Gleason parts separated by smooth curves

Suppose Γ is a simple closed C² curve in the complex plane and let W₁, W₂ be the components of the complement of Γ. Let X be a compact plane set. Necessary and sufficient conditions are given that any two points x₁?X ∩ W₁, and x₂?X ∩ W₂ belong to different Gleason parts for the algebra R(X). We also give an answer to the question: How thin can a nontrivial part for R(X) be ?     

Non-simply connected copes

Let (W⁴,?W⁴) be a 4-manifold. Let f₁,f₂,…,f_k:(D²,?D²)→ (W⁴,?W⁴) be transverse immersions that have spherical duals $\text{α}{}_1\text{,α}{}_2\text{,…,α}{}_{\mathrm{k}}\text{:S}{}^2\text{→}\text{W}\text{?}$. Then there are open disjoint subsets V₁, V₂,…,V_k of W, such that for each 1?i?k, (a) ?V_i=V₁∩?W and ?V_i is an open regular neighborhood of f_i(?D²) in ?W, and (b) (V_i,?V_i,f_i(?D²)) is proper homotopy equivalent to (M, ?M, d)—a standard object in which d bounds an embedded flat disk. If we could get a homeomorphism instead of a proper homotopy equivalence, then we would be able to prove a 5-dimensional s-cobordism theorem.     

Subexponential estimates in the height theorem and estimates on numbers of periodic parts of small periods

This paper is devoted to subexponential estimates in Shirshov’s height theorem. A word W is n-divisible if it can be represented in the form W = W ₀ W ₁ ?W _n , where W ₁ ? W ₂ ??? W _n . If an affine algebra A satisfies a polynomial identity of degree n, then A is spanned by non-n-divisible words of generators a ₁ ??? a _l . A. I. Shirshov proved that the set of non-n-divisible words over an alphabet of cardinality l has bounded height h over the set Y consisting of all words of degree ≤ n ? 1. We show that h < Φ (n, l), where Φ(n, l) = 2⁸⁷ l?n ^{12 log3 n+48}. Let l, n, and d ≥ n be positive integers. Then all words over an alphabet of cardinality l whose length is greater than ψ(n, d, l) are either n-divisible or contain the dth power of a subword, where ψ(n, d, l) = 2¹⁸ l(nd)^{3 log3(nd)+13} d ². In 1993, E. I. Zelmanov asked the following question in the Dniester Notebook: Suppose that F _2,m is a 2-generated associative ring with the identity x ^m = 0. Is it true that the nilpotency degree of F _2,m has exponential growth? We give the definitive answer to E. I. Zelmanov by this result. We show that the nilpotency degree of the l-generated associative algebra with the identity x ^d = 0 is smaller than ψ(d, d, l). This implies subexponential estimates on the nilpotency index of nil-algebras of arbitrary characteristic. Shirshov’s original estimate was just recursive; in 1982 a double exponent was obtained, and an exponential estimate was obtained in 1992. Our proof uses Latyshev’s idea of an application of the Dilworth theorem. We think that Shirshov’s height theorem is deeply connected to problems of modern combinatorics. In particular, this theorem is related to the Ramsey theory. We obtain lower and upper estimates of the number of periods of length 2, 3, n ? 1 in some non-n-divisible word. These estimates differ only by a constant.     

Higher dimensional manifolds with S 1-category 2

A closed topological n-manifold M ⁿ is of S ¹-category 2 if it can be covered by two open subsets W ₁, W ₂ such that the inclusions W _i → M ⁿ factor homotopically through maps W _i → S ¹. We show that for n?>?3, if ${{\rm cat}_{S^1}(M^n )=2}$ then M ⁿ is homeomorphic to S ⁿ or S ^n–1 × S ¹ or the non-orientable S ^n–1-bundle over S ¹. We also obtain an unknotting theorem for locally flat knots of S ^n–2 in S ⁿ and a characterization of S ¹?× S ^n–1.     

Minimal polynomials for compact sets of the complex plane

LetW _N(z)=a_Nz^N+... be a complex polynomial and letT _n be the classical Chebyshev polynomial. In this article it is shown that the polynomials (2a_N)^?n+1T_n(W_N), n ∈N, are minimal polynomials on all equipotential lines for {z∈C:|W _N(z)|≤1 Λ ImW _N(z)=0}     

Topological rearrangement of a function

We extend our results on weak diffeomorphism classes and decompositions of Sobolev functions to a more general framework. We introduce a family of decompositions of Sobolev functions W₀^1,p rich enough that we conjecture it allows decomposition of all Sobolev functions, not just the “craterless” ones considered in [7]. The associated weak diffeomorphism classes of a W₀^1,p Sobolev function are weakly closed when p ≥ n.     

Convergence parameter associated with a Markov chain and a family of functions

The proposed definition of convergence parameter R(W) corresponding to a Markov chain X with a measurable state space (E,?) and any nonempty setW of bounded below measurable functions f: E → ? is wider than the well-known definition of convergence parameter R in the sense of Tweedie or Nummelin. Very often, R(W) < ∞, and there exists a set playing the role of the absorbing set inNummelin’s definition ofR. Special attention is paid to the case in whichE is locally compact, X is a Feller chain on E, and W coincides with the family ? ₀ ⁺ of all compactly supported continuous functions f ≥ 0 (f ? 0). In particular, certain conditions for R(? ₀ ⁺ )^?1 to coincide with the norm of an appropriate modification of the chain transition operator are found.     

A generalized weight for linear codes and a Witt-MacWilliams theorem

Let F be a finite field, H a subgroup of $\text{F}{}^1$ of index ν, and α₁,…, α_ν coset representatives. For each n-tuple u = (u₁,…, u_n) ?Fⁿ define W_H(u) = (w₁(u),…, w_ν(u)), where w_m(u) = #{u_i: u_i?α_mH}. An H-monomial map on Fⁿ is an automorphism of Fⁿ whose matrix with respect to the co-ordinate basis is of the form P · D, where P is a permutation matrix and D is a diagonal matrix with non-zero entries from H. Suppose C is an (n, k) code over F (that is, a k-dimensional subspace of Fⁿ) and that ?: C → Fⁿ is an injective homomorphism which preserves W_H in the sense that W_H(?(u)) = W_H(u) for all u ?C. We prove that ? may be extended to an H-monomial map on Fⁿ. This generalization of a theorem of MacWilliams on the (Hamming) equivalence of codes may be considered an analogue of the Witt theorem of metric vector space theory.     

Boundary noncrossings of additive Wiener fields∗

Let {W _i (t), t ∈ ?₊}, i = 1, 2, be two Wiener processes, and let W ₃ = {W ₃(t), t ∈ ? ₊ ² } be a two-parameter Brownian sheet, all three processes being mutually independent. We derive upper and lower bounds for the boundary noncrossing probability P _f = P{W ₁(t ₁) + W ₂(t ₂) + W ₃(t) + f(t) ≤ u(t), t ∈ ? ₊ ² }, where f, u : ? ₊ ² → ? are two general measurable functions. We further show that, for large trend functions γf > 0, asymptotically, as γ → ∞, P _γf is equivalent to \( {P}_{\gamma}\underset{\bar{\mkern6mu}}{{}_f} \) , where \( \underset{\bar{\mkern6mu}}{f} \) is the projection of f onto some closed convex set of the reproducing kernel Hilbert space of the field W(t) = W ₁(t ₁) + W ₂(t ₂) + W ₃(t). It turns out that our approach is also applicable for the additive Brownian pillow.     

Asymptotic expansions for distributions of latent roots in multivariate analysis

Asymptotic expansions are given for the distributions of latent roots of matrices in three multivariate situations. The distribution of the roots of the matrix S₁(S₁ + S₂)^?1, where S₁ is $\text{W}{}_{\mathrm{<mtext>m</mtext>}}\text{(n}{}_1\text{, Σ, Ω)}$ and S₂ is W_m(n₂, Σ), is studied in detail and asymptotic series for the distribution are obtained which are valid for some or all of the roots of the noncentrality matrix Ω large. These expansions are obtained using partial-differential equations satisfied by the distribution. Asymptotic series are also obtained for the distributions of the roots of n^?1S, where S in W_m(n, Σ), for large n, and S₁S₂^?1, where S₁ is W_m(n₁, Σ) and S₂ is W_m(n₂, Σ), for large n₁ + n₂.     

Pointwise estimates of potentials for higher-order capacities

In a domain D=Ω\E∈ R ⁿ , we consider a nonlinear higher-order elliptic equation such that the corresponding energy space is W _p ^m (D)?W _q ¹ (D), q>mp, and estimate a solution u(x) of this equation satisfying the condition u(x)?kf(x)∈W _p ^m (D)?W _q ¹ (D), where k∈R ¹, f(x)∈ C ₀ ^∞ (Ω), and f(x)=1 for x∈F. We establish a pointwise estimate for u(x) in terms of the higher-order capacity of the set F and the distance from the point x to the set F.     

Lower bounds for the clique and the chromatic numbers of a graph

G is any simple graph with m edges and n vertices where these vertices have vertex degrees d(1)≥d(2)≥?≥d(n), respectively. General algorithms for the exact calculation of χ(G), the chromatic number of G, are almost always of ‘branch and bound’ type; this type of algorithm requires an easily constructed lower bound for χ(G). In this paper it is shown that, for a number of such bounds (many of which are described here for the first time), the bound does not exceed cl(G) where cl(G) is the clique number of G.In 1972 Myers and Liu showed that cl(G)≥n?(n?2m?n). Here we show that cl(G)≥ n?[n?(2m?n)(1+c²_v)^{$\text{1}\text{2}$}], where c_v is the vertex degree coefficient of variation in G, that cl(G)≥ Bondy's constructive lower bound for χ(G), and that cl(G)≥n?(n?W(G)), where W(G) is the largest positive integer such that W(G)≤d(W(G)+1) [W(G)+1 is the Welsh and Powell upper bound for χ(G)]. It is also shown that cl(G)+$\text{1}\text{3}$>n?(n?L(G))≥n?(n?λ₁) and that χ(G)≥ n?(n?λ₁); λ₁ is the largest eigenvalue of A, the adjacency matrix of G, and L(G) is a newly defined single-valued function of G similar to W(G) in its construction from the vertex degree sequence of G [L(G)≥W(G)].Finally, a ‘greedy’ lower bound for cl(G) is constructed from A and it is shown that this lower bound is never less than Bondy's bound; thereafter, for 150 random graphs with varying numbers of vertices and edge densities, the values of lower bounds given in this paper are listed, thereby illustrating that this last greedily obtained lower bound is almost always better than each of the other bounds.     

To the Theory of Sobolev-Type Classes of Functions with Values in a Metric Space

We study some classes of functions with values in a complete metric space which can be considered as analogs of the Sobolev spaces W _p ¹ . Earlier the author considered the case of functions on a domain of ? ⁿ . Here we study the general case of mappings on an arbitrary Lipschitz manifold. We give necessary auxiliary facts, consider some examples, and describe some methods of construction of lower semicontinuous functionals on the classes W _p ¹ (M), where M is a Lipschitz manifold.     

Nonnegative functions in weighted hardy spaces

Let W be a nonnegative summable function whose logarithm is also summable with respect to the Lebesgue measure on the unit circle. For 0?<?p?<?∞ , H^p (W) denotes a weighted Hardy space on the unit circle. When W?≡?1, H ^p(W) is the usual Hardy space H^p . We are interested in H^p ( W)₊ the set of all nonnegative functions in H^p ( W). If p?≥?1/2, H^p ₊ consists of constant functions. However H^p ( W)₊ contains a nonconstant nonnegative function for some weight W. In this paper, if p?≥?1/2 we determine W and describe H^p ( W)₊ when the linear span of H^p ( W)₊ is of finite dimension. Moreover we show that the linear span of H^p (W)₊ is of infinite dimension for arbitrary weight W when 0?<?p?<?1/2.     

Rate of convergence in a class of singular perturbations

Let V_?, W_?, W and X be Hilbert spaces (0 < ? ? 1) with V_? ? W_? ? W ? X algebraically and topologically, each space being dense in the one that follows it. For each t? [0, T] let a_?(t; u, v), b_?(t; u, v) and b(t; u, v) be continuous sesqui-linear forms on V_?, W_? and W, respectively, which satisfy certain ellipticity conditions. Consider the two equations a_?(t; u_?, v) + b_?(t; u_?, v) = 〈f_?, v〉 (v?V_?) and (u′, v)_x + b(t; u, v) = 〈f, v〉 (v?W). Estimates are obtained on the rate of convergence of u_? to u, assuming a_?(t; u, v) → (u, v)_x and b_?(t; u, v) → b(t; u, v) in an appropriate sense. These results are then applied to singular perturbation of a class of parabolic boundary value problems.     

On the points-lines-planes conjecture

It has been conjectured that in any matroid, if W₁, W₂, W₃ denote the number of points, lines, and planes respectively, then W₂² ≥ W₁W₃. We prove this conjecture (and some strengthenings) for matroids in which no line has five or more points, thus generalizing a result of Stonesifer, who proved it for graphic matroids.     

The impact of smooth W-grids in the numerical solution of singular perturbation two-point boundary value problems

This paper develops a semi-analytic technique for generating smooth nonuniform grids for the numerical solution of singularly perturbed two-point boundary value problems. It is based on the usual idea of mapping a uniform grid to the desired nonuniform grid. We introduce the W-grid, which depends on the perturbation parameter ? ? 1. For problems on [0, 1] with a boundary layer at one end point, the local mesh width h_i = x_i+1 − x_i, with 0 = x₀ < x₁ < ? < x_N = 1, is condensed at either 0 or 1. Two simple 2nd order finite element and finite difference methods are combined with the new mesh, and computational experiments demonstrate the advantages of the smooth W-grid compared to the well-known piecewise uniform Shishkin mesh. For small ?, neither the finite difference method nor the finite element method produces satisfactory results on the Shishkin mesh. By contrast, accuracy is vastly improved on the W-grid, which typically produces the nominal 2nd order behavior in L², for large as well as small values of N, and over a wide range of values of ?. We conclude that the smoothness of the mesh is of crucial importance to accuracy, efficiency and robustness.     

Approximation of continuous functions by trigonometric polynomials almost everywhere

We consider the problem of the rate of approximation of continuous 2π-periodic functions of class W^rH[ω]C by trigonometric polynomials of order n on sets of total measure. We prove that when r≥0,ω(δ)δ ^?1 → ∞ (δ → 0) there exists a function f ε W^rH[ω]C such thatf ε W^rH[ω]C and for any sequence {t_{n n=1} ^∞ we have almost everywhere on [0, 2π] $\begin{array}{l} \overline {\mathop {\lim }\limits_{n \to \infty } } \left| {f(x) - t_n (x)} \right|n^r \omega ^{ - 1} (1/n) > C_x > 0, \\ \overline {\mathop {\lim }\limits_{n \to \infty } } \left| {\tilde f(x) - t_n (x)} \right|n^r \omega ^{ - 1} (1/n) > C_x > 0. \\ \end{array}$      

Dalla scoperta dell'interazione debole alla scoperta dei suoi mediatori

After recalling the genesis of the weak interaction and the main steps in our progressive understanding of its structure, the need for the existence of mediator massive «vector bosons» is clarified. Mention is then made of the gauge theories which led to the «electroweak unification» predicting the existence of three vector bosons,W ⁺,W ^?,Z ⁰, of massesM _W ^±,M _Z ⁰ near respectively to 80 GeV/c² and 90 GeV/c². The second part of the article is dedicated to the so-called « \(p\bar p\) project» and to the two experiments (UA1 and UA2) by which physicists working at the European Organization for Nuclear Research (CERN, Geneva) reached the experimental demonstration of the existence ofW ⁺,W ^? andZ ⁰ (the heaviest particles identified so far by man) thus achieving one of the top scientific results of our century.