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Let$K$be the class of trigonometric series of power type, i.e. Taylor series $$\sum\nolimits_{n = 0}^\infty {c_n z^n } $$ for$z$=$e$^{$ix$}, whose partial sums for all$x$in$E$, where$E$is a nondenumerable subset of [0, 2π), lie on a$finite$number of circles (a priori depending on$x$) in the complex plane. The main result of this paper is that for every member of the class$K$, there exist a complex number ω, |ω|=1, and two positive integers $$\nu , \kappa , \nu < \kappa $$ , such that for the coefficients$c$_{$n$}we have: $$c_{\mu + \lambda \left( {\kappa - \nu } \right)} = c_\mu \omega ^\lambda , \mu = \nu ,\nu + 1, \ldots , \kappa - 1, \lambda = 1,2,3, \ldots $$ . Thus, every member of the class$K$has (with minor modifications) a representation of the form: $$P(x)\sum\nolimits_{n = 0}^\infty {e^{iknx} ,} $$ where P($x$) is a suitable trigonometric polynomial and$k$a positive integer. The proof is elementary but rather long. This result is closely related to a theorem of Marcinkiewicz and Zygmund on the circular structure of the set of limit points of the sequence of partial sums of ($C$, 1) summable Taylor series.

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The influences of non-Oberbeck-Boussinesq (NOB) effects on flow instabilities and bifurcation characteristics of Rayleigh-Benard convection are examined. The working fluid is air with reference Prandtl number Pr = 0.71 and contained in two-dimensional rigid cavities of finite aspect ratios. The fluid flow is governed by the low-Mach-number equations, accounting for the NOB effects due to large temperature difference involving flow compressibility and variations of fluid viscosity and thermal conductivity with temperature. The intensity of NOB effects is measured by the dimensionless temperature differential epsilon. Linear stability analysis of the thermal conduction state is performed. An epsilon(2) scaling of the leading-order corrections of critical Rayleigh number Ra-cr and disturbance growth rate sigma due to NOB effects is identified, which is a consequence of an intrinsic symmetry of the system. The influences of weak NOB effects on flow instabilities are further studied by perturbation expansion of linear stability equations with regard to epsilon, and then the influence of aspect ratio A is investigated in detail. NOB effects are found to enhance (weaken) flow stability in large (narrow) cavities. Detailed contributions of compressibility, viscosity and buoyancy actions on disturbance kinetic energy growth are identified quantitatively by energy analysis. Besides, a weakly nonlinear theory is developed based on centre-manifold reduction to investigate the NOB influences on bifurcation characteristics near convection onset, and amplitude equations are constructed for both codimension-one and -two cases. Rich bifurcation regimes are observed based on amplitude equations and also confirmed by direct numerical simulation. Weakly nonlinear analysis is useful for organizing and understanding these simulation results.

Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$, and let $L^k$ be the $k$-th tensor power of a CR complex line bundle $L$ over $X$. Given $q\in\set{0,1,\ldots,n-1}$, let $\Box^{(q)}_{b,k}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$ forms with values in $L^k$. For $\lambda\geq0$, let $\Pi^{(q)}_{k,\leq\lambda}:=E((-\infty,\lambda])$, where $E$ denotes the spectral measure of $\Box^{(q)}_{b,k}$. In this work, we prove that $\Pi^{(q)}_{k,\leq k^{-N_0}}F^*_k$, $F_k\Pi^{(q)}_{k,\leq k^{-N_0}}F^*_k$, $N_0\geq1$, admit asymptotic expansions with respect to $k$ on the non-degenerate part of the characteristic manifold of $\Box^{(q)}_{b,k}$, where $F_k$ is some kind of microlocal cut-off function. Moreover, we show that $F_k\Pi^{(q)}_{k,\leq0}F^*_k$ admits a full asymptotic expansion with respect to $k$ if $\Box^{(q)}_{b,k}$ has small spectral gap property with respect to $F_k$ and $\Pi^{(q)}_{k,\leq0}$ is $k$-negligible away the diagonal with respect to $F_k$. By using these asymptotics, we establish almost Kodaira embedding theorems on CR manifolds and Kodaira embedding theorems on CR manifolds with transversal CR $S^1$ action.