Solutions to Polynomial Generalized Bers–Vekua Equations in Clifford Analysis

In this paper, we mainly study polynomial generalized Vekua-type equation _boxclose)w=0{p(\mathcal{D})w=0} and polynomial generalized Bers–Vekua equation p(D)w=0{p(\mathcal{\underline{D}})w=0} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}} where D{\mathcal{D}} and D{\mathcal{\underline{D}}} mean generalized Vekua-type operator and generalized Bers–Vekua operator, respectively. Using Clifford algebra, we obtain the Fischer-type decomposition theorems for the solutions to these equations including (D-l)^kw=0,(D-l)^kw=0(k ? \mathbbN){\left(\mathcal{D}-\lambda\right)^{k}w=0,\left(\mathcal {\underline{D}}-\lambda\right)^{k}w=0\left(k\in\mathbb{N}\right)} with complex parameter λ as special cases, which derive the Almansi-type decomposition theorems for iterated generalized Bers–Vekua equation and polynomial generalized Cauchy–Riemann equation defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}. Making use of the decomposition theorems we give the solutions to polynomial generalized Bers–Vekua equation defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}} under some conditions. Furthermore we discuss inhomogeneous polynomial generalized Bers–Vekua equation p(D)w=v{p(\mathcal{\underline{D}})w=v} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}, and develop the structure of the solutions to inhomogeneous polynomial generalized Bers–Vekua equation p(D)w=v{p(\mathcal{\underline{D}})w=v} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}.     

An ODE’s Version of the Formula of Baker, Campbell, Dynkin and Hausdorff and the Construction of Lie Groups with Prescribed Lie Algebra

If ${\mathcal{L} = {\sum_{j=1}^m} {X_j^2} + X_0}If L = ?_j=1^m X_j² + X₀{\mathcal{L} = {\sum_{j=1}^m} {X_j^2} + X_0} is a H?rmander partial differential operator in \mathbbR^N{\mathbb{R}^N}, we give sufficient conditions on the vector fields X _j ’s for the existence of a Lie group structure \mathbbG = (\mathbbR^N, *){\mathbb{G} = (\mathbb{R}^N, *)} (and we exhibit its construction), not necessarily nilpotent nor homogeneous, such that L{\mathcal{L}} is left invariant on \mathbbG{\mathbb{G}}. The main tool is a formula of Baker-Campbell-Dynkin-Hausdorff type for the ODE’s naturally related to the system of vector fields {X ₀, . . . , X _m }. We provide a direct proof of this formula in the ODE’s context (which seems to be missing in literature), without invoking any result of Lie group theory, nor the abstract algebraic machinery usually involved in formulas of Baker-Campbell-Dynkin-Hausdorff type. Examples of operators to which our results apply are also furnished.     

Counting Ω-ideals

Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P_\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(Aⁿ) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 2^2à₀{2^{2^{\aleph_0}}} Ω-ideals in P_\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.     

Triangulating Smooth Submanifolds with Light Scaffolding

We propose an algorithm to sample and mesh a k-submanifold M{\mathcal{M}} of positive reach embedded in \mathbbR^d{\mathbb{R}^{d}} . The algorithm first constructs a crude sample of M{\mathcal{M}} . It then refines the sample according to a prescribed parameter e{\varepsilon} , and builds a mesh that approximates M{\mathcal{M}} . Differently from most algorithms that have been developed for meshing surfaces of \mathbbR ³{\mathbb{R} ^3} , the refinement phase does not rely on a subdivision of \mathbbR ^d{\mathbb{R} ^d} (such as a grid or a triangulation of the sample points) since the size of such scaffoldings depends exponentially on the ambient dimension d. Instead, we only compute local stars consisting of k-dimensional simplices around each sample point. By refining the sample, we can ensure that all stars become coherent leading to a k-dimensional triangulated manifold [^(M)]{\hat{\mathcal{M}}} . The algorithm uses only simple numerical operations. We show that the size of the sample is O(e^-k){O(\varepsilon ^{-k})} and that [^(M)]{\hat{\mathcal{M}}} is a good triangulation of M{\mathcal{M}} . More specifically, we show that M{\mathcal{M}} and [^(M)]{\hat{\mathcal{M}}} are isotopic, that their Hausdorff distance is O(e²){O(\varepsilon ^{2})} and that the maximum angle between their tangent bundles is O(e){O(\varepsilon )} . The asymptotic complexity of the algorithm is T(e) = O(e^-k2-k){T(\varepsilon) = O(\varepsilon ^{-k^2-k})} (for fixed M, d{\mathcal{M}, d} and k).     

On Representations of Integers in Thin Subgroups of {{\rm SL}_2({\mathbb {Z}})}

Let ${\Gamma < {\rm SL}(2, {\mathbb Z})}Let G < SL(2, \mathbb Z){\Gamma < {\rm SL}(2, {\mathbb Z})} be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive vectors v₀, w₀ ? \mathbb Z²  \  {0}{v_{0}, w_{0} \in \mathbb {Z}^{2} \, {\backslash} \, \{0\}}. We consider the set S{\mathcal {S}} of all integers occurring in áv₀g, w₀?{\langle v_{0}\gamma, w_{0}\rangle}, for g ? G{\gamma \in \Gamma} and the usual inner product on \mathbb R²{\mathbb {R}^2}. Assume that the critical exponent δ of Γ exceeds 0.99995, so that Γ is thin but not too thin. Using a variant of the circle method, new bilinear forms estimates and Gamburd’s 5/6-th spectral gap in infinite-volume, we show that S{\mathcal {S}} contains almost all of its admissible primes, that is, those not excluded by local (congruence) obstructions. Moreover, we show that the exceptional set \mathfrak E(N){\mathfrak {E}(N)} of integers |n| < N which are locally admissible (n ? S   (mod  q)   for all   q 3 1){(n \in \mathcal {S} \, \, ({\rm mod} \, q) \, \, {\rm for\,all} \,\, q \geq 1)} but fail to be globally represented, n ? S{n \notin \mathcal {S}}, has a power savings, |\mathfrak E(N)| << N^1-e₀{|\mathfrak {E}(N)| \ll N^{1-\varepsilon_{0}}} for some ${\varepsilon_{0} > 0}${\varepsilon_{0} > 0}, as N → ∞.     

Hyperflock determining line sets and totally regular parallelisms of PG (3, \mathbbR){(3,\,{\mathbb{R}})}

By a totally regular parallelism of the real projective 3-space P₃:=PG(3, \mathbb R){\Pi_3:={{\rm PG}}(3, \mathbb {R})} we mean a family T of regular spreads such that each line of Π ₃ is contained in exactly one spread of T. For the investigation of totally regular parallelisms the authors mainly employ Klein’s correspondence λ of line geometry and the polarity π ₅ associated with the Klein quadric H ₅ (for details see Chaps. 1 and 3). The λ-image of a totally regular parallelism T is a hyperflock of H ₅, i.e., a family H of elliptic subquadrics of H ₅ such that each point of H ₅ is on exactly one subquadric of H. Moreover, {p₅(span  l(X))|X ? T}=:H_T{\{\pi_5({{\rm span}} \,\lambda(\mathcal {X}))\vert\mathcal {X}\in\bf{T}\}=:\mathcal {H}_{\bf{T}}} is a hyperflock determining line set, i.e., a set Z{\mathcal {Z}} of 0-secants of H ₅ such that each tangential hyperplane of H ₅ contains exactly one line of Z{\mathcal {Z}} . We say that dim(span H_T)=:d_T{{{\rm dim}}({{\rm span}}\,\mathcal {H}_{\bf{T}})=:d_{\bf{T}}} is the dimension of T and that T is a d _T - parallelism. Clifford parallelisms and 2-parallelisms coincide. The examples of non-Clifford parallelisms exhibited in Betten and Riesinger [Result Math 47:226–241, 2004; Adv Geom 8:11–32, 2008; J Geom (to appear)] are totally regular and of dimension 3. If G{\mathcal{G}} is a hyperflock determining line set, then {l^-1 (p₅(X) ?H₅) | X ? G}{\{\lambda^{-1}\,{\rm (}\pi_5(X){\,\cap H_5)\,|\, X\in\mathcal{G}\}}} is a totally regular parallelism. In the present paper the authors construct examples of topological (see Definition 1.1) 4- and 5-parallelisms via hyperflock determining line sets.     

Intertwining the geodesic flow and the Schr?dinger group on hyperbolic surfaces

We construct an explicit intertwining operator L{\mathcal L} between the Schr?dinger group e^{it \frac\triangle2}{e^{it \frac\triangle2}} and the geodesic flow on certain Hilbert spaces of symbols on the cotangent bundle T*X _Γ of a compact hyperbolic surface X _Γ = Γ\D. We also define Γ-invariant eigendistributions of the geodesic flow PS_{j, k, nj,-nk}{PS_{j, k, \nu_j,-\nu_k}} (Patterson-Sullivan distributions) out of pairs of \triangle{\triangle} -eigenfunctions, generalizing the diagonal case j = k treated in Anantharaman and Zelditch (Ann. Henri Poincaré 8(2):361–426, 2007). The operator L{\mathcal L} maps PS_{j, k, nj,-nk}{PS_{j, k, \nu_j,-\nu_k}} to the Wigner distribution W^G_j,k{W^{\Gamma}_{j,k}} studied in quantum chaos. We define Hilbert spaces H_PS{\mathcal H_{PS}} (whose dual is spanned by {PS_{j, k, nj,-nk}{PS_{j, k, \nu_j,-\nu_k}}}), resp. H_W{\mathcal H_W} (whose dual is spanned by {W^G_j,k}{\{W^{\Gamma}_{j,k}\}}), and show that L{\mathcal L} is a unitary isomorphism from H_W ? H_PS.{\mathcal H_{W} \to \mathcal H_{PS}.}     

On the Second Cohomology of K?hler Groups

Carlson and Toledo conjectured that if an infinite group Γ is the fundamental group of a compact K?hler manifold, then virtually H²(G, \mathbb R) 1 0{H^{2}(\Gamma, {\mathbb R}) \ne 0} . We assume that Γ admits an unbounded reductive rigid linear representation. This representation necessarily comes from a complex variation of Hodge structure ( \mathbbC{\mathbb{C}} -VHS) on the K?hler manifold. We prove the conjecture under some assumption on the \mathbbC{\mathbb{C}} -VHS. We also study some related geometric/topological properties of period domains associated to such a \mathbbC{\mathbb{C}} -VHS.     

Topology and Smooth Structure for Pseudoframes

Given a closed subspace ${\mathcal{S}}Given a closed subspace S{\mathcal{S}} of a Hilbert space H{\mathcal{H}}, we study the sets F_S{\mathcal{F}_\mathcal{S}} of pseudo-frames, CF_S{\mathcal{C}\mathcal{F}_\mathcal{S}} of commutative pseudo-frames and \mathfrakX_S{\tiny{\mathfrak{X}}_{\mathcal{S}}} of dual frames for S{\mathcal{S}}, via the (well known) one to one correspondence which assigns a pair of operators (F, H) to a frame pair ({f_n}_{n ? \mathbbN},{h_n}_{n ? \mathbbN}){(\{f_n\}_{n\in\mathbb{N}},\{h_n\}_{n\in\mathbb{N}})},

F:l2? H,     F({cn}n ? \mathbbN )=?n cn fn,F:\ell^2\to\,\mathcal{H}, \quad F\left(\{c_n\}_{n\in\mathbb{N}} \right)=\sum_n c_n f_n,      10. Platonic polyhedra,topological constraints and periodic solutions of the classical <Emphasis Type="Italic">N</Emphasis>-body problem      G. Fusco  G. F. Gronchi  P. Negrini 《Inventiones Mathematicae》2011,185(2):283-332 We prove the existence of a number of smooth periodic motions u ∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group R\mathcal{R} of one of the five Platonic polyhedra. The number N coincides with the order |R||\mathcal{R}| of R\mathcal{R} and the particles have all the same mass. Our approach is variational and u ∗ is a minimizer of the Lagrangian action A\mathcal{A} on a suitable subset K\mathcal{K} of the H 1 T-periodic maps u:ℝ→ℝ3N . The set K{\mathcal {K}} is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group R\mathcal{R}. There exist infinitely many such cones K{\mathcal {K}}, all with the property that A|K{\mathcal {A}}|_{{\mathcal {K}}} is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N-body problem with a rich geometric–kinematic structure.      11. Unstable structures definable in o-minimal theories      Assaf Hasson  Alf Onshuus 《Selecta Mathematica, New Series》2010,16(1):121-143 Let M{\mathcal {M}} be a dense o-minimal structure, N{\mathcal {N}} an unstable structure interpretable in M{\mathcal {M}}. Then there exists X, definable in Neq{\mathcal {N}^{eq}}, such that X, with the induced N{\mathcal {N}}-structure, is linearly ordered and o-minimal with respect to that ordering. As a consequence we obtain a classification, along the lines of Zilber’s trichotomy, of unstable t-minimal types in structures interpretable in o-minimal theories.      12. On the intersection of a Hermitian curve with a conic      Giorgio Donati  Nicola Durante 《Designs, Codes and Cryptography》2010,57(3):347-360 Let ${\mathcal{H}}${\mathcal{H}} be a Hermitian curve and let Γ be a conic of PG(2, q 2). In this paper we determine the possible intersection configurations between Γ and H{\mathcal{H}} under the hypotheses that Γ and H{\mathcal{H}} either share two points with the same tangent lines or contain a common Baer subconic. Moreover, the intersection configurations between a degenerate Hermitian curve and a conic sharing a Baer subconic are also determined.      13. Entropy bounds and second cohomology      David Fried 《Journal of Fixed Point Theory and Applications》2009,6(1):87-92 When X is a finite complex and p1X\pi_{1}X acts on \mathbbR2{\mathbb{R}}^2 by translations we give criteria involving H2X for an equivariant map F : [(X)\tilde] ? \mathbbR2F : \tilde{X} \rightarrow {\mathbb{R}}^2 to be onto. Following work of Manning and Shub, this leads to entropy bounds related to Shub’s entropy conjecture.      14. An Inverse Theorem for the Uniformity Seminorms Associated with the Action of {{\mathbb {F}^{\infty}_{p}}}      Vitaly Bergelson  Terence Tao  Tamar Ziegler 《Geometric And Functional Analysis》2010,19(6):1539-1596 Let ${\mathbb {F}}Let \mathbb F{\mathbb {F}} a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U k (X) for an ergodic action (Tg)g ? \mathbb Fw{(T_{g})_{{g} \in \mathbb {F}^{\omega}}} of the infinite abelian group \mathbb Fw{\mathbb {F}^{\omega}} on a probability space X = (X, B, m){X = (X, \mathcal {B}, \mu)} is generated by phase polynomials f: X ? S1{\phi : X \to S^{1}} of degree less than C(k) on X, where C(k) depends only on k. In the case where k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for \mathbb Z{\mathbb {Z}} by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} , with a partial result in low characteristic.      15. A Lower Bound of Large-Deviation Probabilities for the Sample Mean under the Cramer Condition      L. V. Rozovsky 《Journal of Mathematical Sciences》2003,118(6):5624-5634 Let X1, X2, ... be i.i.d. random variables satisfying the condition \textE X12 \text elX1 < ￥\text for\text some\text l > 0.{\text{E }}X_1^2 {\text{ }}e^{\lambda X_1 } < \infty {\text{ }}for{\text{ }}some{\text{ }}\lambda >0.      16. Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces      Markus Haase 《Integral Equations and Operator Theory》2006,56(2):197-228 We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( Aa ) ?R( Aa ) )q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as { x  ?  X|t - q\textRea y1 ( tA )x, t - q\textRea y2 ( tA )x ? L*p ( ( 0,￥ );X ) } {\left\{ {x\, \in \,X|t^{{ - \theta {\text{Re}}\alpha }} \psi _{1} {\left( {tA} \right)}x,\,t^{{ - \theta {\text{Re}}\alpha }} \psi _{2} {\left( {tA} \right)}x \in L_{*}^{p} {\left( {{\left( {0,\infty } \right)};X} \right)}} \right\}}      17. Prime radicals of constants of algebraic derivations      Chen-Lian Chuang  Tsiu-Kwen Lee 《Monatshefte für Mathematik》2010,23(6):31-40 Let R be a prime ring with extended centroid F and let δ be an F-algebraic continuous derivation of R with the associated inner derivation ad(b). Factorize the minimal polynomial μ(λ) of b over F into distinct irreducible factors m(l)=?ipi(l)ni{\mu(\lambda)=\prod_i\pi_i(\lambda)^{n_i}} . Set ℓ to be the maximum of n i . Let R(d)=def.{x ? R | d(x)=0}{R^{(\delta)}{\mathop{=}\limits^{{\rm def.}}}\{x\in R\mid \delta(x)=0\}} be the subring of constants of δ on R. Denote the prime radical of a ring A by P(A){{\mathcal{P}}(A)} . It is shown among other things that P(R(d))2l-1=0   \textand   P(R(d))=R(d)?P(CR(b)){\mathcal{P}}(R^{(\delta)})^{2^\ell-1}=0\quad\text{and}\quad{\mathcal{P}}(R^{(\delta)})=R^{(\delta)}\cap {\mathcal{P}}(C_R(b))      18. Minimal rotational hypersurfaces in Minkowski (<Emphasis Type="Italic">α</Emphasis>, <Emphasis Type="Italic">β</Emphasis>)-space      Ningwei?CuiEmail author  Yi-Bing?Shen 《Geometriae Dedicata》2011,151(1):27-39 In Finsler geometry, minimal surfaces with respect to the Busemann-Hausdorff measure and the Holmes-Thompson measure are called BH-minimal and HT-minimal surfaces, respectively. In this paper, we give the explicit expressions of BH-minimal and HT-minimal rotational hypersurfaces generated by plane curves rotating around the axis in the direction of [(b)\tilde]\sharp{\tilde{\beta}^{\sharp}} in Minkowski (α, β)-space (\mathbbVn+1,[(Fb)\tilde]){(\mathbb{V}^{n+1},\tilde{F_b})} , where \mathbbVn+1{\mathbb{V}^{n+1}} is an (n+1)-dimensional real vector space, [(Fb)\tilde]=[(a)\tilde]f([(b)\tilde]/[(a)\tilde]), [(a)\tilde]{\tilde{F_b}=\tilde{\alpha}\phi(\tilde{\beta}/\tilde{\alpha}), \tilde{\alpha}} is the Euclidean metric, [(b)\tilde]{\tilde{\beta}} is a one form of constant length b:=||[(b)\tilde]||[(a)\tilde], [(b)\tilde]\sharp{b:=\|\tilde{\beta}\|_{\tilde{\alpha}}, \tilde{\beta}^{\sharp}} is the dual vector of [(b)\tilde]{\tilde{\beta}} with respect to [(a)\tilde]{\tilde{\alpha}} . As an application, we first give the explicit expressions of the forward complete BH-minimal rotational surfaces generated around the axis in the direction of [(b)\tilde]\sharp{\tilde{\beta}^{\sharp}} in Minkowski Randers 3-space (\mathbbV3,[(a)\tilde]+[(b)\tilde]){(\mathbb{V}^{3},\tilde{\alpha}+\tilde{\beta})} .      19. The Ground State and the Long-Time Evolution in the CMC Einstein Flow      Martin Reiris 《Annales Henri Poincare》2010,10(8):1559-1604 Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume ${\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)}Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume V(k)=(\frac-k3)3Volg(k)(S){\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)} is monotonically decreasing in the expanding direction and bounded below by Vinf=(\frac-16Y(S))\frac32{\mathcal{V}_{\rm \inf}=\left(\frac{-1}{6}Y(\Sigma)\right)^{\frac{3}{2}}}. Inspired by this fact we define the ground state of the manifold Σ as “the limit” of any sequence of CMC states {(g i , K i )} satisfying: (i) k i  = −3, (ii) Viˉ Vinf{\mathcal{V}_{i}\downarrow \mathcal{V}_{\rm inf}}, (iii) Q 0((g i , K i )) ≤ Λ, where Q 0 is the Bel–Robinson energy and Λ is any arbitrary positive constant. We prove that (as a geometric state) the ground state is equivalent to the Thurston geometrization of Σ. Ground states classify naturally into three types. We provide examples for each class, including a new ground state (the Double Cusp) that we analyze in detail. Finally, consider a long time and cosmologically normalized flow ([(g)\tilde],[(K)\tilde])(s)=((\frac-k3)2g,(\frac-k3)K){(\tilde{g},\tilde{K})(\sigma)=\left(\left(\frac{-k}{3}\right)^{2}g,\left(\frac{-k}{3}\right)K\right)}, where s = -ln(-k) ? [a,￥){\sigma=-\ln (-k)\in [a,\infty)}. We prove that if [(E1)\tilde]=E1(([(g)\tilde],[(K)\tilde])) ￡ L{\tilde{\mathcal{E}_{1}}=\mathcal{E}_{1}((\tilde{g},\tilde{K}))\leq \Lambda} (where E1=Q0+Q1{\mathcal{E}_{1}=Q_{0}+Q_{1}}, is the sum of the zero and first order Bel–Robinson energies) the flow ([(g)\tilde],[(K)\tilde])(s){(\tilde{g},\tilde{K})(\sigma)} persistently geometrizes the three-manifold Σ and the geometrization is the ground state if Vˉ Vinf{\mathcal{V}\downarrow \mathcal{V}_{\rm inf}}.      20. On Lie derivations of Lie ideals of prime algebras      K. I. Beidar  M. A. Chebotar 《Israel Journal of Mathematics》2001,123(1):131-148 LetF be a commutative ring with 1, letA, be a primeF-algebra with Martindale extended centroidC and with central closureA c and letR be a noncentral Lie ideal of the algebraA generatingA. Further, letZ(R) be the center ofR, let be the factor Lie algebra and let δ: be a Lie derivation. Suppose that char(A) ≠ 2 andA does not satisfySt 14, the standard identity of degree 14. We show thatR ΩC =Z(R) and there exists a derivation of algebrasD:A →A c such that for allx∈R. Our result solves an old problem of Herstein.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Platonic polyhedra,topological constraints and periodic solutions of the classical <Emphasis Type="Italic">N</Emphasis>-body problem

We prove the existence of a number of smooth periodic motions u _∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group R\mathcal{R} of one of the five Platonic polyhedra. The number N coincides with the order |R||\mathcal{R}| of R\mathcal{R} and the particles have all the same mass. Our approach is variational and u _∗ is a minimizer of the Lagrangian action A\mathcal{A} on a suitable subset K\mathcal{K} of the H ¹ T-periodic maps u:ℝ→ℝ^3N . The set K{\mathcal {K}} is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group R\mathcal{R}. There exist infinitely many such cones K{\mathcal {K}}, all with the property that A|_K{\mathcal {A}}|_{{\mathcal {K}}} is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N-body problem with a rich geometric–kinematic structure.     

Unstable structures definable in o-minimal theories

Let M{\mathcal {M}} be a dense o-minimal structure, N{\mathcal {N}} an unstable structure interpretable in M{\mathcal {M}}. Then there exists X, definable in N^eq{\mathcal {N}^{eq}}, such that X, with the induced N{\mathcal {N}}-structure, is linearly ordered and o-minimal with respect to that ordering. As a consequence we obtain a classification, along the lines of Zilber’s trichotomy, of unstable t-minimal types in structures interpretable in o-minimal theories.     

On the intersection of a Hermitian curve with a conic

Let ${\mathcal{H}}${\mathcal{H}} be a Hermitian curve and let Γ be a conic of PG(2, q ²). In this paper we determine the possible intersection configurations between Γ and H{\mathcal{H}} under the hypotheses that Γ and H{\mathcal{H}} either share two points with the same tangent lines or contain a common Baer subconic. Moreover, the intersection configurations between a degenerate Hermitian curve and a conic sharing a Baer subconic are also determined.     

Entropy bounds and second cohomology

When X is a finite complex and p₁X\pi_{1}X acts on \mathbbR²{\mathbb{R}}^2 by translations we give criteria involving H²X for an equivariant map F : [(X)\tilde] ? \mathbbR²F : \tilde{X} \rightarrow {\mathbb{R}}^2 to be onto. Following work of Manning and Shub, this leads to entropy bounds related to Shub’s entropy conjecture.     

An Inverse Theorem for the Uniformity Seminorms Associated with the Action of {{\mathbb {F}^{\infty}_{p}}}

Let ${\mathbb {F}}Let \mathbb F{\mathbb {F}} a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U ^k (X) for an ergodic action (T_g)_{g ? \mathbb F^w}{(T_{g})_{{g} \in \mathbb {F}^{\omega}}} of the infinite abelian group \mathbb F^w{\mathbb {F}^{\omega}} on a probability space X = (X, B, m){X = (X, \mathcal {B}, \mu)} is generated by phase polynomials f: X ? S¹{\phi : X \to S^{1}} of degree less than C(k) on X, where C(k) depends only on k. In the case where k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for \mathbb Z{\mathbb {Z}} by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} , with a partial result in low characteristic.     

A Lower Bound of Large-Deviation Probabilities for the Sample Mean under the Cramer Condition

Let X₁, X₂, ... be i.i.d. random variables satisfying the condition

Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces

We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( A^a ) ?R( A^a ) )_q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as

Prime radicals of constants of algebraic derivations

Let R be a prime ring with extended centroid F and let δ be an F-algebraic continuous derivation of R with the associated inner derivation ad(b). Factorize the minimal polynomial μ(λ) of b over F into distinct irreducible factors m(l)=?_ip_i(l)^n_i{\mu(\lambda)=\prod_i\pi_i(\lambda)^{n_i}} . Set ℓ to be the maximum of n _i . Let R^(d)=^def.{x ? R | d(x)=0}{R^{(\delta)}{\mathop{=}\limits^{{\rm def.}}}\{x\in R\mid \delta(x)=0\}} be the subring of constants of δ on R. Denote the prime radical of a ring A by P(A){{\mathcal{P}}(A)} . It is shown among other things that

Minimal rotational hypersurfaces in Minkowski (<Emphasis Type="Italic">α</Emphasis>, <Emphasis Type="Italic">β</Emphasis>)-space

In Finsler geometry, minimal surfaces with respect to the Busemann-Hausdorff measure and the Holmes-Thompson measure are called BH-minimal and HT-minimal surfaces, respectively. In this paper, we give the explicit expressions of BH-minimal and HT-minimal rotational hypersurfaces generated by plane curves rotating around the axis in the direction of [(b)\tilde]^\sharp{\tilde{\beta}^{\sharp}} in Minkowski (α, β)-space (\mathbbVⁿ⁺¹,[(F_b)\tilde]){(\mathbb{V}^{n+1},\tilde{F_b})} , where \mathbbVⁿ⁺¹{\mathbb{V}^{n+1}} is an (n+1)-dimensional real vector space, [(F_b)\tilde]=[(a)\tilde]f([(b)\tilde]/[(a)\tilde]), [(a)\tilde]{\tilde{F_b}=\tilde{\alpha}\phi(\tilde{\beta}/\tilde{\alpha}), \tilde{\alpha}} is the Euclidean metric, [(b)\tilde]{\tilde{\beta}} is a one form of constant length b:=||[(b)\tilde]||_[(a)\tilde], [(b)\tilde]^\sharp{b:=\|\tilde{\beta}\|_{\tilde{\alpha}}, \tilde{\beta}^{\sharp}} is the dual vector of [(b)\tilde]{\tilde{\beta}} with respect to [(a)\tilde]{\tilde{\alpha}} . As an application, we first give the explicit expressions of the forward complete BH-minimal rotational surfaces generated around the axis in the direction of [(b)\tilde]^\sharp{\tilde{\beta}^{\sharp}} in Minkowski Randers 3-space (\mathbbV³,[(a)\tilde]+[(b)\tilde]){(\mathbb{V}^{3},\tilde{\alpha}+\tilde{\beta})} .     

The Ground State and the Long-Time Evolution in the CMC Einstein Flow

Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume ${\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)}Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume V(k)=(\frac-k3)³Vol_g(k)(S){\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)} is monotonically decreasing in the expanding direction and bounded below by V_inf=(\frac-16Y(S))^\frac32{\mathcal{V}_{\rm \inf}=\left(\frac{-1}{6}Y(\Sigma)\right)^{\frac{3}{2}}}. Inspired by this fact we define the ground state of the manifold Σ as “the limit” of any sequence of CMC states {(g _i , K _i )} satisfying: (i) k _i  = −3, (ii) V_iˉ V_inf{\mathcal{V}_{i}\downarrow \mathcal{V}_{\rm inf}}, (iii) Q ₀((g _i , K _i )) ≤ Λ, where Q ₀ is the Bel–Robinson energy and Λ is any arbitrary positive constant. We prove that (as a geometric state) the ground state is equivalent to the Thurston geometrization of Σ. Ground states classify naturally into three types. We provide examples for each class, including a new ground state (the Double Cusp) that we analyze in detail. Finally, consider a long time and cosmologically normalized flow ([(g)\tilde],[(K)\tilde])(s)=((\frac-k3)²g,(\frac-k3)K){(\tilde{g},\tilde{K})(\sigma)=\left(\left(\frac{-k}{3}\right)^{2}g,\left(\frac{-k}{3}\right)K\right)}, where s = -ln(-k) ? [a,￥){\sigma=-\ln (-k)\in [a,\infty)}. We prove that if [(E₁)\tilde]=E₁(([(g)\tilde],[(K)\tilde])) ￡ L{\tilde{\mathcal{E}_{1}}=\mathcal{E}_{1}((\tilde{g},\tilde{K}))\leq \Lambda} (where E₁=Q₀+Q₁{\mathcal{E}_{1}=Q_{0}+Q_{1}}, is the sum of the zero and first order Bel–Robinson energies) the flow ([(g)\tilde],[(K)\tilde])(s){(\tilde{g},\tilde{K})(\sigma)} persistently geometrizes the three-manifold Σ and the geometrization is the ground state if Vˉ V_inf{\mathcal{V}\downarrow \mathcal{V}_{\rm inf}}.     

On Lie derivations of Lie ideals of prime algebras

LetF be a commutative ring with 1, letA, be a primeF-algebra with Martindale extended centroidC and with central closureA _c and letR be a noncentral Lie ideal of the algebraA generatingA. Further, letZ(R) be the center ofR, let be the factor Lie algebra and let δ: be a Lie derivation. Suppose that char(A) ≠ 2 andA does not satisfySt ₁₄, the standard identity of degree 14. We show thatR ΩC =Z(R) and there exists a derivation of algebrasD:A →A _c such that for allx∈R. Our result solves an old problem of Herstein.