset A of G is called I \mathcal{I} -thin if gA ?A ? I gA \cap A \in \mathcal{I} for every g ∈ G\{e}. A subset A of G is called I \mathcal{I} -sparse if, for T every infinite subset S of G, there exists a finite subset F ⊂ S such that ?_{g ? F} gA ? F \bigcap\nolimits_{g \in F} {gA \in \mathcal{F}} . An ideal I \mathcal{I} is said to be thin-complete (sparse-complete) if every I \mathcal{I} -thin (I \mathcal{I} -sparse) subset of G belongs to I \mathcal{I} . We define and describe the thin-completion and the sparse-completion of an ideal in P_G {\mathcal{P}_G} .     

The modal logic of continuous functions on the rational numbers

Let L^\square°{{\mathcal L}^{\square\circ}} be a propositional language with standard Boolean connectives plus two modalities: an S4-ish topological modality □ and a temporal modality ◦, understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language L^\square°{{\mathcal L}^{\square\circ}} by interpreting L^\square°{{\mathcal L}^{\square\circ}} in dynamic topological systems, i.e., ordered pairs 〈X, f〉, where X is a topological space and f is a continuous function on X. Artemov, Davoren and Nerode have axiomatized a logic S4C, and have shown that S4C is sound and complete for this semantics. S4C is also complete for continuous functions on Cantor space (Mints and Zhang, Kremer), and on the real plane (Fernández Duque); but incomplete for continuous functions on the real line (Kremer and Mints, Slavnov). Here we show that S4C is complete for continuous functions on the rational numbers.     

Iterates and Hypoellipticity of Partial Differential Operators on Non-Quasianalytic Classes

Let P be a linear partial differential operator with constant coefficients. For a weight function ω and an open subset Ω of \mathbbR^N{\mathbb{R}^N} , the class E_P,{w}(W){\mathcal{E}_{P,\{\omega\}}(\Omega)} of Roumieu type involving the successive iterates of the operator P is considered. The completeness of this space is characterized in terms of the hypoellipticity of P. Results of Komatsu and Newberger-Zielezny are extended. Moreover, for weights ω satisfying a certain growth condition, this class coincides with a class of ultradifferentiable functions if and only if P is elliptic. These results remain true in the Beurling case E_P,(w)(W){\mathcal{E}_{P,(\omega)}(\Omega)}.