Counting sets with small sumset and applications

We study the number of k-element sets A? {1,...,N} with |A+A| ≤ K|A| for some (fixed) K > 0. Improving results of the first author and of Alon, Balogh, Samotij and the second author, we determine this number up to a factor of 2 ^{o ( k )} N ^{o (1)} for most N and k. As a consequence of this and a further new result concerning the number of sets A??/N? with |A+A| ≤ c|A|², we deduce that the random Cayley graph on ?/N? with edge density ½ has no clique or independent set of size greater than (2+o(1)) log₂ N, asymptotically the same as for the Erd?s-Rényi random graph. This improves a result of the first author from 2003 in which a bound of 160log₂ N was obtained. As a second application, we show that if the elements of A ? ? are chosen at random, each with probability 1/2, then the probability that A+A misses exactly k elements of ? is equal to (2+O(1))^?k/2 as k → ∞.     

On Lebesgue Measure of Integral Self-Affine Sets

Let A be an expanding integer n×n matrix and D be a finite subset of ? ⁿ . The self-affine set T=T(A,D) is the unique compact set satisfying the equality \(A(T)=\bigcup_{d\in D}(T+d)\). We present an effective algorithm to compute the Lebesgue measure of the self-affine set T, the measure of the intersection T∩(T+u) for u∈? ⁿ , and the measure of the intersection of self-affine sets T(A,D ₁)∩T(A,D ₂) for different sets D ₁, D ₂?? ⁿ .     

Categorical resolutions of a class of derived categories

We clarify the relation between the subcategory D_(hf)~b(A) of homological finite objects in D~b(A)and the subcategory K~b(P) of perfect complexes in D~b(A), by giving two classes of abelian categories A with enough projective objects such that D_(hf)~b(A) = K~b(P), and finding an example such that D_(hf)~b(A)≠K~b(P). We realize the bounded derived category D~b(A) as a Verdier quotient of the relative derived category D_C~b(A), where C is an arbitrary resolving contravariantly finite subcategory of A. Using this relative derived categories, we get categorical resolutions of a class of bounded derived categories of module categories of infinite global dimension.We prove that if an Artin algebra A of infinite global dimension has a module T with inj.dimT ∞ such that ~⊥T is finite, then D~b(modA) admits a categorical resolution; and that for a CM(Cohen-Macaulay)-finite Gorenstein algebra, such a categorical resolution is weakly crepant.     

The Cauchy problem for operators with injective symbol in the Lebesgue space <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> in a domain

Let D be a bounded domain in ? ⁿ (n ≥ 2) with infinitely smooth boundary ?D. We give some necessary and sufficient conditions for the Cauchy problem to be solvable in the Lebesgue space L ²(D) in D for an arbitrary differential operator A having an injective principal symbol. Furthermore, using bases with double orthogonality, we construct Carleman’s formula that restores a (vector-)function in L ²(D) from the Cauchy data given on a relatively open connected set Γ ? ?D and the values Au in D whenever the data belong to L ²(Γ) and L ²(D) respectively.     

Perturbed Evolution Problems with Continuous Bounded Variation in Time and Applications

This paper is devoted to the study of evolution problems of the form \(-\frac {du}{dr}(t) \in A(t)u(t) + f(t, u(t))\) in a new setting, where, for each t, A(t) : D(A(t)) → 2 ^H is a maximal monotone operator in a Hilbert space H and the mapping t?A(t) has continuous bounded or Lipschitz variation on [0, T], in the sense of Vladimirov’s pseudo-distance. The measure dr gives an upper bound of that variation. The perturbation f is separately integrable on [0, T] and separately Lipschitz on H. Several versions and new applications are presented.     

Elliptic function of level 4

The article is devoted to the theory of elliptic functions of level n. An elliptic function of level n determines a Hirzebruch genus called an elliptic genus of level n. Elliptic functions of level n are also of interest because they are solutions of the Hirzebruch functional equations. The elliptic function of level 2 is the Jacobi elliptic sine function, which determines the famous Ochanine–Witten genus. It is the exponential of the universal formal group of the form F(u, v) = (u² ? v²)/(uB(v) ? vB(u)), B(0) = 1. The elliptic function of level 3 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uA(v)² ? vA(u)²), A(0) = 1, A″(0) = 0. In the present study we show that the elliptic function of level 4 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uB(v) ? vB(u)), where A(0) = B(0) = 1 and for B′(0) = A″(0) = 0, A′(0) = A₁, and B″(0) = 2B₂ the following relation holds: (2B(u) + 3A₁u)² = 4A(u)³ ? (3A₁² ? 8B₂)u²A(u)². To prove this result, we express the elliptic function of level 4 in terms of the Weierstrass elliptic functions.     

G-matrices,J-orthogonal matrices,and their sign patterns

A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D₁ and D₂ such that A^?T = D₁AD₂, where A^?T denotes the transpose of the inverse of A. Denote by J = diag(±1) a diagonal (signature) matrix, each of whose diagonal entries is +1 or ?1. A nonsingular real matrix Q is called J-orthogonal if Q^TJQ = J. Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation of a J-orthogonal matrix. An investigation into the sign patterns of the J-orthogonal matrices is initiated. It is observed that the sign patterns of the G-matrices are exactly the column permutations of the sign patterns of the J-orthogonal matrices. Some interesting constructions of certain J-orthogonal matrices are exhibited. It is shown that every symmetric staircase sign pattern matrix allows a J-orthogonal matrix. Sign potentially J-orthogonal conditions are also considered. Some examples and open questions are provided.     

Mappings connected with the gradient of conformal radius

In this paper we prove the following conformity criterion for the gradient of conformal radius ?R(D, z) of a convex domain D: the boundary ?D has to be a circumference. We calculate coefficients K(r) for K(r)-quasiconformal mappings ?R(D(r), z), D(r) ? D, 0 < r < 1, and complete the results obtained by F. G. Avkhadiev and K.-J. Wirths for the structure of boundary elements of quasiconformal mappings of the domain D.     

A note on generalized Lie derivations of prime rings

Let R be a prime ring of characteristic not 2, A be an additive subgroup of R, and F, T, D, K: A-R be additive maps such that F([x, y]) = F(x)_y-yK(x)-T(y)_{x + x}D(y) for all x, yEA. Our aim is to deal with this functional identity when A is R itself or a noncentral Lie ideal of R. Eventually, we are able to describe the forms of the mappings F, T, D, and K in case A = R with deg(R) > 3 and also in the case A is a noncentral Lie ideal and deg(R) > 9. These enable us in return to characterize the forms of both generalized Lie derivations, D-Lie derivations and Lie centralizers of R under some mild assumptions. Finally, we give a generalization of Lie homomorphisms on Lie ideals.     

Inversion and subspaces of a finite field

Consider two F _q -subspaces A and B of a finite field, of the same size, and let A ^?1 denote the set of inverses of the nonzero elements of A. The author proved that A ^?1 can only be contained in A if either A is a subfield, or A is the set of trace zero elements in a quadratic extension of a field. Csajbók refined this to the following quantitative statement: if A ^?1 ? B, then the bound |A ^?1∩B| ≤ 2|B|/q ? 2 holds. He also gave examples showing that his bound is sharp for |B| ≤ q ³. Our main result is a proof of the stronger bound |A ^?1 ∩ B| ≤ |B|/q · (1 + O _d (q ^?1/2)), for |B| = q ^d with d > 3. We also classify all examples with |B| ≤ q ³ which attain equality or near-equality in Csajbók’s bound.     

On the Nitsche conjecture for harmonic mappings in ℝ<Superscript>2</Superscript> and ℝ<Superscript>3</Superscript>

We give the new inequality related to the J. C. C. Nitsche conjecture (see [6]). Moreover, we consider the two- and three-dimensional case. LetA(r, 1)={z:r<|z|<1}. Nitsche's conjecture states that if there exists a univalent harmonic mapping from an annulusA(r, 1), to an annulusA(s, 1), thens is at most 2r/(r ²+1).Lyzzaik's result states thats<t wheret is the length of the Grötzsch's ring domain associated withA(r, 1) (see [5]). Weitsman's result states thats≤1/(1+1/2(r logr)²) (see [8]).Our result for two-dimensional space states thats≤1/(1+1/2 log² r) which improves Weitsman's bound for allr, and Lyzzaik's bound forr close to 1. For three-dimensional space the result states thats≤1/(r?logr).     

Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions

Let D be an open connected subset of the complex plane C with sufficiently smooth boundary ?D. Perturbing the Cauchy problem for the Cauchy–Riemann system ??u = f in D with boundary data on a closed subset S ? ?D, we obtain a family of mixed problems of the Zaremba-type for the Laplace equation depending on a small parameter ε ∈ (0, 1] in the boundary condition. Despite the fact that the mixed problems include noncoercive boundary conditions on ?D\S, each of them has a unique solution in some appropriate Hilbert space H ⁺(D) densely embedded in the Lebesgue space L ₂(?D) and the Sobolev–Slobodetski? space H ^1/2?δ(D) for every δ > 0. The corresponding family of the solutions {u _ε} converges to a solution to the Cauchy problem in H ⁺(D) (if the latter exists). Moreover, the existence of a solution to the Cauchy problem in H ⁺(D) is equivalent to boundedness of the family {u _ε} in this space. Thus, we propose solvability conditions for the Cauchy problem and an effective method of constructing a solution in the form of Carleman-type formulas.     

On Homogenization for Non-Self-Adjoint Periodic Elliptic Operators on an Infinite Cylinder

We consider an operator A^ε on L₂(\({\mathbb{R}^{{d_1}}} \times {T^{{d_2}}}\)) (d₁ is positive, while d₂ can be zero) given by A^ε = ?div A(ε^?1x₁,x₂)?, where A is periodic in the first variable and smooth in a sense in the second. We present approximations for (A^ε ? μ)^?1 and ?(A^ε ? μ)^?1 (with appropriate μ) in the operator norm when ε is small. We also provide estimates for the rates of approximation that are sharp with respect to the order.     

Multiplying a conjugacy class by its inverse in a finite group

Suppose that G is a finite group and K is a non-trivial conjugacy class of G such that KK^?1 = 1 ∪ D ∪ D^?1 with D a conjugacy class of G. We prove that G is not a non-abelian simple group and we give arithmetical conditions on the class sizes determining the solvability and the structure of 〈K〉 and 〈D〉.     

Augmentation quotients for burnside rings of generalized dihedral groups

Let H be a finite abelian group of odd order, D be its generalized dihedral group, i.e., the semidirect product of C₂ acting on H by inverting elements, where C₂ is the cyclic group of order two. Let Ω (D) be the Burnside ring of D, Δ(D) be the augmentation ideal of Ω (D). Denote by Δⁿ(D) and Q_n(D) the nth power of Δ(D) and the nth consecutive quotient group Δⁿ(D)/Δⁿ⁺¹(D), respectively. This paper provides an explicit Z-basis for Δⁿ(D) and determines the isomorphism class of Q_n(D) for each positive integer n.     

Upper recollements and triangle expansions

By using the stable t-structure induced by an adjoint pair, we extend several results concerning recollements to upper (resp. lower) recollements. These include the fundamental results of Parshall and Scott on comparisons of recollements, Wiedemann’s result on the global dimension and Happel’s result on the finitistic dimension, occurring in a recollement (D ^b (A′),D ^b (A),D ^b (A″)) of bounded derived categories of Artin algebras. We introduce and describe a triangle expansion of a triangulated category and illustrate it by examples.     

Sturm–Liouville problems in weighted spaces in domains with non-smooth edges. II

We consider a (generally, noncoercive) mixed boundary value problem in a bounded domain D of Rⁿ for a second order elliptic differential operator A(x, ?). The differential operator is assumed to be of divergent form in D and the boundary operator B(x, ?) is of Robin type on ?D. The boundary of D is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset Y ? ?D and control the growth of solutions near Y. We prove that the pair (A, B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set Y. Moreover, we prove the completeness of root functions related to L.     

The Folded (2<Emphasis Type="Italic">D</Emphasis> + 1)-cube and Its Uniform Posets

Let Γ denote the folded (2D + 1)-cube with vertex set X and diameter D ≥ 3. Fix x ∈ X. We first define a partial order ≤ on X as follows. For y, z ∈ X let y ≤ z whenever ?(x, y) + ?(y, z) = ?(x, z). Let R (resp. L) denote the raising matrix (resp. lowering matrix) of Γ. Next we show that there exists a certain linear dependency among RL², LRL,L²R and L for each given Q-polynomial structure of Γ. Finally, we determine whether the above linear dependency structure gives this poset a uniform structure or strongly uniform structure.     

Endomorphism Algebras of 2-term Silting Complexes

We study possible values of the global dimension of endomorphism algebras of 2-term silting complexes. We show that for any algebra A whose global dimension gl.dim A ≤ 2 and any 2-term silting complex P in the bounded derived category D ^b (A) of A, the global dimension of \(\text {End}_{{D^b(A)}}(\mathbf {P})\) is at most 7. We also show that for each n > 2, there is an algebra A with gl.dim A = n such that D ^b (A) admits a 2-term silting complex P with \(\mathrm {gl. dim~}\text {End}_{{D^b(A)}}(\mathbf {P})\) infinite.     

Coefficient estimates of negative powers and inverse coefficients for certain starlike functions

For ?1≤B<A≤1, let \(\mathcal {S}^{*}(A,B)\) denote the class of normalized analytic functions \(f(z)= z+{\sum }_{n=2}^{\infty }a_{n} z^{n}\) in |z|<1 which satisfy the subordination relation z f ^′(z)/f(z)?(1 + A z)/(1 + B z) and Σ^?(A,B) be the corresponding class of meromorphic functions in |z|>1. For \(f\in \mathcal {S}^{*}(A,B)\) and λ>0, we shall estimate the absolute value of the Taylor coefficients a _n (?λ,f) of the analytic function (f(z)/z)^?λ . Using this we shall determine the coefficient estimate for inverses of functions in the classes \(\mathcal {S}^{*}(A,B)\) and Σ^?(A,B).