On Projections of Semi-Algebraic Sets Defined by Few Quadratic Inequalities

Let S⊂ℝ ^k+m be a compact semi-algebraic set defined by P ₁≥0,…,P _ℓ ≥0, where P _i ∈ℝ[X ₁,…,X _k ,Y ₁,…,Y _m ], and deg (P _i )≤2, 1≤i≤ℓ. Let π denote the standard projection from ℝ ^k+m onto ℝ ^m . We prove that for any q>0, the sum of the first q Betti numbers of π(S) is bounded by (k+m) ^{O(q ℓ)}. We also present an algorithm for computing the first q Betti numbers of π(S), whose complexity is . For fixed q and ℓ, both the bounds are polynomial in k+m. The author was supported in part by an NSF Career Award 0133597 and a Sloan Foundation Fellowship.     

Geometry of finite-dimensional normed spaces,and continuous functions on the Euclidean sphere

Let ℝⁿ be the n-dimensional Euclidean space, and let { · } be a norm in Rⁿ. Two lines ℓ₁ and ℓ₂ in ℝⁿ are said to be { · }-orthogonal if their { · }-unit direction vectors e ₁ and e ₂ satisfy {e ₁ + e ₂} = {e ₁ − e ₂}. It is proved that for any two norms { · } and { · }′ in ℝⁿ there are n lines ℓ₁, ..., ℓ_n that are { · }-and { · }′-orthogonal simultaneously. Let be a continuous function on the unit sphere with center O. It is proved that there exists an (n − 1)-cube C centered at O, inscribed in , and such that all sums of values of f at the vertices of (n − 3)-faces of C are pairwise equal. If the function f is even, then there exists an n-cube with the same properties. Furthermore, there exists an orthonormal basis e ₁, ..., e _n such that for 1 ≤ i ≤ j ≤ n we have . Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 107–117.     

Surjective partial differential operators on real analytic functions defined on open convex sets

Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on a covex open set Ω⊂ℝ ⁿ . Let L(P _m ) denote the localizations at ∞ (in the sense of H?rmander) of the principal part P _m . Then Q(x+iτN)≠ 0 for (x,τ)∈ℝ ⁿ ×(ℝ\{ 0}) for any Q∈L(P _m ) if N is a normal to δΩ which is noncharacteristic for Q. Under additional assumptions this implies that P _m must be locally hyperbolic. Received: 24 January 2000     

Construction of multivariate compactly supported orthonormal wavelets

We propose a constructive method to find compactly supported orthonormal wavelets for any given compactly supported scaling function φ in the multivariate setting. For simplicity, we start with a standard dilation matrix 2I_2×2 in the bivariate setting and show how to construct compactly supported functions ψ₁,. . .,ψ_n with n>3 such that {2^kψ_j(2^kx−ℓ,2^ky−m), k,ℓ,m∈Z, j=1,. . .,n} is an orthonormal basis for L₂(ℝ²). Here, n is dependent on the size of the support of φ. With parallel processes in modern computer, it is possible to use these orthonormal wavelets for applications. Furthermore, the constructive method can be extended to construct compactly supported multi-wavelets for any given compactly supported orthonormal multi-scaling vector. Finally, we mention that the constructions can be generalized to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C15, 42C30.     

Multiderivations of Coxeter arrangements

Let V be an ℓ-dimensional Euclidean space. Let G⊂O(V) be a finite irreducible orthogonal reflection group. Let ? be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H∈? choose α _H ∈V ^* such that H=ker(α _H ). For each nonnegative integer m, define the derivation module D ^(m) (?)={θ∈Der _S |θ(α _H )∈Sα ^m _H }. The module is known to be a free S-module of rank ℓ by K. Saito (1975) for m=1 and L. Solomon-H. Terao (1998) for m=2. The main result of this paper is that this is the case for all m. Moreover we explicitly construct a basis for D ^(m) (?). Their degrees are all equal to mh/2 (when m is even) or are equal to ((m−1)h/2)+m _i (1≤i≤ℓ) (when m is odd). Here m ₁≤···≤m _ℓ are the exponents of G and h=m _ℓ+1 is the Coxeter number. The construction heavily uses the primitive derivation D which plays a central role in the theory of flat generators by K. Saito (or equivalently the Frobenius manifold structure for the orbit space of G). Some new results concerning the primitive derivation D are obtained in the course of proof of the main result. Oblatum 27-XI-2001 & 4-XII-2001?Published online: 18 February 2002     

Affinely equivalent complete flat manifolds

Let E _Aff(Γ,G, m) be the set of affine equivalence classes of m-dimensional complete flat manifolds with a fixed fundamental group Γ and a fixed holonomy group G. Let n be the dimension of a closed flat manifold whose fundamental group is isomorphic to Γ. We describe E _Aff(Γ,G, m) in terms of equivalence classes of pairs (ε, ρ), consisting of epimorphisms of Γ onto G and representations of G in ℝ ^m-n . As an application we give some estimates of card E _Aff(Γ,G, m).     

Characterization of separating couples

Let A, B be two archimedean ℓ-algebras and let U,V be two positive linear maps from A to B. We call that the couple (U,V) is separating with respect to A and B if |a||b| = 0 in A implies |U (a)||V (b)| = 0 in B. In this paper, we prove that if A is an f-algebra with unit elment e, if B is an ℓ-algebra and if (U,V) is a separating couple with respect to A and B then (U ^∼∼,V ^∼∼), where U ^∼∼ (resp V ^∼∼) is the bi-adjoint of U (resp of V), is again a separating couple with respect to the order continuous order biduals (A′)′ _n and (B′)′ _n of A and B respectively furnished with their Arens products respectively. Moreover, in the case where B′ separates the points of B, we give a characterization of any separating couple with respect to A and B.      

Hardy spaces via distribution spaces

Let ℐ(ℝⁿ) be the Schwartz class on ℝⁿ and ℐ_∞(ℝⁿ) be the collection of functions ϕ ∊ ℐ(ℝⁿ) with additional property that

On the Bethe-Sommerfeld conjecture for higher-order elliptic operators

 We consider the elliptic operator P(D)+V in ℝ ^d , d≥2 where P(D) is a constant coefficient elliptic pseudo-differential operator of order 2ℓ with a homogeneous convex symbol P(ξ), and V is a real periodic function in L ^∞(ℝ ^d ). We show that the number of gaps in the spectrum of P(D)+V is finite if 4ℓ>d+1. If in addition, V is smooth and the convex hypersurface {ξℝ ^d :P(ξ)=1} has positive Gaussian curvature everywhere, then the number of gaps in the spectrum of P(D)+V is finite, provided 8ℓ>d+3 and 9≥d≥2, or 4ℓ>d−3 and d≥10. Received: 10 October 2001 / Published online: 28 March 2003 Mathematics Subject Classification (1991): 35J10 Research supported in part by NSF Grant DMS-9732894.     

A homological characterization of hyperbolic groups

A finitely presented group G is hyperbolic iff H ⁽¹⁾ ₁(G,ℝ)=0=⁽¹⁾ ₂(G, ℝ), where H ⁽¹⁾ _* (resp. ⁽¹⁾ _*) denotes the ℓ₁-homology (resp. reduced ℓ₁-homology). If Γ is a graph, then every ℓ₁ 1-cycle in Γ with real coefficients can be approximated by 1-cycles of compact support. A 1-relator group G is hyperbolic iff H ⁽¹⁾ ₁(G,ℝ)=0. Oblatum: 30-IV-1997 & 14-V-1998 / Published online: 14 January 1999     

Non-isogenous superelliptic Jacobians

Let ℓ be an odd prime. Let K be a field of characteristic zero with algebraic closure K_a. Let n, m ≥ 4 be integers that are not divisible by ℓ. Let f(x), h(x) ∈ K[x] be irreducible separable polynomials of degree n and m respectively. Suppose that the Galois group Gal(f) of f acts doubly transitively on the set of roots of f and that Gal(h) acts doubly transitively on as well. Let J(C_f,ℓ) and J(C_h,ℓ) be the Jacobians of the superelliptic curves C_f,ℓ:y^ℓ=f(x) and C_h,ℓ:y^ℓ=h(x) respectively. We prove that J(C_f,ℓ) and J(C_h,ℓ) are not isogenous over K_a if the splitting fields of f and h are linearly disjoint over K and K contains a primitive ℓth root of unity.     

Existence of limits of maximal means

For the class II(ℝ ^m ) of continuous almost periodic functionsf: ℝ ^m → ℝ, we consider the problem of the existence of the limit

Boundedness of Multilinear Operators in Herz-type Hardy Space

Let κ∈ℕ. We prove that the multilinear operators of finite sums of products of singular integrals on ℝⁿ are bounded from HK ^α1,p1 _q1 (ℝⁿ) ×···×HK ^αk,pk _qk (ℝⁿ) into HK ^α,p _q (ℝⁿ) if they have vanishing moments up to a certain order dictated by the target spaces. These conditions on vanishing moments satisfied by the multilinear operators are also necessary when α_j≥ 0 and the singular integrals considered here include the Calderón-Zygmund singular integrals and the fractional integrals of any orders. Received September 6, 1999, Revised November 17, 1999, Accepted December 9, 1999     

Lines Pinning Lines

A line ℓ is a transversal to a family F of convex polytopes in ℝ³ if it intersects every member of F. If, in addition, ℓ is an isolated point of the space of line transversals to F, we say that F is a pinning of ℓ. We show that any minimal pinning of a line by polytopes in ℝ³ such that no face of a polytope is coplanar with the line has size at most eight. If in addition the polytopes are pairwise disjoint, then it has size at most six.     

A distortion problem for the space ℓ ∞ c (ω 1)

The Banach space ℓ _∞ ^c (ω ₁) is the space of boundedω ₁-sequences of countable support. A pointwise-closed subspaceV≤ℓ _∞ ^c (ω ₁) will be calledunbounded if lcub;min(supp(υ)):υ ∈Vrcub; is unbounded inω ₁. It is shown that there are Lipshitz functionsf: Sph(ℓ _∞ ^c (ω ₁)) → ℝ which have large variation on the unit sphere of any unbounded subspace. This answers a question implicit in Partington [P 80].     

On some new generalized difference sequence spaces on seminormed spaces defined by a sequence of Orlicz functions

In this paper we define the sequence space ℓ _M (Δ _υ ^m , p, q, s) on a seminormed complex linear space, by using a sequence of Orlicz functions. We study some algebraic and topological properties. We prove some inclusion relations involving ℓ _M (Δ _υ ^m , p, q, s). spaces     

Dilations and Completions for Gabor Systems

Let be a full rank time-frequency lattice in ℝ ^d ×ℝ ^d . In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pair for the whole space L ²(ℝ ^d ) when the volume v(Λ) of the lattice Λ satisfies the condition v(Λ)≤1, and to a dual Gabor Riesz basis pair for a Λ-shift invariant subspace containing M when v(Λ)>1. This generalizes the dilation result in Gabardo and Han (J. Fourier Anal. Appl. 7:419–433, [2001]) to both higher dimensions and dual subspace Gabor frame pairs. Secondly, for any fixed positive integer N, we investigate the problem whether any Bessel–Gabor family G(g,Λ) can be completed to a tight Gabor (multi-)frame G(g,Λ)∪(∪ _j=1 ^N G(g _j ,Λ)) for L ²(ℝ ^d ). We show that this is true whenever v(Λ)≤N. In particular, when v(Λ)≤1, any Bessel–Gabor system is a subset of a tight Gabor frame G(g,Λ)∪G(h,Λ) for L ²(ℝ ^d ). Related results for affine systems are also discussed. Communicated by Chris Heil.     

Approximation of Functions of Few Variables in?High?Dimensions

Let f be a continuous function defined on Ω:=[0,1] ^N which depends on only ℓ coordinate variables, f(x₁,?,x_N)=g(x_i1,?,x_il)f(x_{1},\ldots,x_{N})=g(x_{i_{1}},\ldots,x_{i_{\ell}}). We assume that we are given m and are allowed to ask for the values of f at m points in Ω. If g is in Lip1 and the coordinates i ₁,…,i _ℓ are known to us, then by asking for the values of f at m=L ^ℓ uniformly spaced points, we could recover f to the accuracy |g|_Lip1 L ⁻¹ in the norm of C(Ω). This paper studies whether we can obtain similar results when the coordinates i ₁,…,i _ℓ are not known to us. A prototypical result of this paper is that by asking for C(ℓ)L ^ℓ (log ₂ N) adaptively chosen point values of f, we can recover f in the uniform norm to accuracy |g|_Lip1 L ⁻¹ when g∈Lip1. Similar results are proven for more general smoothness conditions on g. Results are also proven under the assumption that f can be approximated to some tolerance ε (which is not known) by functions of ℓ variables.     

Subsets of rectifiable curves in Hilbert space-the analyst’s TSP

We study one dimensional sets (Hausdorff dimension) lying in a Hilbert space. The aim is to classify subsets of Hilbert spaces that are contained in a connected set of finite Hausdorff length. We do so by extending and improving results of Peter Jones and Kate Okikiolu for sets in ℝ^d. Their results formed the basis of quantitative rectifiability in ℝ^d. We prove a quantitative version of the following statement: a connected set of finite Hausdorff length (or a subset of one), is characterized by the fact that inside balls at most scales aroundmost points of the set, the set lies close to a straight line segment (which depends on the ball). This is done via a quantity, similar to the one introduced in [Jon90], which is a geometric analogue of the Square function. This allows us to conclude that for a given set K, the ℓ₂ norm of this quantity (which is a function of K) has size comparable to a shortest (Hausdorff length) connected set containing K. In particular, our results imply that, with a correct reformulation of the theorems, the estimates in [Jon90, Oki92] are independent of the ambient dimension.     

Simplicial arrangements associated with the 24-cell

Let = A ₄ ¹ (49) be the 4-arrangement in real projective 4-space that includes all 24 facet hyperplanes of the 24-cellT, its 24 hyperplanes of mirror symmetry, and the hyperplane at infinity. In this paper we show that is theonly simplicial 4-arrangement formed by including some of the hyperplanes of mirror symmetry and possibly the hyperplane at infinity with the 24 facet hyperplanes ofT. Only one other simplicial 4-arrangement is known that does not lie in a natural sequence of arrangements that are simplicial in each dimension. Two of the simplicial 3-arrangements induced by X were not previously in the literature, so 19 joinirreducible simplicial 3-arrangements are currently known.Most of the results reported in this paper were presented August 5, 1987 at a Special Session on Discrete Geometry and Convexity at the 90th Summer Meeting of the American Mathematical Society in Salt Lake City, Utah.