Square function inequalities for non-commutative martingales

We prove a non-commutative version of the weak-type (1,1) boundedness of square functions of martingales. More precisely, we prove that there is an absolute constantK with the following property: ifM is a semifinite von Neumann algebra with a faithful normal traceτ and (M _n ) _n=1 ^∞ is an increasing filtration of von Neumann subalgebras of (M then for any martingalex= _n=1 ^∞ inL ¹(M,τ), adapted to (M _n ) _n=1 ^∞ , there is a decomposition into two sequences (x _n ) _n=1 ^∞ and (z _n ) _n=1 ^∞ withx _n=y _n+z _nfor everyn≥1 and such that . This generalizes a result of Burkholder from classical martingale theory to non-commutative martingales. We also include some applications to martingale Hardy spaces. Supported in part by NSF grant DMS-0096696.     

Compactly Supported Tight Affine Frames with Integer Dilations and Maximum Vanishing Moments

When a cardinal B-spline of order greater than 1 is used as the scaling function to generate a multiresolution approximation of L ²=L ²(R) with dilation integer factor M2, the standard matrix extension approach for constructing compactly supported tight frames has the limitation that at least one of the tight frame generators does not annihilate any polynomial except the constant. The notion of vanishing moment recovery (VMR) was introduced in our earlier work (and independently by Daubechies et al.) for dilation M=2 to increase the order of vanishing moments. This present paper extends the tight frame results in the above mentioned papers from dilation M=2 to arbitrary integer M2 for any compactly supported M-dilation scaling functions. It is shown, in particular, that M compactly supported tight frame generators suffice, but not M–1 in general. A complete characterization of the M-dilation polynomial symbol is derived for the existence of M–1 such frame generators. Linear spline examples are given for M=3,4 to demonstrate our constructive approach.     

Construction of orthonormal multi-wavelets with additional vanishing moments

An iterative scheme for constructing compactly supported orthonormal (o.n.) multi-wavelets with vanishing moments of arbitrarily high order is established. Precisely, let φ=[φ₁,. . .,φ_r]^⊤ be an r-dimensional o.n. scaling function vector with polynomial preservation of order (p.p.o.) m, and ψ=[ψ₁,. . .,ψ_r]^⊤ an o.n. multi-wavelet corresponding to φ, with two-scale symbols P and Q, respectively. Then a new (r+1)-dimensional o.n. scaling function vector φ^♯:=[φ^⊤,φ_r+1]^⊤ and some corresponding o.n. multi-wavelet ψ^♯ are constructed in such a way that φ^♯ has p.p.o.=n>m and their two-scale symbols P^♯ and Q^♯ are lower and upper triangular block matrices, respectively, without increasing the size of the supports. For instance, for r=1, if we consider the mth order Daubechies o.n. scaling function φ_m^D, then φ^♯:=[φ_m^D,φ₂]^⊤ is a scaling function vector with p.p.o. >m. As another example, for r=2, if we use the symmetric o.n. scaling function vector φ in our earlier work, then we obtain a new pair of scaling function vector φ^♯=[φ^⊤,φ₃]^⊤ and multi-wavelet ψ^♯ that not only increase the order of vanishing moments but also preserve symmetry. Dedicated to Charles A. Micchelli in Honor of His Sixtieth Birthday Mathematics subject classifications (2000) 42C15, 42C40. Charles K. Chui: Supported in part by NSF grants CCR-9988289 and CCR-0098331 and Army Research Office under grant DAAD 19-00-1-0512. Jian-ao Lian: Supported in part by Army Research Office under grant DAAD 19-01-1-0739.     

Reverse hypercontractivity over manifolds

Suppose thatX is a vector field on a manifoldM whose flow, exptX, exists for all time. If μ is a measure onM for which the induced measuresμ _t ≡(exptX)_* μ are absolutely continuous with respect to μ, it is of interest to establish bounds on theL ^p (μ) norm of the Radon-Nikodym derivativedμ _t /dμ. We establish such bounds in terms of the divergence of the vector fieldX. We then specilizeM to be a complex manifold and derive reverse hypercontractivity bounds and reverse logarithmic Sololev inequalities in some holomorphic function spaces. We give examples onC ^m and on the Riemann surface forz ^1/n . Research supported in part by CONACyT, Mexico, grant 32725-E. Research supported in part by CONACyT, Mexico, grant 32146-E.     

On the limit cycles available from polynomial perturbations of the Bogdanov-Takens Hamiltonian

The displacement map related to small polynomial perturbations of the planar Hamiltonian systemdH=0 is studied in the elliptic caseH=1/2y ²+1/2x ²−1/3x ³. An estimate of the number of isolated zeros for each of the successive Melnikov functionsM _k(h),k=1, 2,…is given in terms of the orderk and the maximal degreen of the perturbation. This sets up an upper bound to the number of limit cycles emerging from the periodic orbits of the Hamiltonian system under polynomial perturbations. Research partially supported by grant MM810/98 from the NSF of Bulgaria and MURST, Italy.     

A remark on a maximal function over a Cantor set of directions

LetM _e ⁰ be the maximal operator over segments of length 1 with directions belonging to a Cantor set. It has been conjectured that this operator is bounded onL ². We consider a sequence of operators over finite sets of directions converging toM _e ⁰ . We improve the previous estimate for the (L ²,L ²)-norm of these particular operators. We also prove thatM _e ⁰ is bounded from some subsets ofL ² toL ². These subsets are composed of positive functions whose Fourier transforms have a very weak decay or are supported in a vertical strip. Partially supported by Spanish DGICYT grant no. PB90-0187.     

Weakly symmetric spaces and bounded symmetric domains

LetG be a connected, simply-connected, real semisimple Lie group andK a maximal compactly embedded subgroup ofG such thatD=G/K is a hermitian symmetric space. Consider the principal fiber bundleM=G/K _s G/K, whereK _s is the semisimple part ofK=K _s ·Z _K ⁰ andZ _K ⁰ is the connected center ofK. The natural action ofG onM extends to an action ofG ¹=G×Z _K ⁰ . We prove as the main result thatM is weakly symmetric with respect toG ¹ and complex conjugation. In the case whereD is an irreducible classical bounded symmetric domain andG is a classical matrix Lie group under a suitable quotient, we provide an explicit construction ofM=D×S ¹ and determine a one-parameter family of Riemannian metrics onM invariant underG ¹. Furthermore,M is irreducible with respect to . As a result, this provides new examples of weakly symmetric spaces that are nonsymmetric, including those already discovered by Selberg (cf. [M]) for the symplectic case and Berndt and Vanhecke [BV1] for the rank-one case.Research partially supported by an NSF grant. The author wishes to thank the International Erwin Schroedinger Institute for its hospitality during the preparation of this paper.     

The moduli space of complete embedded constant mean curvature surfaces

We examine the space of finite topology surfaces in ³ which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the spaceM _k of all such surfaces withk ends (where surfaces are identified if they differ by an isometry of ³) is locally a real analytic variety. When the linearization of the quasilinear elliptic equation specifying mean curvature equal to one has noL ²-nullspace, we prove thatM _k is locally the quotient of a real analytic manifold of dimension 3k–6 by a finite group (i.e. a real analytic orbifold), fork 3. This finite group is the isotropy subgroup of the surface in the group of Euclidean motions. It is of interest to note that the dimension ofM _k is independent of the genus of the underlying punctured Riemann surface to which is conformally equivalent. These results also apply to hypersurfaces of H ⁿ⁺¹ with nonzero constant mean curvature greater than that of a horosphere and whose ends are cylindrically bounded.Research of the first author supported in part by NSF grant # DMS9404278 and an NSF Postdoctoral Fellowship, of the second auther by NSF Young Investigator Award, a Sloan Foundation Postdoctoral Fellowship and NSF grant # DMS9303236, and of the third author by NSF grant # DMS9022140 and an NSF Postdoctoral Fellowship.     

Confidence Bands for Isotonic Median Curves Using Sign Tests

Suppose that one observes independent random variables (X₁, Y₁), (X₂, Y₂), …, (X_n, Y_n) in R² with unknown distributions, except that Median(Y_i | X_i = M(x) for some unknown isotonic function M. We describe an explicit algorithm for the computation of confidence bands for the median function M whose running time is of order O(n²). The bands rely on multiscale sign tests and are shown to have desirable asymptotic properties.     

Dupin hypersurfaces with four principal Curvatures,II

If M is an isoparametric hypersurface in a sphere S ⁿ with four distinct principal curvatures, then the principal curvatures κ₁, . . . , κ₄ can be ordered so that their multiplicities satisfy m ₁ = m ₂ and m ₃ = m ₄, and the cross-ratio r of the principal curvatures (the Lie curvature) equals −1. In this paper, we prove that if M is an irreducible connected proper Dupin hypersurface in R ⁿ (or S ⁿ ) with four distinct principal curvatures with multiplicities m ₁ = m ₂ ≥ 1 and m ₃ = m ₄ = 1, and constant Lie curvature r = −1, then M is equivalent by Lie sphere transformation to an isoparametric hypersurface in a sphere. This result remains true if the assumption of irreducibility is replaced by compactness and r is merely assumed to be constant.      

Convex mappings on some Reinhardt domains

In this paper, we consider the following Reinhardt domains. Let M = （M1, M2,..., Mn） ： [0,1] → [0,1]^n be a C2-function and Mj（0） = 0, Mj（1） = 1, Mj″ 〉 0, C1jr^pj-1 〈 Mj′（r） 〈 C2jr^pj-1, r∈ （0, 1）, pj 〉 2, 1 ≤ j ≤ n, 0 〈 C1j 〈 C2j be constants. Define
DM={z=（z1,z2,…,Zn）^T∈C^n：n∑j=1 Mj（｜zj｜）〈1}
Then DM C^n is a convex Reinhardt domain. We give an extension theorem for a normalized biholomorphic convex mapping f ： DM -→ C^n.     

A constructive method of solving the Liapounov equation for complex matrices

Summary This paper describes a method of solving the Liapounov equation (1)HM+M ^* H=2D, M in upper Hessenberg form,D diagonal. Initialising the first row of the matrixA arbitrarily, one can find (by solving equations with one unknown) the unknown elements ofA such that (2)AM+M ^* A ^* =2F, whereA differs from a Hermitian matrix only in that its diagonal elements need not be real.F is a diagonal matrix which is uniquely determined by the first row ofA. By solving Eq. (2) for several initial values one may generate several matricesA andF (in the most unfavourable case 2n–1A's andF's are needed) and superpose them to getn linearly independent Hermitian matricesH _j andD _j respectively for whichH _j M+M ^* H _j =2D _j is valid. Then one can solve the real system to obtain the solution of Eq. (1).This work was performed under the terms of the agreement on association between the Max-Planck-Institut für Plasmaphysik and Euratom.     

Sharp Estimate of Global Coulomb Gauge

Let A be a W^{1, 2} -connection on a principal SU(2) -bundle P over a compact 4 -manifold M whose curvature F_A satisfies . Our main result is the existence of a global section σ : M → P with finite singularities on M such that the connection form σ^*A satisfies the Coulomb equation d^*(σ^*A) = 0 and admits a sharp estimate . Here ℒ^{4, ∞} is a new function space we introduce in this paper that satisfies L⁴(M) ⊊ ℒ^{4, ∞}(M) ⊊ L^{4 − ϵ}(M) for all ϵ > 0 . More precisely, ℒ^{4, ∞}(M) is the collection of measurable function u such that , where L^{4, ∞} is the classical Lorentz space and s_u is the L⁴ -integrability radius function associated to u defined by Briefly speaking, we achieve the estimate of by showing that σ^*A is effectively L⁴ -integrable away from controllably many points on M . © 2020 Wiley Periodicals LLC     

On the prime radical and Baer’s lower nilradical of modules

Let M be a left R-module. In this paper a generalization of the notion of m-system set of rings to modules is given. Then for a submodule N of M, we define := { m ε M: every m-system containing m meets N}. It is shown that is the intersection of all prime submodules of M containing N. We define rad _R (M) = . This is called Baer-McCoy radical or prime radical of M. It is shown that if M is an Artinian module over a PI-ring (or an FBN-ring) R, then M/rad _R (M) is a Noetherian R-module. Also, if M is a Noetherian module over a PI-ring (or an FBN-ring) R such that every prime submodule of M is virtually maximal, then M/rad _R (M) is an Artinian R-module. This yields if M is an Artinian module over a PI-ring R, then either rad _R (M) = M or rad _R (M) = ∩ _i=1 ⁿ for some maximal ideals of R. Also, Baer’s lower nilradical of M [denoted by Nil_* ( _R M)] is defined to be the set of all strongly nilpotent elements of M. It is shown that, for any projective R-module M, rad _R (M) = Nil_*( _R M) and, for any module M over a left Artinian ring R, rad _R (M) = Nil_*( _R M) = Rad(M) = Jac(R)M. This research was in part supported by a grant from IPM (No. 85130016). Also this work was partially supported by IUT (CEAMA). The author would like to thank the anonymous referee for a careful checking of the details and for helpful comments that improved this paper.     

Exact emulation of a priority queue with a switch and delay lines

All-optical packet switched networking is hampered by the problem of realizing viable queues for optical packets. Packets can be buffered in delay lines, but delay lines do not functionally emulate queues from an input-output point of view. In this paper we consider the problem of exact emulation of a priority queue of size K using a switching system comprised of a switch of size (M + 1) × (M + 1), which has one distinguished input for external arrivals, one distinguished output for external departures, and fixed-length delay lines of lengths L₁, L₂, ..., L_M connecting the other inputs and outputs in pairs. We measure the complexity of such an emulation by M + 1. We prove that and present a construction which works with ; further, in our construction . We also sketch an idea for an all-optical packet switched communication network architecture based on approximate emulation of priority queues of finite size using switches and delay lines, with erasure control coding at the packet level. AMS 2000 subject classifications: Primary: 60K25; Secondary: 90B22 · 90B36 · 68R99 The work of A. D. Sarwate is supported by an NDSEG Graduate Research Fellowship which is sponsored by the U.S. Department of Defense. The work of V. Anantharam is supported by ONR grant No. N00014-1-0637, DARPA grant No. N66001-00-C-8062, and by NSF grant No. ECS 0123512.     

Analytic Discs and Extension of CR Functions

Let M be a manifold of X = C ⁿ , A a small analytic disc attached to M, z ^o a point of A where A is tangent to M, z ¹ another point of A where M extends to a germ of manifold M ₁ with boundary M. We prove that CR functions on M which extend to M ₁ at z ¹ also extend at z ^o to a new manifold M ₂. The directions M ₁ and M ₂ point to, are related by a sort of connection associated to A which is dual to the connection obtained by attaching 'partial analytic lifts' of A to the co-normal bundle to M in X.     

The variance of a partial match retrieval in a multidimensional symmetric trie

Let N records be stored in an M-dimensional trie (M-d trie). It is shown that under the symmetric Bernoulli model the variance of the cost C_N^ω of a partial match retrieval is with continuous periodic function τ_ω, where u is the number of unspecified components in a query ω of size M. This confirms in quantitative manner a conjecture by Kirschenhofer, Prodinger, and Szpankowski [Int. J. Found. Comput. Sci., 4 , 69–84 (1993)], who presented a detailed analysis of Var C_N^ω in the M = 2, u = 1-case, but used in their proof a transformation formula of Ramanujan, which does not seem to have an analogue for M > 2. Our analysis is based on exponential approximations and Mellin inversion.     

Asymtotics of <Emphasis Type="Italic">M</Emphasis>-estmation in Non-linear Regression

Consider the standard non-linear regression model y _i = g(x _i , θ ₀)+ε _i , i = 1, ... ,n where g(x, θ) is a continuous function on a bounded closed region X × Θ, θ ₀ is the unknown parameter vector in Θ ⊂ R ^p , {x ₁, x ₂, ... , x _n } is a deterministic design of experiment and {ε₁, ε₂, ... , ε _n } is a sequence of independent random variables. This paper establishes the existences of M-estimates and the asymptotic uniform linearity of M-scores in a family of non-linear regression models when the errors are independent and identically distributed. This result is then used to obtain the asymptotic distribution of a class of M-estimators for a large class of non-linear regression models. At the same time, we point out that Theorem 2 of Wang (1995) (J. of Multivariate Analysis, vol. 54, pp. 227–238, Corrigenda. vol. 55, p. 350) is not correct. This research was supported by the Natural Science Foundation of China (Grant No. 19831010 and grant No. 39930160) and the Doctoral Foundation of China     

Classification of manifolds with weakly 1/4-pinched curvatures

We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds M with the property that M × R ² has non-negative isotropic curvature. The first author was partially supported by a Sloan Foundation Fellowship and by NSF grant DMS-0605223. The second author was partially supported by NSF grant DMS-0604960.     

Automatic Smoothing Spline Projection Pursuit

Abstract

A highly flexible nonparametric regression model for predicting a response y given covariates {x_k}^d _k=1 is the projection pursuit regression (PPR) model ? = h(x) = β₀ + Σ_jβ_jf_j(α^T _jx) where the f_j , are general smooth functions with mean 0 and norm 1, and Σ^d _k=1α² _kj=1. The standard PPR algorithm of Friedman and Stuetzle (1981) estimates the smooth functions f_j using the supersmoother nonparametric scatterplot smoother. Friedman's algorithm constructs a model with M _max linear combinations, then prunes back to a simpler model of size M ≤ M _max, where M and M _max are specified by the user. This article discusses an alternative algorithm in which the smooth functions are estimated using smoothing splines. The direction coefficients α_j, the amount of smoothing in each direction, and the number of terms M and M _max are determined to optimize a single generalized cross-validation measure.