Asymptotic Behavior of Variable-Coefficient Toeplitz Determinants

For a continuous function s\sigma defined on [0,1]×\mathbbT[0,1]\times\mathbb{T}, let \ops\op\sigma stand for the (n+1)×(n+1)(n+1)\times(n+1) matrix whose (j,k)(j,k)-entries are equal to \frac1 2pò₀^2p s( \fracjn,e^iq) e^-i(j-k)q  dq,        j,k = 0,1,...,n . \displaystyle \frac{1} {2\pi}\int_0^{2\pi} \sigma \left( \frac{j}{n},e^{i\theta}\right) e^{-i(j-k)\theta} \,d\theta, \qquad j,k =0,1,\dots,n~. These matrices can be thought of as variable-coefficient Toeplitz matrices or as the discrete analogue of pseudodifferential operators. Under the assumption that the function s\sigma possesses a logarithm which is sufficiently smooth on [0,1]×\mathbbT[0,1]\times\mathbb{T}, we prove that the asymptotics of the determinants of \ops\op\sigma are given by det[\ops] ~ G[s]⁽ⁿ⁺¹⁾E[s]     \text as   n?￥ , \det \left[\op\sigma\right] \sim G[\sigma]^{(n+1)}E[\sigma] \quad \text{ as \ } n\to\infty~, where G[s]G[\sigma] and E[s]E[\sigma] are explicitly determined constants. This formula is a generalization of the Szegö Limit Theorem. In comparison with the classical theory of Toeplitz determinants some new features appear.     

Embeddings ofl p m intol ∞ n , 1≦p≦2

Isomorphic embeddings ofl _l ^m intol _∞ ⁿ are studied, and ford(n, k)=inf{‖T ‖ ‖T ⁻¹ ‖;T varies over all isomorphic embeddings ofl ₁ ^[klog2n] intol _∞ ⁿ we have that lim _n→∞ d(n, k)=γ(k)⁻¹,k>1, whereγ(k) is the solution of (1+γ)ln(1+γ)+(1 −γ)ln(1 −γ)=k ⁻¹ln4. Here [x] denotes the integer part of the real numberx.     

Additive polylogarithms and their functional equations

Let ${k[\varepsilon]_{2}:=k[\varepsilon]/(\varepsilon^{2})}Let k[e]₂:=k[e]/(e²){k[\varepsilon]_{2}:=k[\varepsilon]/(\varepsilon^{2})} . The single valued real analytic n-polylogarithm L_n: \mathbbC ? \mathbbR{\mathcal{L}_{n}: \mathbb{C} \to \mathbb{R}} is fundamental in the study of weight n motivic cohomology over a field k, of characteristic 0. In this paper, we extend the construction in ünver (Algebra Number Theory 3:1–34, 2009) to define additive n-polylogarithms li_n:k[e]₂? k{li_{n}:k[\varepsilon]_{2}\to k} and prove that they satisfy functional equations analogous to those of L_n{\mathcal{L}_{n}}. Under a mild hypothesis, we show that these functions descend to an analog of the nth Bloch group B_n￠(k[e]₂){B_{n}' (k[\varepsilon]_{2})} defined by Goncharov (Adv Math 114:197–318, 1995). We hope that these functions will be useful in the study of weight n motivic cohomology over k[ε]₂.     

Path numbers of tournaments

A family of simple (that is, cycle-free) paths is a path decomposition of a tournament T if and only if partitions the acrs of T. The path number of T, denoted pn(T), is the minimum value of | | over all path decompositions of T. In this paper it is shown that if n is even, then there is a tournament on n vertices with path number k if and only if n/2 k n²/4, k an integer. It is also shown that if n is odd and T is a tournament on n vertices, then (n + 1)/2 pn(T) (n² − 1)/4. Moreover, if k is an integer satisfying (i) (n + 1)/2 k n − 1 or (ii) n < k (n² − 1)/4 and k is even, then a tournament on n vertices having path number k is constructed. It is conjectured that there are no tournaments of odd order n with odd path number k for n k < (n² − 1)/4.     

Korovkin-type theorems and applications

Let {T _n } be a sequence of linear operators on C[0,1], satisfying that {T _n (e _i )} converge in C[0,1] (not necessarily to e _i ) for i = 0,1,2, where e _i = t ⁱ . We prove Korovkin-type theorem and give quantitative results on C ²[0,1] and C[0,1] for such sequences. Furthermore, we define King’s type q-Bernstein operator and give quantitative results for the approximation properties of such operators.      

When is a ring of torus invariants a polynomial ring?

Let ρ:T→GL(V) be a finite dimensional rational representation of a torus over an algebraically closed fieldk. We give necessary and sufficient conditions on the arrangement of the weights ofV within the character lattice ofT for the ring of invariants,k[V] ^T , to have a homogeneous system of parameters consisting of monomials (Theorem 4.1). Using this we give two simple constructive criteria each of which gives necessary and sufficient conditions fork[V] ^T to be a polynomial ring (Theorem 5.8 and Theorem 5.10). Research supported in part by NSERC Grant OGP 137522     

Polynomials of best approximation inC[−1,1]

Equivalences between the condition |P _n ^(k) (x)|≦K(n ⁻¹√1−x ²+1/n ²) ^k n ^-a, whereP _n(x) is the bestn-th degree polynomial approximation tof(x), and the Peetre interpolation space betweenC[−1,1] and the space (1−x ²) ^k f ^(2k)(x)∈C[−1,1] is established. A similar result is shown forE _n(f)= ‖f−P _n‖ _C[−1,1]. Rates other thann ^-a are also discussed. Supported by NSERC grant A4816 of Canada.     

Remarks on rings of constants of derivations II

Let k be a field of characteristic p>0 and D≠0 a family of k-derivations of k[x,y]. It is proved in [1] that k[x,y]^D, the ring of constants with respect to D, can be generated, as a k[x ^p,y ^p]-algebra, by p - 1 elements. In this note we prove that p - 1 is the sharp upper bound of numbers of generators.     

On Skew Triangular Matrix Rings

For a ring R, endomorphism α of R and positive integer n we define a skew triangular matrix ring T _n (R, α). By using an ideal theory of a skew triangular matrix ring T _n (R, α) we can determine prime, primitive, maximal ideals and radicals of the ring R[x; α]/ ? x ⁿ  ?, for each positive integer n, where R[x; α] is the skew polynomial ring, and ? x ⁿ  ? is the ideal generated by x ⁿ .     

Notes on the Kernels of Locally Finite Higher Derivations in Polynomial Rings

Let A = k^[3] be the polynomial ring in three variables over a field k, and let D be a nontrivial locally finite iterative higher derivation on A. Let A^D denote the kernel of D. In this note, we prove that, if chark > 0 and ML(A^D) ≠ A^D, then A^D ? k^[2]. As a consequence of this result, we give another proof of the cancellation theorem for k^[2] over any field k of positive characteristic.     

Distribution of exponential functions with k-full exponent modulo a prime

For a real x -1 we denote by S_k[X] the set of k-full integers n x, that is, the set of positive integers n x such that ℓ^k|n for any prime divisor ℓ|n. We estimate exponential sums of the form where is a fixed integer with gcd (, p) = 1, and apply them to studying the distribution of the powers ⁿ, n S_k[x], in the residue ring modulo p 1.     

Constructive proofs of theorems relating to:F(x) = y,with applications

Some theorems are given which relate to approximating and establishing the existence of solutions to systemsF(x) = y ofn equations inn unknowns, for variousy, in a region of euclideann-space E ⁿ . They generalize known theorems.Viewing complementarity problems and fixed-point problems as examples, known results or generalizations of known results are obtained.A familiar use is made of homotopies H: E ⁿ × [0, 1]E ⁿ of the formH(x, t) = (1 –t)F ⁰ (x) + t[F(x) – y] where theF ⁰ in this paper is taken to be linear. Simplicial subdivisionsT ^k of E ⁿ × [0, 1] furnish piecewise linear approximatesG ^k toH. The basic computation is via the generation of piecewise linear curvesP ^k which satisfyG ^k (x, t) = 0. Visualizing a sequence {T ^k } of such subdivisions, with mesh size going to zero, arguments are made on connected, compact limiting curvesP on whichH(x, t) = 0.This paper builds upon and continues recent work of C.B. Garcia.The authors respectively: A. Charnes, research partially supported by Proj. No. NR047-021 Contract N00014-75-C-0269; C.B. Garcia, C.E. Lemke, research partially supported by NSF Grant No. MPS75-09443.     

On the Equations De.ning Curves in a Polynomial Algebra

Let A be a commutative Noetherian ring of dimension n (n ≥ 3). Let I be a local complete intersection ideal in A[T] of height n. Suppose I/I ² is free A[T]/I-module of rank n and (A[T]/I) is torsion in K ₀(A[T]). It is proved in this article that I is a set theoretic complete intersection ideal in A[T] if one of the following conditions holds: (1) n ≥ 5, odd; (2) n is even, and A contains the field of rational numbers; (3) n = 3, and A contains the field of rational numbers.     

A straightforward generalization of Diliberto and Straus' algorithm does not work

An algorithm for best approximating in the sup-norm a function f C[0, 1]² by functions from tensor-product spaces of the form π_k C[0, 1] + C[0, 1] π_l, is considered. For the case k = L = 0 the Diliberto and Straus algorithm is known to converge. A straightforward generalization of this algorithm to general k, l is formulated, and an example is constructed demonstrating that this algorithm does not, in general, converge for k² + l² > 0.     

Cohen–Macaulayness and Limit Behavior of Depth for Powers of Cover Ideals

Let 𝕂 be a field, and let R = 𝕂[x ₁,…, x _n ] be the polynomial ring over 𝕂 in n indeterminates x ₁,…, x _n . Let G be a graph with vertex-set {x ₁,…, x _n }, and let J be the cover ideal of G in R. For a given positive integer k, we denote the kth symbolic power and the kth bracket power of J by J ^(k) and J ^[k], respectively. In this paper, we give necessary and sufficient conditions for R/J ^k , R/J ^(k), and R/J ^[k] to be Cohen–Macaulay. We also study the limit behavior of the depths of these rings.     

Nikol’skii’s inequality for different metrics and properties of the sequence of norms of the Fourier sums of a function in the Lorentz space

Let (X, Y) be a pair of normed spaces such that X ? Y ? L ₁[0, 1] ⁿ and {e _k } _k be an expanding sequence of finite sets in ? ⁿ with respect to a scalar or vector parameter k, k ∈ ? or k ∈ ? ⁿ . The properties of the sequence of norms $\{ \left\| {S_{e_k } (f)} \right\|x\} _k $ of the Fourier sums of a fixed function f ∈ Y are studied. As the spaces X and Y, the Lebesgue spaces L _p [0, 1], the Lorentz spaces L _p,q [0, 1], L _p,q [0, 1] ⁿ , and the anisotropic Lorentz spaces L _p,q*[0, 1] ⁿ are considered. In the one-dimensional case, the sequence {e _k } _k consists of segments, and in the multidimensional case, it is a sequence of hyperbolic crosses or parallelepipeds in ? ⁿ . For trigonometric polynomials with the spectrum given by step hyperbolic crosses and parallelepipeds, various types of inequalities for different metrics in the Lorentz spaces L _p,q [0, 1] ⁿ and L _p,q*[0, 1] ⁿ are obtained.     

A family of counterexamples in ergodic theory

LetT _α be the translationx↦x+α (mod 1) of [0, 1), α irrational. LetT be the Lebesgue measure-preserving automorphism ofX=[0, 3/2) defined byTx = x + 1 forx∈[0, 1/2),Tx=T _α(x−1) forx∈[1,3/2) andTx = T _α x forx∈[1/2, 1), i.e.T isT _α with a tower of height one built over [0, 1/2). If α is poorly approximable by rationals (there does not exist {p _n /q _n } with |α−p _n /q _n |=o(q _n ⁻²)) and λ is a measure onX ^k all of whose one-dimensional marginals are Lebesgue and which is ⊗ _{i − 1} ^k T ¹ invariant and ergodic (l>0) then λ is a product of off-diagonal measures. This property suffices for many purposes of counterexample construction. A connection is established with the POD (proximal orbit dense) condition in topological dynamics. Research supported in part by NSF contract MCS-8003038.     

A reversal index for tournament matrices *

Given a tournament matrix T, its reversal indexi_R (T), is the minimum k such that the reversal of the orientation of k arcs in the directed graph associated with T results in a reducible matrix. We give a formula for i_R (T) in terms of the score vector of T which generalizes a simple criterion for a tournament matrix to be irreducible. We show that i_R (T)≤[(n?1)/2] for any tournament matrix T of order n, with equality holding if and only if T is regular or almost regular, according as n is odd or even. We construct, for each k between 1 and [(n?1)/2], a tournament matrix of order n whose reversal index is k. Finally, we suggest a few problems.     

Low-Light Trees, and Tight Lower Bounds for Euclidean Spanners

We show that for every n-point metric space M and positive integer k, there exists a spanning tree T with unweighted diameter O(k) and weight w(T)=O(k⋅n ^1/k )⋅w(MST(M)), and a spanning tree T′ with weight w(T′)=O(k)⋅w(MST(M)) and unweighted diameter O(k⋅n ^1/k ). These trees also achieve an optimal maximum degree. Furthermore, we demonstrate that these trees can be constructed efficiently.     

Roots of the affine Cremona group

Let k ^[n] = k[x ₁,…, x _n ] be the polynomial algebra in n variables and let \mathbbAⁿ = \textSpec  \boldk^{[ n ]} {\mathbb{A}^n} = {\text{Spec}}\;{{\bold{k}}^{\left[ n \right]}} . In this note we show that the root vectors of \textAu\textt^*( \mathbbAⁿ ) {\text{Au}}{{\text{t}}^*}\left( {{\mathbb{A}^n}} \right) , the subgroup of volume preserving automorphisms in the affine Cremona group \textAut( \mathbbAⁿ ) {\text{Aut}}\left( {{\mathbb{A}^n}} \right) , with respect to the diagonal torus are exactly the locally nilpotent derivations x ^α (∂/∂x _i ), where x ^α is any monomial not depending on x _i . This answers a question posed by Popov.