An isometrical ℂP<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>-theorem

Let M ⁿ (n ? 3) be a complete Riemannian manifold with sec _M ? 1, and let \(M_i^{n_i }\) (i = 1, 2) be two complete totally geodesic submanifolds in M. We prove that if n₁ + n₂ = n ? 2 and if the distance |M₁M₂| ? π/2, then M _i is isometric to \(\mathbb{S}^{n_i } /\mathbb{Z}_h\), \(\mathbb{C}P^{n_i /2}\), or \(\mathbb{C}P^{n_i /2} /\mathbb{Z}_2 \) with the canonical metric when n _i > 0; and thus, M is isometric to S ⁿ /? _h , ?P^n/2, or ?P^n/2/?₂ except possibly when n = 3 and \(M_1 (or M_2 )\mathop \cong \limits^{iso} \mathbb{S}^1 /\mathbb{Z}_h \) with h ? 2 or n = 4 and \(M_1 (or M_2 )\mathop \cong \limits^{iso} \mathbb{R}P^2 \).     

Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Let A :=(A_1, A_2) be a pair of expansive dilations and φ : R~n×R~m×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space H~φ_A(R~n× R~m) via the anisotropic Lusin-area function and establish its atomic characterization, the g-function characterization, the g_λ~*-function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of H~φ_A(R~n× R~m) is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet(φ, q, s), if T is a sublinear operator and maps all(φ, q, s)-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from H~φ_A(R~n× R~m) to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from H~φ_A(R~n× R~m) to L~φ(R~n× R~m)and from H~φ_A(R~n×R~m) to itself, whose kernels are adapted to the action of A. The results of this article essentially extend the existing results for weighted product Hardy spaces on R~n× R~m and are new even for classical product Orlicz-Hardy spaces.     

Kähler geometry of hyperbolic type on the manifold of nondegenerate <Emphasis Type="Italic">m</Emphasis>-pairs

A nondegenerate m-pair (A, Ξ) in an n-dimensional projective space ?P _n consists of an m-plane A and an (n ? m ? 1)-plane Ξ in ?P _n , which do not intersect. The set \(\mathfrak{N}_m^n \) of all nondegenerate m-pairs ?P _n is a 2(n ? m)(n ? m ? 1)-dimensional, real-complex manifold. The manifold \(\mathfrak{N}_m^n \) is the homogeneous space \(\mathfrak{N}_m^n = {{GL(n + 1,\mathbb{R})} \mathord{\left/ {\vphantom {{GL(n + 1,\mathbb{R})} {GL(m + 1,\mathbb{R})}}} \right. \kern-\nulldelimiterspace} {GL(m + 1,\mathbb{R})}} \times GL(n - m,\mathbb{R})\) equipped with an internal Kähler structure of hyperbolic type. Therefore, the manifold \(\mathfrak{N}_m^n \) is a hyperbolic analogue of the complex Grassmanian ?G _m,n = U(n+1)/U(m+1) × U(n?m). In particular, the manifold of 0-pairs \(\mathfrak{N}_m^n {{GL(n + 1,\mathbb{R})} \mathord{\left/ {\vphantom {{GL(n + 1,\mathbb{R})} {GL(1,\mathbb{R})}}} \right. \kern-\nulldelimiterspace} {GL(1,\mathbb{R})}} \times GL(n,\mathbb{R})\) is a hyperbolic analogue of the complex projective space ?P _n = U(n+1)/U(1) × U(n). Similarly to ?P _n , the manifold \(\mathfrak{N}_m^n \) is a Kähler manifold of constant nonzero holomorphic sectional curvature (relative to a hyperbolic metrics). In this sense, \(\mathfrak{N}_0^n \) is a hyperbolic spatial form. It was proved in [6] that the manifold of 0-pairs \(\mathfrak{N}_0^n \) is globally symplectomorphic to the total space T*?P _n of the cotangent bundle over the projective space ?P _n . A generalization of this result (see [7]) is as follows: the manifold of nondegenerate m-pairs \(\mathfrak{N}_m^n \) is globally symplectomorphic to the total space T*?G _m,n of the cotangent bundle over the Grassman manifold ?G _m,n of m-dimensional subspaces of the space ?P _n .In this paper, we study the canonical Kähler structure on \(\mathfrak{N}_m^n \). We describe two types of submanifolds in \(\mathfrak{N}_m^n \), which are natural hyperbolic spatial forms holomorphically isometric to manifolds of 0-pairs in ?P _m +1 and in ?P _n?m , respectively. We prove that for any point of the manifold \(\mathfrak{N}_m^n \), there exist a 2(n ? m)-parameter family of 2(m + 1)-dimensional hyperbolic spatial forms of first type and a 2(m + 1)-parameter family of 2(n ? m)-dimensional hyperbolic spatial forms of second type passing through this point. We also prove that natural hyperbolic spatial forms of first type on \(\mathfrak{N}_m^n \) are in bijective correspondence with points of the manifold \(\mathfrak{N}_{m + 1}^n \) and natural hyperbolic spatial forms of second type on \(\mathfrak{N}_m^n \) are in bijective correspondence with points of the manifolds \(\mathfrak{N}_{m + 1}^n \).     

The cohomology of orbit spaces of certain free circle group actions

Suppose that \(G =\mathbb{S}^1\) acts freely on a finitistic space X whose (mod p) cohomology ring is isomorphic to that of a lens space \(L^{2m-1}(p;q_1,\ldots,q_m)\) or \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\). The mod p index of the action is defined to be the largest integer n such that α ⁿ ?≠?0, where \(\alpha \,\epsilon\, H^2(X/G;\mathbb{Z}_p)\) is the nonzero characteristic class of the \(\mathbb{S}^1\)-bundle \(\mathbb{S}^1\hookrightarrow X\rightarrow X/G\). We show that the mod p index of a free action of G on \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\) is p???1, when it is defined. Using this, we obtain a Borsuk–Ulam type theorem for a free G-action on \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\). It is note worthy that the mod p index for free G-actions on the cohomology lens space is not defined.     

Classification of smooth structures on a homotopy complex projective space

We classify, up to diffeomorphism, all closed smooth manifolds homeomorphic to the complex projective n-space \(\mathbb {C}\textbf {P}^{n}\), where n=3 and 4. Let M²ⁿ be a closed smooth 2n-manifold homotopy equivalent to \(\mathbb {C}\textbf {P}^{n}\). We show that, up to diffeomorphism, M⁶ has a unique differentiable structure and M⁸ has at most two distinct differentiable structures. We also show that, up to concordance, there exist at least two distinct differentiable structures on a finite sheeted cover N²ⁿ of \(\mathbb {C}\textbf {P}^{n}\) for n=4,7 or 8 and six distinct differentiable structures on N¹⁰.     

Several extremal approximation problems for the characteristic function of a spherical layer

We discuss three interrelated extremal problems on the set P _n,m of algebraic polynomials of a given degree n on the unit sphere \(\mathbb{S}^{m - 1}\) of the Euclidean space ? ^m of dimension m ≥ 2. (1) Find the norm of the functional \(F\left( \eta \right) = F_h P_n = \int_{\mathbb{G}\left( \eta \right)} {P_n (x)dx}\), which is the integral over the spherical layer \(\mathbb{G}\left( \eta \right) = \left\{ {x = \left( {x_1 , \ldots ,x_m } \right) \in \mathbb{S}^{m - 1} :h' \leqslant x_m \leqslant h''} \right\}\) defined by a pair of real numbers η = (h′, h″), ?1 ≤ h′ < h″ ≤ 1, on the set P _n,m with the norm of the space \(L\left( {\mathbb{S}^{m - 1} } \right)\) of functions summable on the sphere. (2) Find the best approximation in \(L_\infty \left( {\mathbb{S}^{m - 1} } \right)\) of the characteristic function χ _η of the layer \(\mathbb{G}\left( \eta \right)\) by the subspace P _n,m ^⊥ of functions from \(L_\infty \left( {\mathbb{S}^{m - 1} } \right)\) that are orthogonal to the space of polynomials P _n,m . (3) Find the best approximation in the space \(L\left( {\mathbb{S}^{m - 1} } \right)\) of the function χ _η by the space of polynomials P _n,m . We present a solution of all three problems for the values h′ and h″ that are neighboring roots of the polynomial in a single variable of degree n + 1 that deviates least from zero in the space L ₁ ^φ (?1, 1) of functions summable on the interval (?1, 1) with ultraspherical weight φ(t) = (1 ? t ²) ^α , α = (m ? 3)/2. We study the respective one-dimensional problems in the space of functions summable on (?1, 1) with an arbitrary not necessarily ultraspherical weight.     

Sobolev spaces and approximation by affine spanning systems

We develop conditions on a Sobolev function \(\psi \in W^{m,p}({\mathbb{R}}^d)\) such that if \(\widehat{\psi}(0) = 1\) and ψ satisfies the Strang–Fix conditions to order m ? 1, then a scale averaged approximation formula holds for all \(f \in W^{m,p}({\mathbb{R}}^d)\) :

$ f(x) = \lim_{J \to \infty} \frac{1}{J} \sum_{j=1}^{J} \sum_{k \in {{\mathbb{Z}}}^d} c_{j,k}\psi(a_j x - k) \quad {\rm in} W^{m, p}({{\mathbb{R}}}^d).$

The dilations { a _j } are lacunary, for example a _j =  2 ^j , and the coefficients c _j,k are explicit local averages of f, or even pointwise sampled values, when f has some smoothness. For convergence just in \({W^{m - 1,p}({\mathbb{R}}^d)}\) the scale averaging is unnecessary and one has the simpler formula \(f(x) = \lim_{j \to \infty} \sum_{k \in {\mathbb{Z}}^d} c_{j,k}\psi(a_j x - k)\) . The Strang–Fix rates of approximation are recovered. As a corollary of the scale averaged formula, we deduce new density or “spanning” criteria for the small scale affine system \(\{\psi(a_j x - k) : j > 0, k \in {\mathbb{Z}}^d \}\) in \(W^{m,p}({\mathbb{R}}^d)\) . We also span Sobolev space by derivatives and differences of affine systems, and we raise an open problem: does the Gaussian affine system span Sobolev space?

     

Entropy and Widths of Multiplier Operators on Two-Point Homogeneous Spaces

In this article we continue the development of methods of estimating n-widths and entropy of multiplier operators begun in 1992 by A. Kushpel (Fourier Series and Their Applications, pp. 49–53, 1992; Ukr. Math. J. 45(1):59–65, 1993). Our main aim is to give an unified treatment for a wide range of multiplier operators Λ on symmetric manifolds. Namely, we investigate entropy numbers and n-widths of decaying multiplier sequences of real numbers \(\varLambda=\{\lambda_{k}\}_{k=1}^{\infty}\), |λ ₁|≥|λ ₂|≥?, \(\varLambda:L_{p}(\mathbb{M}^{d}) \rightarrow L_{q}(\mathbb{M}^{d})\) on two-point homogeneous spaces \(\mathbb{M}^{d}\): \(\mathbb{S}^{d}\), ? ^d (?), ? ^d (?), ? ^d (?), ?¹⁶(Cay). In the first part of this article, general upper and lower bounds are established for entropy and n-widths of multiplier operators. In the second part, different applications of these results are presented. In particular, we show that these estimates are order sharp in various important situations. For example, sharp order estimates are found for function sets with finite and infinite smoothness. We show that in the case of finite smoothness (i.e., |λ _k |?k ^?γ (lnk)^?ζ, γ/d>1, ζ≥0, k→∞), we have \(e_{n}(\varLambda U_{p}(\mathbb{S}^{d}), L_{q}(\mathbb{S}^{d})) \ll d_{n}(\varLambda U_{p}(\mathbb{S}^{d}), L_{q}(\mathbb{S}^{d}))\), n→∞, but in the case of infinite smoothness (i.e., \(|\lambda_{k}| \asymp e^{-\gamma k^{r}}\), γ>0, 0<r≤1, k→∞), we have \(e_{n}(\varLambda U_{p}(\mathbb{S}^{d}), L_{q}(\mathbb{S}^{d})) \gg d_{n}(\varLambda U_{p}(\mathbb{S}^{d}), L_{q}(\mathbb{S}^{d}))\), n→∞ for different p and q, where \(U_{p}(\mathbb{S}^{d})\) denotes the closed unit ball of \(L_{p}(\mathbb{S}^{d})\).     

Geometric and probabilistic analysis of convex bodies with unconditional structures,and associated spaces of operators

Let \({{\|\cdot\|}}\) be a norm on \({\mathbb{R}^n}\) and \({\|.\|_*}\) be the dual norm. If \({\|\cdot\|}\) has a normalized 1-symmetric basis \({\{e_i\}_{i=1}^n}\) then the following inequalities hold: for all \({x,y\in \mathbb{R}^n}\), \({\|x\|\cdot\|y\|_*\le \max(\|x\|_1\cdot\|y\|_\infty,\|x\|_\infty\cdot\|y\|_1)}\) and if the basis is only 1-unconditional and normalized then for all \({x \in \mathbb{R}^n}\) , \({\|x\|+\|x\|_{*}\leq \|x\|_1+\|x\|_\infty}\) . We consider other geometric generalizations and apply these results to get, as a special case, estimates on best random embeddings of k-dimensional Hilbert spaces in the spaces of nuclear operators \({{\mathcal N}(K,K)}\) of dimension n ², for all k = [λn ²] and 0 < λ < 1. We obtain universal upper bounds independent on the 1-symmetric norm \({\|.\|}\) for the products of pth moments

$\left( {\mathbb{E}} \left\|\sum_{i=1}^n f_i(\omega)\,e_i\right\|^p\cdot\, \mathbb {E} \left\|\sum_{i=1}^n f_i(\omega)\,e_i\right\|_*^p\right)^{1/p}$

for independent random variables {f _i (ω)}, and 1 ≤ p < ∞.

     

On Causal Invertibility with Respect to a Cone of Integral-Difference Operators in Vector Function Spaces

Let \(\mathbb{S}\) be a cone in ? ⁿ . A bounded linear operator T: L _p (? ⁿ ) → L _p (? ⁿ ) is said to be causal with respect to \(\mathbb{S}\) if the implication x(s) = 0 (s ε W ? \(\mathbb{S}\)) ? (Tx) (s) = 0 (s ε W ? \(\mathbb{S}\)) is valid for any x ε L _p (? ⁿ ) and any open subset W\(\subseteq\) ? ⁿ . The set of all causal operators is a Banach algebra. We describe the spectrum of the operator

$(Tx)(t) = \sum\limits_{n = 1}^\infty {a_n x(t - t_n )} + \int {\mathbb{S}g(s)x(t - s)ds,} \quad t \in \mathbb{R}^n ,$

in this algebra. Here x ranges in a Banach space \(\mathbb{E}\), the a _n are bounded linear operators in \(\mathbb{E}\), and the function g ranges in the set of bounded operators in \(\mathbb{E}\).

     

Nonarchimedean Green functions and dynamics on projective space

Let \({\varphi: \mathbb{P}^N_K\to\mathbb{P}^N_K}\) be a morphism of degree d ≥ 2 defined over a field K that is algebraically closed field and complete with respect to a nonarchimedean absolute value. We prove that a modified Green function \({\hat{g}_\varphi}\) associated to \({\varphi}\) is Hölder continuous on \({\mathbb{P}^N(K)}\) and that the Fatou set \({\mathcal{F}(\varphi)}\) of \({\varphi}\) is equal to the set of points at which \({\hat{g}_\Phi}\) is locally constant. Further, \({\hat{g}_\varphi}\) vanishes precisely on the set of points P such that \({\varphi}\) has good reduction at every point in the forward orbit \({\mathcal{O}_\varphi(P)}\) of P. We also prove that the iterates of \({\varphi}\) are locally uniformly Lipschitz on \({\mathcal{F}(\varphi)}\) .     

Anisotropic Hardy-Lorentz spaces and their applications

Let p ∈(0, 1], q ∈(0, ∞] and A be a general expansive matrix on Rn. We introduce the anisotropic Hardy-Lorentz space H~(p,q)_A(R~n) associated with A via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic and the molecular decompositions, the radial and the non-tangential maximal functions, and the finite atomic decompositions. All these characterizations except the ∞-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on Rn.As applications, we first prove that Hp,q A(Rn) is an intermediate space between H~(p1,q1)_A(Rn) and H~(p2,q2)_A(R~n) with 0 p1 p p2 ∞ and q1, q, q2 ∈(0, ∞], and also between H~(p,q1)_A(Rn) and H~(p,q2)_A(R~n) with p ∈(0, ∞)and 0 q1 q q2 ∞ in the real method of interpolation. We then establish a criterion on the boundedness of sublinear operators from H~(p,q)_A(R~n) into a quasi-Banach space; moreover, we obtain the boundedness of δ-type Calder′on-Zygmund operators from H~(p,∞)_A(R~n) to the weak Lebesgue space L~(p,∞)(R~n)(or to H~p_A(R~n)) in the ln λcritical case, from H~(p,q)_A(R~n) to L~(p,q)(R~n)(or to H~(p,q)_A(R~n)) with δ∈(0,(lnλ)/(ln b)], p ∈(1/(1+,δ),1] and q ∈(0, ∞], as well as the boundedness of some Calderon-Zygmund operators from H~(p,q)_A(R~n) to L~(p,∞)(R~n), where b := | det A|,λ_:= min{|λ| : λ∈σ(A)} and σ(A) denotes the set of all eigenvalues of A.     

Interpolation of sum and intersection spaces of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-type and applications to the Stokes problem in general unbounded domains

In a general unbounded uniform C ²-domain \({\Omega \subset \mathbb{R}^n, n \geq 3}\) , and \({1\leq q\leq \infty}\) consider the spaces \({\tilde{L}^q(\Omega)}\) defined by \({\tilde{L^q}(\Omega) := \left\{\begin{array}{ll}L^q(\Omega)+L^2(\Omega),\quad q < 2, \\ L^q(\Omega)\cap L^2(\Omega),\quad q\geq 2, \end{array}\right.}\) and corresponding subspaces of solenoidal vector fields, \({\tilde{L}^q_\sigma(\Omega)}\) . By studying the complex and real interpolation spaces of these we derive embedding properties for fractional order spaces related to the Stokes problem and L ^p ? L ^q -type estimates for the corresponding semigroup.     

Arens regularity of projective tensor products

For completely contractive Banach algebras A and B (respectively operator algebras A and B), the necessary and sufficient conditions for the operator space projective tensor product \({A\widehat{\otimes}B}\) (respectively the Haagerup tensor product \({A\otimes^{h}B}\)) to be Arens regular are obtained. Using the non-commutative Grothendieck inequality, we show that, for C*-algebras A and B, \({A\otimes^{\gamma} B}\) is Arens regular if \({A\widehat{\otimes}B}\) and \({A\widehat{\otimes}B^{op}}\) are Arens regular whereas \({A\widehat{\otimes}B}\) is Arens regular if and only if \({A\otimes^{h}B}\) and \({B\otimes^{h}A}\) are, where \({\otimes^h}\), \({\otimes^{\gamma}}\), and \({\widehat{\otimes}}\) are the Haagerup, the Banach space projective tensor norm, and the operator space projective tensor norm, respectively.     

On projective N-curvature inheritance

The concept of a Lie recurrence was introduced by the first author?[6]. It is an infinitesimal transformation $\overline{x}^{i}={x}^{i}+\varepsilon {v}^{i}({x}^{j})$ with respect to which the Lie derivative of a curvature tensor is proportional to itself. Apart from other results related to a Lie recurrence, it was established that the Weyl projective curvature tensor is Lie recurrent with respect to a Lie recurrence but its converse is not necessarily true. However, an infinitesimal transformation with respect to which the Weyl projective curvature tensor and the Ricci tensor are Lie recurrent, is necessarily a Lie recurrence. Singh?[12] studied an infinitesimal transformation with respect to which the Lie derivative of the curvature tensor is proportional to itself and called such transformation as curvature inheritance. Obviously, a curvature inheritance is nothing but a Lie recurrence. Singh?[13] also considered a curvature inheritance which is a projective motion and called it a projective curvature inheritance. Gatoto and Singh [1,2] studied $\widetilde{K}$ -curvature inheritance and projective $\widetilde{K}$ -curvature inheritance. Pandey and Pandey?[9] studied $\widetilde{K}$ projective Lie recurrence. Mishra and Yadav?[3] studied projective curvature inheritance in an NP-F _n . In the present paper we have established that an infinitesimal transformation in a Finsler space is Lie recurrence if and only if the normal projective curvature tensor is Lie recurrent. A part from this result we have generalized almost all theorems of Mishra and Yadav?[3].     

Infinitesimal form boundedness and Trudinger’s subordination for the Schrödinger operator

We give explicit analytic criteria for two problems associated with the Schrödinger operator H=-Δ+Q on L²(? ⁿ ) where Q∈D’(? ⁿ ) is an arbitrary real- or complex-valued potential.

First, we obtain necessary and sufficient conditions on Q so that the quadratic form \(\langle{Q}\cdot,\ \cdot\rangle\) has zero relative bound with respect to the Laplacian. For Q∈L¹_loc(? ⁿ ), this property can be expressed in the form of the integral inequality:

$\left\vert\int_{\mathbb{R}^n} |u(x)|^2 Q(x) dx \right\vert\leq\epsilon\| \nabla u \|^2_{L^2(\mathbb{R}^n)} + C(\epsilon) \|u \|^2_{L^2(\mathbb{R}^n)}, \quad\forall u \in C^{\infty}_0(\mathbb{R}^n),$

for an arbitrarily small ε>0 and some C(ε)>0. One of the major steps here is the reduction to a similar inequality with nonnegative function \(|\nabla(1-\Delta)^{-1} Q|^2 + |(1-\Delta)^{-1} Q|\) in place of Q. This provides a complete solution to the infinitesimal form boundedness problem for the Schrödinger operator, and leads to new broad classes of admissible distributional potentials Q, which extend the usual L ^p and Kato classes, as well as those based on the well-known conditions of Fefferman–Phong and Chang–Wilson–Wolff.

Secondly, we characterize Trudinger’s subordination property where C(ε) in the above inequality is subject to the condition C(ε)≤cε^-β(β>0) as ε→+0. Such quadratic form inequalities can be understood entirely in the framework of Morrey–Campanato spaces, using mean oscillations of \(\nabla(1-\Delta)^{-1}Q\) and \((1-\Delta)^{-1}Q\) on balls or cubes. A version of this condition where ε∈(0,+∞) is equivalent to the multiplicative inequality:

$\left\vert\int_{\mathbb{R}^n} |u(x)|^2Q(x)dx\right\vert\leq{C}\|\nabla{u}\|^{2p}_{L^2(\mathbb{R}^n)}\|u\|^{2(1-p)}_{L^2(\mathbb{R}^n)},\quad\forall{u}\in{C}^\infty_0(\mathbb{R}^n),$

with \(p=\frac\beta{1 + \beta}\in(0,1)\). We show that this inequality holds if and only if \(\nabla\Delta^{-1} Q \in{BMO}(\mathbb{R}^n)\) if \(p=\frac{1}{2}\). For \(0 < p < \frac{1}{2}\), it is valid whenever \(\nabla\Delta^{-1}Q\) is Hölder-continuous of order 1-2p, or respectively lies in the Morrey space \(\mathcal{L}^{2,\lambda}\) with λ=n+2-4p if \(\frac{1}{2} < p < 1\). As a consequence, we characterize completely the class of those Q which satisfy an analogous multiplicative inequality of Nash’s type, with \(\|u\|_{L^1(\mathbb{R}^n)}\) in placeof \(\|u\|_{L^2(\mathbb{R}^n)}\).

These results are intimately connected with spectral theory and dynamics of the Schrödinger operator, and elliptic PDE theory.     

On height isotopy classes of embeddings in the plane of a Morse function of a circle

Let \(\mathrm{SM}_{2n}(S^1,\mathbb {R})\) be a set of stable Morse functions of an oriented circle such that the number of singular points is \(2n\in \mathbb {N}\) and the order of singular values satisfies the particular condition. For an orthogonal projection \(\pi :\mathbb {R}^2\rightarrow \mathbb {R}\), let \({\tilde{f}}_0\) and \({\tilde{f}}_1:S^1\rightarrow \mathbb {R}^2\) be embedding lifts of f. If there is an ambient isotopy \(\tilde{\varphi }_t:\mathbb {R}^2\rightarrow \mathbb {R}^2\) \((t\in [0,1])\) such that \({\pi \circ \tilde{\varphi }}_t(y_1,y_2)=y_1\) and \(\tilde{\varphi }_1\circ {\tilde{f}}_0={\tilde{f}}_1\), we say that \({\tilde{f}}_0\) and \({\tilde{f}}_1\) are height isotopic. We define a function \(I:\mathrm{SM}_{2n}(S^1,\mathbb {R})\rightarrow \mathbb {N}\) as follows: I(f) is the number of height isotopy classes of embeddings such that each rotation number is one. In this paper, we determine the maximal value of the function I equals the n-th Baxter number and the minimal value equals \(2^{n-1}\).     

Lie Algebras of Slow Growth and Klein-Gordon PDE

We discuss the notion of characteristic Lie algebra of a hyperbolic PDE. The integrability of a hyperbolic PDE is closely related to the properties of the corresponding characteristic Lie algebra χ. We establish two explicit isomorphisms:

1. 1)
   the first one is between the characteristic Lie algebra \(\chi (\sinh {u})\) of the sinh-Gordon equation \(u_{xy}=\sinh {u}\) and the non-negative part \({\mathcal {L}}({\mathfrak {sl}}(2,{\mathbb {C}}))^{\ge 0}\) of the loop algebra of \({\mathfrak {sl}}(2,{\mathbb {C}})\) that corresponds to the Kac-Moody algebra \(A_{1}^{(1)}\)

   $$\chi(\sinh{u})\cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(2,{\mathbb{C}}))^{\ge 0}={\mathfrak{s}\mathfrak{l}}(2, {\mathbb{C}}) \otimes {\mathbb{C}}[t]. $$
    
2. 2)
   the second isomorphism is for the Tzitzeica equation u_xy = e^u + e^??2u

   $$\chi(e^{u}{+}e^{-2u}) \cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(3,{\mathbb{C}}), \mu)^{\ge0}=\bigoplus_{j = 0}^{+\infty}{\mathfrak{g}}_{j (\text{mod} \; 2)} \otimes t^{j}, $$

   where \({\mathcal {L}}({\mathfrak {sl}}(3,{\mathbb {C}}), \mu )=\bigoplus _{j \in {\mathbb {Z}}}{\mathfrak {g}}_{j (\text {mod} \; 2)} \otimes t^{j}\) is the twisted loop algebra of the simple Lie algebra \({\mathfrak {sl}}(3,{\mathbb {C}})\) that corresponds to the Kac-Moody algebra \(A_{2}^{(2)}\).
    

Hence the Lie algebras \(\chi (\sinh {u})\) and χ(e^u + e^??2u) are slowly linearly growing Lie algebras with average growth rates \(\frac {3}{2}\) and \(\frac {4}{3}\) respectively.     

Frame completions with prescribed norms: local minimizers and applications

Let \(\mathcal {F}_{0}=\{f_{i}\}_{i\in \mathbb {I}_{n_{0}}}\) be a finite sequence of vectors in \(\mathbb {C}^{d}\) and let \(\mathbf {a}=(a_{i})_{i\in \mathbb {I}_{k}}\) be a finite sequence of positive numbers, where \(\mathbb {I}_{n}=\{1,\ldots , n\}\) for \(n\in \mathbb {N}\). We consider the completions of \(\mathcal {F}_{0}\) of the form \(\mathcal {F}=(\mathcal {F}_{0},\mathcal {G})\) obtained by appending a sequence \(\mathcal {G}=\{g_{i}\}_{i\in \mathbb {I}_{k}}\) of vectors in \(\mathbb {C}^{d}\) such that ∥g _i ∥² = a _i for \(i\in \mathbb {I}_{k}\), and endow the set of completions with the metric \(d(\mathcal {F},\tilde {\mathcal {F}}) =\max \{ \,\|g_{i}-\tilde {g}_{i}\|: \ i\in \mathbb {I}_{k}\}\) where \(\tilde {\mathcal {F}}=(\mathcal {F}_{0},\,\tilde {\mathcal {G}})\). In this context we show that local minimizers on the set of completions of a convex potential P _φ , induced by a strictly convex function φ, are also global minimizers. In case that φ(x) = x ² then P _φ is the so-called frame potential introduced by Benedetto and Fickus, and our work generalizes several well known results for this potential. We show that there is an intimate connection between frame completion problems with prescribed norms and frame operator distance (FOD) problems. We use this connection and our results to settle in the affirmative a generalized version of Strawn’s conjecture on the FOD.