On simultaneous diophantine approximations. Vectors of given diophantine type

For any monotone functionψ(y)=O(y ^1/s), we prove the existence of a continual family of vectors (α₁...,α_s) admitting infinitely many simultaneous ψ-approximations, but nocψ-approximations with some constantc>0. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 706–716, May, 1997. Translated by S. K. Lando     

Multi-bump solutions to Carrier's problem

We study the existence of well-known singularly perturbed BVP problem ε²y″=1−y²−2b(1−x²)y, y(−1)=y(1)=0 introduced by G.F. Carrier. In particular, we show that there exist multi-spike solutions, and the locations of interior spikes are clustered near x=0 and are separated by an amount of O(ε|lnε|), while only single spikes are allowed near the boundaries x=±1.     

Coincidence sets in quasilinear elliptic problems of monostable type

This paper concerns the formation of a coincidence set for the positive solution of the boundary value problem: −εΔpu=uq−1f(a(x)−u) in Ω with u=0 on ∂Ω, where ε is a positive parameter, Δpu=div(|∇u|^p−2∇u), 1<q?p<∞, f(s)∼|s|^θ−1s(s→0) for some θ>0 and a(x) is a positive smooth function satisfying Δpa=0 in Ω with infΩ|∇a|>0. It is proved in this paper that if 0<θ<1 the coincidence set Oε={x∈Ω:uε(x)=a(x)} has a positive measure for small ε and converges to Ω with order O(ε1/p) as ε→0. Moreover, it is also shown that if θ?1, then Oε is empty for any ε>0. The proofs rely on comparison theorems and the energy method for obtaining local comparison functions.     

Planar zero divisor graphs of partially ordered sets

We associate a graph G ^?(P) to a partially ordered set (poset, briefly) with the least element?0, as an undirected graph with vertex set P ^?=P?{0} and, for two distinct vertices x and y, x is adjacent to?y in?G ^?(P) if and only if {x,y} ^? ={0}, where, for a subset?S of?P, S ^? is the set of all elements x∈P with x≦s for all s∈S. We study some basic properties of?G ^?(P). Also, we completely investigate the planarity of?G ^?(P).     

Second order linear O.D.E. and Riccati equations

We prove that approximate solutions of the Riccati equation ?′ + ?² = a(x) yield asymptotic solutions y = e^∝x?(s)ds of the second order linear equation y″ = a(x)y. We show that the iterative scheme ?₀ = a, ?_{n + 1}² = a ? ?_n′ leads to asymptotic solutions of the cited linear equation in many interesting cases.     

Generalized Brouncker’s continued fractions and their logarithmic derivatives

In this paper, we study the continued fraction y(s,r) which satisfies the equation y(s,r)y(s+2r,r)=(s+1)(s+2r?1) for $r > \frac{1}{2}$ . This continued fraction is a generalization of the Brouncker’s continued fraction b(s). We extend the formulas for the first and the second logarithmic derivatives of b(s) to the case of y(s,r). The asymptotic series for y(s,r) at ∞ are also studied. The generalizations of some Ramanujan’s formulas are presented.     

Regular graphs with high edge degree

For any vertex x of a graph G let Δ(x) denote the set of vertices adjacent to x. We seek to describe the connected graphs G which are regular of valence n and in which for all adjacent vertices x and y |Δ(x) ∩ Δ(y)| = n ? 1 ? s. It is known that the complete graphs are the graphs for which s = 0. For any s, any complete many-partite graph, each part containing s + 1 vertices, is such a graph. We show that these are the only such graphs for which the valence exceeds 2s² ? s + 1. The graphs satisfying these conditions for s = 1 or 2 are characterized (up to the class of trivalent triangle-free graphs.)     

Conditional expectations and operator decompositions

For aC ^*-algebraA with a conditional expectation Φ:A → A onto a subalgebraB we have the linear decompositionA=B⊕H whereH=ker(Φ). Since Φ preserves adjoints, it is also clear that a similar decomposition holds for the selfadjoint parts:A ^s =B ^s ⊕H ^s (we useV ^s ={aεV;a ^*=a} for any subspaceV of A). Apply now the exponential function to each of the three termsA ^s ,B ^s , andH ^s . The results are: the setG ⁺ of positive invertible elements ofA, the setB ⁺ of positive invertible elements ofB, and the setC={e^h;h ^*=h, Φ(h)=0}, respectively. We consider here the question of lifting the decompositionA ^s =B ^s ⊕H ^s to the exponential sets. Concretely, is every element ofG ⁺ the product of elements ofB ⁺ andC, respectively, just as any selfadjoint element ofA is the sum of selfadjoint elements ofB andH? The answer is yes in the following sense: Eacha ε G ⁺ is the positive part of a productbe of elementsb ε B ⁺ and c εC, and bothb andc are uniquely determined and depend analytically ona. This can be rephrased as follows: The map (6, c) →(bc) ⁺ is an analytic diffeomorphism fromB ⁺ x C ontoG ⁺, where for any invertiblex ε A we denote with x⁺ the positive square root ofxx ^*. This result can be expressed equivalently as: The map (b, c) →bcb is a diffeomorphism between the same spaces. Notice that combining the polar decomposition with these results we can write every invertibleg ε A asg=bcu, whereb ε B ⁺,c ε C, andu is unitary. This decomposition is unique and the factorsb, c, u depend analytically ofg. In the case of matrix algebras with Φ=trace/dimension, the factorization corresponds tog=| det(g)|cu withc > 0,det(c)=1, andu unitary. This paper extends some results proved by G. Corach and the authors in [2]. Also, Theorem 2 states that the reductive homogeneous space resulting from a conditional expectation satisfies the regularity hypothesis introduced by L. Mata-Lorenzo and L. Recht in [5], Definition 11.1. The situation considered here is the ”general context” for regularity indicated in the introduction of the last mentioned paper.     

Linkage between Adiabatic Invariance and Symmetry of Solutions

In this article, for the symmetric pendulum equation and the symmetric bisuperlinear equation respectively, we show that there are two one-parameter families of solutions, y_s and y_a, so that one is adiabatically symmetric, y_s(?t)=y_s(t)+o(ε^k) for all k≥0, and the other adiabatically antisymmetric, y_a(?t)=?y_a(t)+o(ε^k) for all k≥0. By using the techniques of exponential asymptotics to calculate y′_s(0) and y_a(0), we demonstrate that, in general, they are not genuinely symmetric or antisymmetric, because these quantities are in fact exponentially small. Finally, after establishing a relationship between the total change in the leading-order adiabatic invariant and the quantity y′_s(0) for the family of solutions y_s of the bisuperlinear equation, we are able to reveal explicitly how the behavior of the adiabatic invariant depends on the complex singularities of the equation.      

A conjecture of Kippenhahn about the characteristic polynomial of a pencil generated by two Hermitian matrices. II

Let A be an n × n matrix; write A = H+iK, where i²=—1 and H and K are Hermitian. Let f(x,y,z) = det(zI?xH?yK). We first show that a pair of matrices over an algebraically closed field, which satisfy quadratic polynomials, can be put into block, upper triangular form, with diagonal blocks of size 1×1 or 2×2, via a simultaneous similarity. This is used to prove that if $\text{f(x,y,z) = [g(x,y,z)]}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}$, where g has degree 2, then for some unitary matrix U, the matrix U¹AU is the direct sum of $\text{n}\text{2}$ copies of a 2×2 matrix A₁, where A₁ is determined, up to unitary similarity, by the polynomial g(x,y,z). We use the connection between f(x,y,z) and the numerical range of A to investigate the case where f(x,y,z) has the form (z?αax? βy)^r[g(x,y,z)]^s, where g(x,y,z) is irreducible of degree 2.     

A new infeasible-interior-point algorithm for linear programming over symmetric cones

In this paper we present an infeasible-interior-point algorithm, based on a new wide neighbourhood N(τ₁, τ₂, η), for linear programming over symmetric cones. We treat the classical Newton direction as the sum of two other directions. We prove that if these two directions are equipped with different and appropriate step sizes, then the new algorithm has a polynomial convergence for the commutative class of search directions. In particular, the complexity bound is O(r^1.5logε^?1) for the Nesterov-Todd (NT) direction, and O(r²logε^?1) for the xs and sx directions, where r is the rank of the associated Euclidean Jordan algebra and ε > 0 is the required precision. If starting with a feasible point (x⁰, y⁰, s⁰) in N(τ₁, τ₂, η), the complexity bound is \(O\left( {\sqrt r \log {\varepsilon ^{ - 1}}} \right)\) for the NT direction, and O(rlogε^?1) for the xs and sx directions. When the NT search direction is used, we get the best complexity bound of wide neighborhood interior-point algorithm for linear programming over symmetric cones.     

On quasigroups rich in associative triples

Let G be a group and $\text{G(1)}$ a quasigroup on the same underlying set. Let dist($\text{G, G(1)}$) denote the number of pairs (x, y) ?G² such that $\text{xy ≠ x 1 y}$. For a finite quasigroup Q, n = card(Q), let t = dist(Q) = min dist(G, Q), where G runs through all groups with the same underlying set, and s = s(Q) the number of non-associative triples. Then 4tn?2t²?24t?s?4tn. If 1 ? s < 3n²/32, then 3tn < s holds as well. Let n ? 168 be an even integer and let σ = min s(Q), where Q runs through all non-associative quasigroups of order n. Then σ = 16n?64.     

On homogenization for non-self-adjoint locally periodic elliptic operators

In this note we consider the homogenization problem for a matrix locally periodic elliptic operator on R ^d of the form A ^ε = ?divA(x, x/ε)?. The function A is assumed to be Hölder continuous with exponent s ∈ [0, 1] in the “slow” variable and bounded in the “fast” variable. We construct approximations for (A ^ε ? μ)^?1, including one with a corrector, and for (?Δ) ^s/2(A ^ε ? μ)^?1 in the operator norm on L ₂(R ^d ) ⁿ . For s ≠ 0, we also give estimates of the rates of approximation.     

What Is the Complexity of Stieltjes Integration?

We study the complexity of approximating the Stieltjes integral ∫¹₀ f(x) dg(x) for functions f having r continuous derivatives and functions g whose sth derivative has bounded variation. Let r(n) denote the nth minimal error attainable by approximations using at most n evaluations of f and g, and let comp(ε) denote the ε-complexity (the minimal cost of computing an ε-approximation). We show that r(n)≍n^{−min{r, s+1}} and that comp(ε)≍ε^{−1/min{r, s+1}}. We also present an algorithm that computes an ε-approximation at nearly minimal cost.     

Bessel (Riesz) potentials on banach function spaces and their applications I theory

In this paper, we shall introduce the concept of the Bessel (Riesz) potential Köthe function spacesX ^s (X ^s ) and give some dual estimates for a class of operators determined by a semi-group in the spacesL ^q (?T, T; X ^s ) (L ^q (?T, T; X ^s )). Moreover, some time-spaceL ^p ?L ^p′ estimates for the semi-group exp(it(-Δ) ^m/2) and the operatorA:=∫ ₀ ^t exp(i(t-τ)(-Δ) ^m/2)·dτ in the Lebesgue-Besov spacesL ^q (?T,T;B _p,2 ^s are given. On the basis of these results, in a subsequent paper we shall present some further applications to a class of nonlinear wave equations.     

The number of positive integers ≤x and free of prime factors >y

The number defined by the title is denoted by Ψ(x, y). Let $\text{u =}\text{log}\text{x}\text{log}\text{y}$ and let ?(u) be the function determined by ?(u) = 1, 0 ≤ u ≤ 1, u?′(u) = ? ?(u ? 1), u > 1. We prove the following:Theorem. For x sufficiently large and log y ≥ (log log x)², Ψ(x,y) ? x?(u) while for 1 + log log x ≤ log y ≤ (log log x)², and ε > 0, $\text{Ψ(x, y) ?}{}_{\mathrm{\epsilon }}\text{x?(u)}\text{exp}\text{(?u e}\text{xp}\text{(?(}\text{log}\text{y)}{}^{\mathrm{<mtext>(</mtext><mtext>3</mtext><mtext>5</mtext><mtext>\; ?\; \epsilon )</mtext>}}\text{))}$.The proof uses a weighted lower approximation to Ψ(x, y), a reinterpretation of this sum in probability terminology, and ultimately large-deviation methods plus the Berry-Esseen theorem.     

Proper partial geometries with Singer groups and pseudogeometric partial difference sets

A partial geometry admitting a Singer group G is equivalent to a partial difference set in G admitting a certain decomposition into cosets of line stabilizers. We develop methods for the classification of these objects, in particular, for the case of abelian Singer groups. As an application, we show that a proper partial geometry Π=pg(s+1,t+1,2) with an abelian Singer group G can only exist if t=2(s+2) and G is an elementary abelian 3-group of order ³(s+1) or Π is the Van Lint-Schrijver partial geometry. As part of the proof, we show that the Diophantine equation (m³−1)/2=(²rw−1)/(r²−1) has no solutions in integers m,r?1, w?2, settling a case of Goormaghtigh's equation.     

Strong edge colorings of uniform graphs

For a graph G=(V(G),E(G)), a strong edge coloring of G is an edge coloring in which every color class is an induced matching. The strong chromatic index of G, χ_s(G), is the smallest number of colors in a strong edge coloring of G. The strong chromatic index of the random graph G(n,p) was considered in Discrete Math. 281 (2004) 129, Austral. J. Combin. 10 (1994) 97, Austral. J. Combin. 18 (1998) 219 and Combin. Probab. Comput. 11 (1) (2002) 103. In this paper, we consider χ_s(G) for a related class of graphs G known as uniform or ε-regular graphs. In particular, we prove that for 0<ε?d<1, all (d,ε)-regular bipartite graphs G=(U∪V,E) with |U|=|V|?n₀(d,ε) satisfy χ_s(G)?ζ(ε)Δ(G)², where ζ(ε)→0 as ε→0 (this order of magnitude is easily seen to be best possible). Our main tool in proving this statement is a powerful packing result of Pippenger and Spencer (Combin. Theory Ser. A 51(1) (1989) 24).