Quasi-convex Mappings on the Unit Polydisk in $\mathbb{C}^{n}$

In this paper, the sharp estimates of all homogeneous expansions for f are established, where f(z) = (f ₁(z), f ₂(z), …, f _n (z))′ is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in ℂ ⁿ and

Open orbifold Gromov-Witten invariants of {[\mathbb{C}^3/\mathbb{Z}_n]}: localization and mirror symmetry

We develop a mathematical framework for the computation of open orbifold Gromov-Witten invariants of [\mathbbC³/\mathbbZ_n]{[\mathbb{C}^3/\mathbb{Z}_n]} and provide extensive checks with predictions from open string mirror symmetry. To this aim, we set up a computation of open string invariants in the spirit of Katz-Liu [23], defining them by localization. The orbifold is viewed as an open chart of a global quotient of the resolved conifold, and the Lagrangian as the fixed locus of an appropriate anti-holomorphic involution. We consider two main applications of the formalism. After warming up with the simpler example of [\mathbbC³/\mathbbZ₃]{[\mathbb{C}^3/\mathbb{Z}_3]} , where we verify physical predictions of Bouchard, Klemm, Mari?o and Pasquetti [4,5], the main object of our study is the richer case of [\mathbbC³/\mathbbZ₄]{[\mathbb{C}^3/\mathbb{Z}_4]} , where two different choices are allowed for the Lagrangian. For one choice, we make numerical checks to confirm the B-model predictions; for the other, we prove a mirror theorem for orbifold disc invariants, match a large number of annulus invariants, and give mirror symmetry predictions for open string invariants of genus ≤ 2.     

Combinatorics of character formulas for the lie superalgebra \mathfrak{g}\mathfrak{l}\left( {m,n} \right)

Let \mathfrakg \mathfrak{g} be the Lie superalgebra \mathfrakg\mathfrakl( m,n ) \mathfrak{g}\mathfrak{l}\left( {m,n} \right) . Algorithms for computing the composition factors and multiplicities of Kac modules for \mathfrakg \mathfrak{g} were given by the second author, [12] and by J. Brundan [1]. We give a combinatorial proof of the equivalence between the two algorithms. The proof uses weight and cap diagrams introduced by Brundan and C. Stroppel, and cancelations between paths in a graph G \mathcal{G} defined using these diagrams. Each vertex of G \mathcal{G} corresponds to a highest weight of a finite dimensional simple module, and each edge is weighted by a nonnegative integer. If E \mathcal{E} is the subgraph of G \mathcal{G} obtained by deleting all edges of positive weight, then E \mathcal{E} is the graph that describes nonsplit extensions between simple highest weight modules. We also give a procedure for finding the composition factors of any Kac module, without cancelation. This procedure leads to a second proof of the main result.     

$\mathcal{P}$-kernel normal systems for $\mathcal{P}$-inversive semigroups

As a generalization of Preston’s kernel normal systems, P\mathcal{P}-kernel normal systems for P\mathcal{P}-inversive semigroups are introduced, and strongly regular P\mathcal{P}-congruences on P\mathcal{P}-inversive semigroups in terms of their P\mathcal{P}-kernel normal systems are characterized. These results generalize the corresponding results for P\mathcal{P}-regular semigroups and P\mathcal{P}-inversive semigroups.     

Period functions for $ {\mathcal{C}^0} $- and $ {\mathcal{C}^1} $-flows

Let F:M ×\mathbbR ? M {\mathbf{F}}:M \times \mathbb{R} \to M be a continuous flow on a manifold M, let V ⊂ M be an open subset, and let x:V ? \mathbbR \xi :V \to \mathbb{R} be a continuous function. We say that ξ is a period function if F(x, ξ(x)) = x for all x ∈ V. Recently, for any open connected subset V ⊂ M; the author has described the structure of the set P(V) of all period functions on V. Assume that F is topologically conjugate to some C¹ {\mathcal{C}^1} -flow. It is shown in this paper that, in this case, the period functions of F satisfy some additional conditions that, generally speaking, are not satisfied for general continuous flows.     

Frames and their associated $\emph{H}_{{\kern-2pt}\emph{F}}^{\emph{p}}$-subspaces

Given a frame F = {f _j } for a separable Hilbert space H, we introduce the linear subspace H^p_FH^{p}_{F} of H consisting of elements whose frame coefficient sequences belong to the ℓ ^p -space, where 1 ≤ p < 2. Our focus is on the general theory of these spaces, and we investigate different aspects of these spaces in relation to reconstructions, p-frames, realizations and dilations. In particular we show that for closed linear subspaces of H, only finite dimensional ones can be realized as H^p_FH^{p}_{F}-spaces for some frame F. We also prove that with a mild decay condition on the frame F the frame expansion of any element in H_F^pH_{F}^{p} converges in both the Hilbert space norm and the ||·|| _{F, p} -norm which is induced by the ℓ ^p -norm.     

A hyperbolicity criterion for subgroups of $\mathcal{RF}(G)$

This paper continues the investigation of the groups RF(G)\mathcal{RF}(G) first introduced in the forthcoming book of Chiswell and Müller “A Class of Groups Universal for Free ℝ-Tree Actions” and in the article by Müller and Schlage-Puchta (Abh. Math. Semin. Univ. Hambg. 79:193–227, 2009). We establish a criterion for a family {H_s}\{\mathcal{H}_{\sigma}\} of hyperbolic subgroups H_s ￡ RF(G)\mathcal{H}_{\sigma}\leq\mathcal{RF}(G) to generate a hyperbolic subgroup isomorphic to the free product of the H_s\mathcal{H}_{\sigma} (Theorem 1.2), as well as a local-global principle for local incompatibility (Theorem 4.1). In conjunction with the theory of test functions as developed by Müller and Schlage-Puchta (Abh. Math. Semin. Univ. Hambg. 79:193–227, 2009), these results allow us to obtain a necessary and sufficient condition for a free product of real groups to embed as a hyperbolic subgroup in RF(G)\mathcal{RF}(G) for a given group G (Corollary 5.4). As a further application, we show that the centralizers associated with a family of pairwise locally incompatible cyclically reduced functions in RF(G)\mathcal{RF}(G) generate a hyperbolic subgroup isomorphic to the free product of these centralizers (Corollary 5.2).     

A Class of Integral Operators on the Unit Ball of \mathbb{C}^{n}

For real parameters a, b, c, and t, where c is not a nonpositive integer, we determine exactly when the integral operator

Dimension Free Estimates for Riesz Transforms Associated to the Harmonic Oscillator on \mathbb{R}^{n}

Let L=?Δ+|ξ|² be the harmonic oscillator on $\mathbb{R}^{n}Let L=−Δ+|ξ|² be the harmonic oscillator on \mathbbRⁿ\mathbb{R}^{n} , with the associated Riesz transforms R_2j−1=(∂/∂ξ_j)L^−1/2,R_2j=ξ_jL^−1/2. We give a shorter proof of a recent result of Harboure, de Rosa, Segovia, Torrea: For 1<p<∞ and a dimension free constant C_p,

||(?k=12n|Rk(f)|2)1/2||Lp(\mathbbRn,dx)\leqslant Cp||f||Lp(\mathbbRn,dx).\bigg\Vert \bigg(\sum_{k=1}^{2n}\vert R_{k}(f)\vert ^{2}\bigg)^{{1}/{2}}\bigg\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}\leqslant C_{p}\Vert f\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}.      11. Characterizations of Bent and Almost Bent Function on {\mathbb{Z}_p^2}      Xiyong Zhang  Hua Guo  Zongsheng Gao 《Graphs and Combinatorics》2011,27(4):603-620 Bent and almost-bent functions on \mathbbZp2{\mathbb{Z}_p^2} are studied in this paper. By calculating certain exponential sum and using a technique due to Hou (Finite Fields Appl 10:566–582, 2004), we obtain a degree bound for quasi-bent functions, and prove that almost-bent functions on \mathbbZp2{\mathbb{Z}_p^2} are equivalent to a degenerate quadratic form. From the viewpoint of relative difference sets, we also characterize bent functions on \mathbbZp2{\mathbb{Z}_p^2} in two classes of M{\mathcal{M}} ’s and PS{\mathcal{PS}} ’s, and show that the graph set corresponding to a bent function on \mathbbZp2{\mathbb{Z}_p^2} can be written as the sum of a graph set of M{\mathcal{M}} ’s type bent function and another group ring element. By using our characterization and some technique of permutation polynomial, we obtain the result: a bent function must be of M{\mathcal{M}} ’s type if its corresponding set contains more than (p − 3)/2 flats. A problem proposed by Ma and Pott (J Algebra 175:505–525, 1995) is therefore partially answered.      12. On the Convergence of Products γ^-sh1hn in the Adams Spectral Sequence      Xiu Gui LIU 《数学学报(英文版)》2007,23(6):1025-1032 Abstract Let A be the mod p Steenrod algebra and S the sphere spectrum localized at p, where p is an odd prime. In 2001 Lin detected a new family in the stable homotopy of spheres which is represented by （b0hn-h1bn-1）∈ ExtA^3,（p^n＋p）q（Zp,Zp） in the Adams spectral sequence. At the same time, he proved that i.（hlhn） ∈ExtA^2,（p^n＋P）q（H^＊M, Zp） is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element ξn∈π（p^n＋p）q-2M. In this paper, with Lin＇s results, we make use of the Adams spectral sequence and the May spectral sequence to detect a new nontrivial family of homotopy elements jj′j^-γsi^-i′ξn in the stable homotopy groups of spheres. The new one is of degree p^nq ＋ sp^2q ＋ spq ＋ （s - 2）q ＋ s - 6 and is represented up to a nonzero scalar by hlhnγ-s in the E2^s＋2,＊-term of the Adams spectral sequence, where p ≥ 7, q = 2（p - 1）, n ≥ 4 and 3 ≤ s 〈 p.      13. The Cesàro operator on the Bergman space $ A^{2} (\mathbb{D}) $      V. Glass Miller  T. L. Miller 《Archiv der Mathematik》2002,78(5):409-416 The Cesàro operator is shown to be subdecomposable on the Bergman spaces Ap (\mathbbD) A^{p} (\mathbb{D}) for p \geqq 2 p \geqq 2 , extending a result of [12] to the case that p < 4. For A2 (\mathbbD) A^{2} (\mathbb{D}) , we show that Cesàro operator is in fact subscalar, but in contrast to the situation in the Hardy space, C |A2 C |_{A^2} fails to be hyponormal.      14. Heat flows for a nonconvex Signorini type problem in $$ {\mathbb{R}^{N}} $$      A. Arkhipova 《Journal of Mathematical Sciences》2011,176(6):732-758 We prove the existence of a global heat flow u : Ω ×  \mathbbR+ ? \mathbbRN {\mathbb{R}^{+}} \to {\mathbb{R}^{N}}, N > 1, satisfying a Signorini type boundary condition u(∂Ω ×  \mathbbR+ {\mathbb{R}^{+}}) ⊂  \mathbbRn {\mathbb{R}^{n}}), n \geqslant 2 n \geqslant 2 , and \mathbbRN {\mathbb{R}^{N}}) with boundary ∂ [`(W)] \bar{\Omega } such that φ(∂Ω) ⊂ \mathbbRN {\mathbb{R}^{N}} is given by a smooth noncompact hypersurface S. Bibliography: 30 titles.      15. The Kneser-Tits conjecture for groups with Tits-index {\text{E}}_{8,2}^{66} over an arbitrary field      R. Parimala  J.-P. Tignol  R. M. Weiss 《Transformation Groups》2012,17(1):209-231 We prove: (1) The group of multipliers of similitudes of a 12-dimensional anisotropic quadratic form over a field K with trivial discriminant and split Clifford invariant is generated by norms from quadratic extensions E/K such that q E is hyperbolic. (2) If G is the group of K-rational points of an absolutely simple algebraic group whose Tits index is \textE8,266 {\text{E}}_{8,2}^{66} , then G is generated by its root groups, as predicted by the Kneser-Tits conjecture.      16. Ring of invariants of general linear group over local ring \mathbb{Z}_{p^m }      Jizhu Nan  Yin Chen 《Frontiers of Mathematics in China》2011,6(5):887-899 Let \mathbbZpm \mathbb{Z}_{p^m } be the ring of integers modulo p m , where p is a prime and m ⩾ 1. The general linear group GL n ( \mathbbZpm \mathbb{Z}_{p^m } ) acts naturally on the polynomial algebra A n := \mathbbZpm \mathbb{Z}_{p^m } [x 1, …, x n ]. Denote by AnGL2 (\mathbbZpm ) A_n^{GL_2 (\mathbb{Z}_{p^m } )} the corresponding ring of invariants. The purpose of the present paper is to calculate this invariant ring. Our results also generalize the classical Dickson’s theorem.      17. On p-chief factors and extensions of $ \mathbb{K} $ G-modules      Conchita  Martínez-Pérez 《Archiv der Mathematik》2003,80(1):25-36 The subject of this paper is the relationship between the set of chief factors of a finite group G and extensions of an irreducible \mathbbK \mathbb{K} G-module U ( \mathbbK \mathbb{K} a field). Let H / L be a p-chief factor of G. We prove that, if H / L is complemented in a vertex of U, then there is a short exact sequence of Ext-functors for the module U and any \mathbbK \mathbb{K} G-module V. In some special cases, we prove the converse, which is false in general. We also consider the intersection of the centralizers of all the extensions of U by an irreducible module and provide new bounds for this group.      18. $$ H^{p}_{w} - L^{p}_{w} $$ Boundedness of Marcinkiewicz Integral      Chin-Cheng Lin  Ying-Chieh Lin 《Integral Equations and Operator Theory》2007,58(1):87-98 The Marcinkiewicz integral is essentially a Littlewood-Paley g-function, which plays a very important role in harmonic analysis. In this paper we give weaker smoothness conditions assumed on Ω to imply the boundedness of the Marcinkiewicz integral operator μΩ, where w belongs to the Muckenhoupt weight class.      19. Analytic criterion for linear convexity of Hartogs domains with smooth boundary in $ {\mathbb{H}^2} $      T. M. Osipchuk  M. V. Tkachuk 《Ukrainian Mathematical Journal》2011,63(2):266-277 We establish a criterion for the local linear convexity of sets in the two-dimensional quaternion space \mathbbH2 {\mathbb{H}^2} that are analogs of bounded Hartogs domains with smooth boundary in the two-dimensional complex space \mathbbC2 {\mathbb{C}^2} .      20. Truncated Complex Moment Problems with a $ \widetilde{zz} $ Relation      Lawrence A. Fialkow 《Integral Equations and Operator Theory》2003,45(4):405-435 We solve the truncated complex moment problem for measures supported on the variety K o \mathcal{K}\equiv { z ? \in C: z [(z)\tilde]\widetilde{z} = A+Bz+C [(z)\tilde]\widetilde{z} +Dz 2 ,D 1 \neq 0}. Given a doubly indexed finite sequence of complex numbers g o g(2n):g00,g01,g10,?,g0,2n,g1,2n-1,?,g2n-1,1,g2n,0 \gamma\equiv\gamma^{(2n)}:\gamma_{00},\gamma_{01},\gamma_{10},\ldots,\gamma_{0,2n},\gamma_{1,2n-1},\ldots,\gamma_{2n-1,1},\gamma_{2n,0} , there exists a positive Borel measure m\mu supported in K \mathcal{K} such that gij=ò[`(z)]izj dm (0 ￡ 1+j ￡ 2n) \gamma_{ij}=\int\overline{z}^{i}z^{j}\,d\mu\,(0\leq1+j\leq2n) if and only if the moment matrix M(n)( g\gamma ) is positive, recursively generated, with a column dependence relation Z [(Z)\tilde]\widetilde{Z} = A1+BZ +C [(Z)\tilde]\widetilde{Z} +DZ 2, and card V(g) 3\mathcal{V}(\gamma)\geq rank M(n), where V(g)\mathcal{V}(\gamma) is the variety associated to g \gamma . The last condition may be replaced by the condition that there exists a complex number gn,n+1 \gamma_{n,n+1} satisfying gn+1,n o [`(g)]n,n+1=Agn,n-1+Bgn,n+Cgn+1,n-1+Dgn,n+1 \gamma_{n+1,n}\equiv\overline{\gamma}_{n,n+1}=A\gamma_{n,n-1}+B\gamma_{n,n}+C\gamma_{n+1,n-1}+D\gamma_{n,n+1} . We combine these results with a recent theorem of J. Stochel to solve the full complex moment problem for K \mathcal{K} , and we illustrate the connection between the truncated and full moment problems for other varieties as well, including the variety z k = p(z, [(Z)\tilde] \widetilde{Z} ), deg p < k.

Characterizations of Bent and Almost Bent Function on {\mathbb{Z}_p^2}

Bent and almost-bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} are studied in this paper. By calculating certain exponential sum and using a technique due to Hou (Finite Fields Appl 10:566–582, 2004), we obtain a degree bound for quasi-bent functions, and prove that almost-bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} are equivalent to a degenerate quadratic form. From the viewpoint of relative difference sets, we also characterize bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} in two classes of M{\mathcal{M}} ’s and PS{\mathcal{PS}} ’s, and show that the graph set corresponding to a bent function on \mathbbZ_p²{\mathbb{Z}_p^2} can be written as the sum of a graph set of M{\mathcal{M}} ’s type bent function and another group ring element. By using our characterization and some technique of permutation polynomial, we obtain the result: a bent function must be of M{\mathcal{M}} ’s type if its corresponding set contains more than (p − 3)/2 flats. A problem proposed by Ma and Pott (J Algebra 175:505–525, 1995) is therefore partially answered.     

On the Convergence of Products γ^-sh1hn in the Adams Spectral Sequence

Abstract Let A be the mod p Steenrod algebra and S the sphere spectrum localized at p, where p is an odd prime. In 2001 Lin detected a new family in the stable homotopy of spheres which is represented by （b0hn-h1bn-1）∈ ExtA^3,（p^n＋p）q（Zp,Zp） in the Adams spectral sequence. At the same time, he proved that i.（hlhn） ∈ExtA^2,（p^n＋P）q（H^＊M, Zp） is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element ξn∈π（p^n＋p）q-2M. In this paper, with Lin＇s results, we make use of the Adams spectral sequence and the May spectral sequence to detect a new nontrivial family of homotopy elements jj′j^-γsi^-i′ξn in the stable homotopy groups of spheres. The new one is of degree p^nq ＋ sp^2q ＋ spq ＋ （s - 2）q ＋ s - 6 and is represented up to a nonzero scalar by hlhnγ-s in the E2^s＋2,＊-term of the Adams spectral sequence, where p ≥ 7, q = 2（p - 1）, n ≥ 4 and 3 ≤ s 〈 p.     

The Cesàro operator on the Bergman space $ A^{2} (\mathbb{D}) $

The Cesàro operator is shown to be subdecomposable on the Bergman spaces A^p (\mathbbD) A^{p} (\mathbb{D}) for p \geqq 2 p \geqq 2 , extending a result of [12] to the case that p < 4. For A² (\mathbbD) A^{2} (\mathbb{D}) , we show that Cesàro operator is in fact subscalar, but in contrast to the situation in the Hardy space, C |_A² C |_{A^2} fails to be hyponormal.     

Heat flows for a nonconvex Signorini type problem in $$ {\mathbb{R}^{N}} $$

We prove the existence of a global heat flow u : Ω ×  \mathbbR⁺ ? \mathbbR^N {\mathbb{R}^{+}} \to {\mathbb{R}^{N}}, N > 1, satisfying a Signorini type boundary condition u(∂Ω ×  \mathbbR⁺ {\mathbb{R}^{+}}) ⊂  \mathbbRⁿ {\mathbb{R}^{n}}), n \geqslant 2 n \geqslant 2 , and \mathbbR^N {\mathbb{R}^{N}}) with boundary ∂ [`(W)] \bar{\Omega } such that φ(∂Ω) ⊂ \mathbbR^N {\mathbb{R}^{N}} is given by a smooth noncompact hypersurface S. Bibliography: 30 titles.     

The Kneser-Tits conjecture for groups with Tits-index {\text{E}}_{8,2}^{66} over an arbitrary field

We prove: (1) The group of multipliers of similitudes of a 12-dimensional anisotropic quadratic form over a field K with trivial discriminant and split Clifford invariant is generated by norms from quadratic extensions E/K such that q _E is hyperbolic. (2) If G is the group of K-rational points of an absolutely simple algebraic group whose Tits index is \textE_8,2⁶⁶ {\text{E}}_{8,2}^{66} , then G is generated by its root groups, as predicted by the Kneser-Tits conjecture.     

Ring of invariants of general linear group over local ring \mathbb{Z}_{p^m }

Let \mathbbZ_p^m \mathbb{Z}_{p^m } be the ring of integers modulo p ^m , where p is a prime and m ⩾ 1. The general linear group GL _n ( \mathbbZ_p^m \mathbb{Z}_{p^m } ) acts naturally on the polynomial algebra A _n := \mathbbZ_p^m \mathbb{Z}_{p^m } [x ₁, …, x _n ]. Denote by A_n^{GL₂ (\mathbbZ_pm )} A_n^{GL_2 (\mathbb{Z}_{p^m } )} the corresponding ring of invariants. The purpose of the present paper is to calculate this invariant ring. Our results also generalize the classical Dickson’s theorem.     

On p-chief factors and extensions of $ \mathbb{K} $ G-modules

The subject of this paper is the relationship between the set of chief factors of a finite group G and extensions of an irreducible \mathbbK \mathbb{K} G-module U ( \mathbbK \mathbb{K} a field). Let H / L be a p-chief factor of G. We prove that, if H / L is complemented in a vertex of U, then there is a short exact sequence of Ext-functors for the module U and any \mathbbK \mathbb{K} G-module V. In some special cases, we prove the converse, which is false in general. We also consider the intersection of the centralizers of all the extensions of U by an irreducible module and provide new bounds for this group.     

$$ H^{p}_{w} - L^{p}_{w} $$ Boundedness of Marcinkiewicz Integral

The Marcinkiewicz integral is essentially a Littlewood-Paley g-function, which plays a very important role in harmonic analysis. In this paper we give weaker smoothness conditions assumed on Ω to imply the boundedness of the Marcinkiewicz integral operator μ_Ω, where w belongs to the Muckenhoupt weight class.     

Analytic criterion for linear convexity of Hartogs domains with smooth boundary in $ {\mathbb{H}^2} $

We establish a criterion for the local linear convexity of sets in the two-dimensional quaternion space \mathbbH² {\mathbb{H}^2} that are analogs of bounded Hartogs domains with smooth boundary in the two-dimensional complex space \mathbbC² {\mathbb{C}^2} .     

Truncated Complex Moment Problems with a $ \widetilde{zz} $ Relation

We solve the truncated complex moment problem for measures supported on the variety K o \mathcal{K}\equiv { z ? \in C: z [(z)\tilde]\widetilde{z} = A+Bz+C [(z)\tilde]\widetilde{z} +Dz ² ,D 1 \neq 0}. Given a doubly indexed finite sequence of complex numbers g o g⁽²ⁿ⁾:g₀₀,g₀₁,g₁₀,?,g_0,2n,g_1,2n-1,?,g_2n-1,1,g_2n,0 \gamma\equiv\gamma^{(2n)}:\gamma_{00},\gamma_{01},\gamma_{10},\ldots,\gamma_{0,2n},\gamma_{1,2n-1},\ldots,\gamma_{2n-1,1},\gamma_{2n,0} , there exists a positive Borel measure m\mu supported in K \mathcal{K} such that g_ij=ò[`(z)]<sup>i</sup>z<sup>j</sup> dm (0 ￡ 1+j ￡ 2n) \gamma_{ij}=\int\overline{z}^{i}z^{j}\,d\mu\,(0\leq1+j\leq2n) if and only if the moment matrix M(n)( g\gamma ) is positive, recursively generated, with a column dependence relation Z [(Z)\tilde]\widetilde{Z} = A1+BZ +C [(Z)\tilde]\widetilde{Z} +DZ <sup>2</sup>, and card V(g) 3\mathcal{V}(\gamma)\geq rank M(n), where V(g)\mathcal{V}(\gamma) is the variety associated to g \gamma . The last condition may be replaced by the condition that there exists a complex number g<sub>n,n+1</sub> \gamma_{n,n+1} satisfying g<sub>n+1,n</sub> o [`(g)]_n,n+1=Ag_n,n-1+Bg_n,n+Cg_n+1,n-1+Dg_n,n+1 \gamma_{n+1,n}\equiv\overline{\gamma}_{n,n+1}=A\gamma_{n,n-1}+B\gamma_{n,n}+C\gamma_{n+1,n-1}+D\gamma_{n,n+1} . We combine these results with a recent theorem of J. Stochel to solve the full complex moment problem for K \mathcal{K} , and we illustrate the connection between the truncated and full moment problems for other varieties as well, including the variety z ^k = p(z, [(Z)\tilde] \widetilde{Z} ), deg p < k.