Complete catalogue of graphs of maximum degree 3 and defect at most 4

We consider graphs of maximum degree 3, diameter D≥2 and at most 4 vertices less than the Moore bound M_3,D, that is, (3,D,−?)-graphs for ?≤4.We prove the non-existence of (3,D,−4)-graphs for D≥5, completing in this way the catalogue of (3,D,−?)-graphs with D≥2 and ?≤4. Our results also give an improvement to the upper bound on the largest possible number N_3,D of vertices in a graph of maximum degree 3 and diameter D, so that N_3,D≤M_3,D−6 for D≥5.     

On local convexity of nonlinear mappings between Banach spaces

We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ? < c the image f(B _?(x)) of each ?-ball B _?(x) ? U is convex. We give a lower bound on c via the second order Lipschitz constant Lip₂(f), the Lipschitz-open constant Lip_o(f) of f, and the 2-convexity number conv₂(X) of the Banach space X.     

Weighted Berezin transform in the polydisc

For c>−1, let νc denote a weighted radial measure on C normalized so that νc(D)=1. If f is harmonic and integrable with respect to νc over the open unit disc D, then for every ψ∈Aut(D). Equivalently f is invariant under the weighted Berezin transform; Bcf=f. Conversely, does the invariance under the weighted Berezin transform imply the harmonicity of a function? In this paper, we prove that for any 1?p<∞ and c₁,c₂>−1, a function f∈Lp(D²,νc₁×νc₂) which is invariant under the weighted Berezin transform; Bc₁,c₂f=f needs not be 2-harmonic.     

A slight improvement to Korenblum's constant

Let A²(D) be the Bergman space over the open unit disk D in the complex plane. Korenblum conjectured that there is an absolute constant c∈(0,1) such that whenever |f(z)|?|g(z)| in the annulus c<|z|<1, then ‖f(z)‖?‖g(z)‖. This conjecture had been solved by Hayman [W.K. Hayman, On a conjecture of Korenblum, Analysis (Munich) 19 (1999) 195-205. [1]], but the constant c in that paper is not optimal. Since then, there are many papers dealing with improving the upper and lower bounds for the best constant c. For example, in 2004 C. Wang gave an upper bound on c, that is, c<0.67795, and in 2006 A. Schuster gave a lower bound, c>0.21. In this paper we slightly improve the upper bound for c.     

On the parity of the partition function

Let N be the set of all positive integers and D a subset of N. Let p(D,n) be the number of partitions of n with parts in D and let |D(x)| denote the number of elements of D not exceeding x. It is proved that if D is an infinite subset of N such that p(D,n) is even for all n?n₀, then |D(x)|?logx/log2−logn₀/log2. Moreover, if D is an infinite subset of N such that p(D,n) is odd for all n?n₀ and , then |D(x)|?logx/log2−logn₀/log2. These lower bounds are essentially the best possible.     

Isometries of some F-algebras of holomorphic functions on the upper half plane

Linear isometries of N ^p (D) onto N ^p (D) are described, where N ^p (D), p > 1, is the set of all holomorphic functions f on the upper half plane D = {z ∈ ?: Im z > 0} such that sup _y >0 ∫_? ln ^p (1 + |(x + iy)|) dx < +∞. Our result is an improvement of the results by D.A. Efimov.     

Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains

For D, a bounded Lipschitz domain in Rⁿ, n ? 2, the classical layer potentials for Laplace's equation are shown to be invertible operators on L²(?D) and various subspaces of L²(?D). For 1 < p ? 2 and data in L^p(?D) with first derivatives in L^p(?D) it is shown that there exists a unique harmonic function, u, that solves the Dirichlet problem for the given data and such that the nontangential maximal function of ▽u is in L^p(?D). When n = 2 the question of the invertibility of the layer potentials on every L^p(?D), 1 < p < ∞, is answered.     

Maximum modulus sets in pseudoconvex boundaries

LedD be a strictly pseudoconvex domain in ? ⁿ withC ^∞ boundary. We denote byA ^∞(D) the set of holomorphic functions inD that have aC ^∞ extension to \(\bar D\) . A closed subsetE of ?D is locally a maximum modulus set forA ^∞(D) if for everyp∈E there exists a neighborhoodU ofp andf∈A ^∞(D∩U) such that |f|=1 onE∩U and |f|<1 on \(\bar D \cap U\backslash E\) . A submanifoldM of ?D is an interpolation manifold ifT _p (M)?T _p ^c (?D) for everyp∈M, whereT _p ^c (?D) is the maximal complex subspace of the tangent spaceT _p (?D). We prove that a local maximum modulus set forA ^∞ (D) is locally contained in totally realn-dimensional submanifolds of ?D that admit a unique foliation by (n?1)-dimensional interpolation submanifolds. LetD =D ₁ x ... xD _r ? ? ⁿ whereD _i is a strictly pseudoconvex domain withC ^∞ boundary in ? ⁿ _i ,i=1,…,r. A submanifoldM of ?D ₁×…×?D _r verifies the cone condition if \(II_p (T_p (M)) \cap \bar C[Jn_1 (p),...,Jn_r (p)] = \{ 0\} \) for everyp∈M, wheren _i (p) is the outer normal toD _i atp, J is the complex structure of ? ⁿ , \(\bar C[Jn_1 (p),...,Jn_r (p)]\) is the closed positive cone of the real spaceV _p generated byJ _{n 1}(p),…,J _{n r}(p), and II _p is the orthogonal projection ofT _p (?D) onV _p . We prove that a closed subsetE of ?D ₁×…×?D _r which is locally a maximum modulus set forA ^∞(D) is locally contained inn-dimensional totally real submanifolds of ?D ₁×…×?D _r that admit a foliation by (n?1)-dimensional submanifolds such that each leaf verifies the cone condition at every point ofE. A characterization of the local peak subsets of ?D ₁×…×?D _r is also given.     

A refined Schwarz inequality on the boundary

We consider a function f holomorphic in the unit disc D with f(D)???D and f(0)?=?f(z ₀)?=?0, for 0?<?|z ₀|?<?1. We obtain sharp lower bounds on the angular derivative f′(c) at the point c where |c|?=?|f(c)|?=?1.     

Quantitative characterization of the difference between Birkhoff orthogonality and isosceles orthogonality

In this paper we introduce a new geometry constant D(X) to give a quantitative characterization of the difference between Birkhoff orthogonality and isosceles orthogonality. We show that 1 and is the upper and lower bound for D(X), respectively, and characterize the spaces of which D(X) attains the upper and lower bounds. We calculate D(X) when X=(R²,‖⋅p_‖) and when X is a symmetric Minkowski plane respectively, we show that when X is a symmetric Minkowski plane D(X)=D(X^∗).     

Lp-bounds on curvature,elliptic estimates and rectifiability of singular setsBornes Lp sur la courbure,estimées elliptiques et rectifiabilité d'ensembles singuliers

We announce results on rectifiability of singular sets of pointed metric spaces which are pointed Gromov–Hausdorff limits on sequences of Riemannian manifolds, satisfying uniform lower bounds on Ricci curvature and volume, and uniform L_p-bounds on curvature. The rectifiability theorems depend on estimates for |Hess_h|_L2p, (|?Hess_h·|Hess_h|^p?2)_L2, where Δh=c, for some constant c. We also observe that (absent any integral bound on curvature) in the Kähler case, given a uniform 2-sided bound on Ricci curvature, the singular set has complex codimension 2. To cite this article: J. Cheeger, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 195–198.     

The Bishop-Phelps-Bollobás Theorem for operators from c 0 to uniformly convex spaces

We show that the pair of Banach spaces (c ₀, Y) has the Bishop-Phelps-Bollobás property when Y is uniformly convex. Further, when Y is strictly convex, if (c ₀, Y) has the Bishop-Phelps-Bollobás property then Y is uniformly convex for the case of real Banach spaces. As a corollary, we show that the Bishop-Phelps-Bollobás theorem holds for bilinear forms on c ₀ × ? _p (1 < p < ∞).     

Edge search in graphs with restricted test sets

Suppose a graph G(V,E) contains one defective edge e. We search for the endpoints of e by asking questions of the form “Is at least one of the vertices of X an endpoint of e?”, where X is a subset of V with cardinality at most p. Then what is the minimum number c_p(G) of questions, which are needed in the worst case to find e?We solve this search problem suggested by M. Aigner in [M. Aigner, Combinatorial Search, Teubner, 1988] by deriving lower and sharp upper bounds for c_p(G). For the case that G is the complete graph K_n the problem described above is equivalent to the (2,n) group testing problem with test sets of cardinality at most p. We present sharp upper and lower bounds for the worst case number c_p of tests for this group testing problem and show that the maximum difference between the upper and the lower bounds is 3.     

Asymptotics of the complexity of systems of integer linear forms for additive computations

The computational complexity of integer linear forms is studied. By l ₂(A) we denote the minimal number of the additions and subtractions required for computing the system of p linear forms in q variables x ₁, x ₂, …, x _q that are defined by an integer matrix A of size p × q (repeated use of the results of intermediate computation is permitted). We show that l ₂(A) ? log D(A), where D(A) is the maximum of the absolute values of the minors of A over all minors from order 1 to order min (p, q) (Theorem 1). Moreover, for each sequence of matrices A(n) of size p(n) × q(n) satisfying the condition p + q = o ((log log D(A))^1/2) as n → ∞ the bound l ₂(A) ? log D(A) + o(log D(A)) is valid (Theorem 2). Hence, for all fixed (and even weakly increasing) sizes of matrices that determine a system of integer linear forms, the upper bound on the computational complexity of this system is asymptotically equal to the lower bound.     

Uniqueness of linear combinations (mod p)

In an earlier paper [1] one of us examined the question: Given a prime p what is the maximal number n=f(p) in any set of residues {a₁, a₂, …, a_n} (mod p) so that at least one of the differences a_i−a_j is incongruent to all other differences a_i′−a_j′?It was determined that the answer is of the order of magnitude log p. That is, there exist constants 0<c₁<c₂ so that c₁ log p<f(p)<c₂ log p for all p.In this note we examine a natural generalization, where instead of differences we examine the set of linear expressions with prescribed coefficients.     

Some paranormed sequence spaces of non-absolute type derived by weighted mean

The sequence spaces ?_∞(p), c(p) and c₀(p) were introduced and studied by Maddox [I.J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc. 64 (1968) 335-340]. In the present paper, the sequence spaces λ(u,v;p) of non-absolute type which are derived by the generalized weighted mean are defined and proved that the spaces λ(u,v;p) and λ(p) are linearly isomorphic, where λ denotes the one of the sequence spaces ?_∞, c or c₀. Besides this, the β- and γ-duals of the spaces λ(u,v;p) are computed and the basis of the spaces c₀(u,v;p) and c(u,v;p) is constructed. Additionally, it is established that the sequence space c₀(u,v) has AD property and given the f-dual of the space c₀(u,v;p). Finally, the matrix mappings from the sequence spaces λ(u,v;p) to the sequence space μ and from the sequence space μ to the sequence spaces λ(u,v;p) are characterized.     

On embedding expanders into ℓ p spaces

In this note we show that the minimum distortion required to embed alln-point metric spaces into the Banach space ℓ _p is between (c ₁/p) logn and (c ₂/p) logn, wherec ₂>c ₁>0 are absolute constants and 1≤p<logn. The lower bound is obtained by a generalization of a method of Linial et al. [LLR95], by showing that constant-degree expanders (considered as metric spaces) cannot be embedded any better. Research supported by Czech Republic Grant GAČR 201/94/2167 and Charles University grants No. 351 and 361.     

Harmonic measure on simply connected domains of fixed inradius

LetD?C be a simply connected domain that contains 0 and does not contain any disk of radius larger than 1. ForR>0, letω _D (R) denote the harmonic measure at 0 of the set {z:|z|?R}??D. Then it is shown thatthere exist β>0and C>0such that for each such D,ω _D (R)≤Ce ^?βR ,for every R>0. Thus a natural question is: What is the supremum of all β′s , call it β₀, for which the above inequality holds for every suchD? Another formulation of the problem involves hyperbolic metric instead of harmonic measure. Using this formulation a lower bound for β₀ is found. Upper bounds for β₀ can be obtained by constructing examples of domainsD. It is shown that a certain domain whose boundary consists of an infinite number of vertical half-lines, i.e. a comb domain, gives a good upper bound. This bound disproves a conjecture of C. Bishop which asserted that the strips of width 2 are extremal domains. Harmonic measures on comb domains are also studied.     

On cardinality constrained cycle and path polytopes

Given a directed graph D = (N, A) and a sequence of positive integers ${1 \leq c_1 < c_2 < \cdots < c_m \leq |N|}Given a directed graph D = (N, A) and a sequence of positive integers 1 ￡ c₁ < c₂ < ? < c_m ￡ |N|{1 \leq c_1 < c_2 < \cdots < c_m \leq |N|}, we consider those path and cycle polytopes that are defined as the convex hulls of the incidence vectors simple paths and cycles of D of cardinality c _p for some p ? {1,?,m}{p \in \{1,\ldots,m\}}, respectively. We present integer characterizations of these polytopes by facet defining linear inequalities for which the separation problem can be solved in polynomial time. These inequalities can simply be transformed into inequalities that characterize the integer points of the undirected counterparts of cardinality constrained path and cycle polytopes. Beyond we investigate some further inequalities, in particular inequalities that are specific to odd/even paths and cycles.