In this paper we continue our investigation in [5, 7, 8] on multipeak solutions to the problem

formula here

where Δ = [sum ]Ni=1δ2/δx2i is the Laplace operator in ℝN, 2 < q < ∞ for N = 1, 2, 2 < q < 2N/(N−2) for N[ges ]3, and Q(x) is a bounded positive continuous function on ℝN satisfying the following conditions.

(Q1) Q has a strict local minimum at some point x0∈ℝN, that is, for some δ > 0

formula here

for all 0 < [mid ]x−x0[mid ] < δ.

(Q2) There are constants C, θ > 0 such that

formula here

for all [mid ]x−x0[mid ] [les ] δ, [mid ]y−y0[mid ] [les ] δ.

Our aim here is to show that corresponding to each strict local minimum point x0 of Q(x) in ℝN, and for each positive integer k, (1.1) has a positive solution with k-peaks concentrating near x0, provided ε is sufficiently small, that is, a solution with k-maximum points converging to x0, while vanishing as ε → 0 everywhere else in ℝN.

This paper estimates from below the attractor dimension of the dynamical system determined from a chemotaxis growth model which was presented by Mimura and Tsujikawa. It is already known that the dynamical system has exponential attractors and it is also known by numerical computations that the model contains various pattern solutions. This paper is then devoted to estimating the attractor dimension from below and in fact to showing that, as the parameter of chemotaxis increases and tends to infinity, so does the attractor dimension. Such a result is in a good correlation with the numerical results.

When S is a discrete subsemigroup of a discrete group G such that G = S−1S, it is possible to extend circle-valued multipliers from S to G, to dilate (projective) isometric representations of S to (projective) unitary representations of G, and to dilate/extend actions of S by injective endomorphisms of a C*-algebra to actions of G by automorphisms of a larger C*-algebra. These dilations are unique provided they satisfy a minimality condition. The (twisted) semigroup crossed product corresponding to an action of S is isomorphic to a full corner in the (twisted) crossed product by the dilated action of G. This shows that crossed products by semigroup actions are Morita equivalent to crossed products by group actions, making powerful tools available to study their ideal structure and representation theory. The dilation of the system giving the Bost–Connes Hecke C*-algebra from number theory is constructed explicitly as an application: it is the crossed product C0([Aopf ]f)[rtimes ]ℚ*+, corresponding to the multiplicative action of the positive rationals on the additive group [Aopf ]f of finite adeles.

For a variety X of dimension n in ${\mathbb P}^r,\ r\geq n(k+1)+k$, the kth secant order of X is the number $\mu_k(X)$ of $(k+1)$-secant k-spaces passing through a general point of the kth secant variety. We show that, if $r>n(k+1)+k$, then $\mu_k(X)=1$ unless X is k-weakly defective. Furthermore we give a complete classification of surfaces $X\subset{\mathbb P}^r,\ r>3k+2$, for which $\mu_k(X)>1$.

Inverse Sturm–Liouville problems with eigenparameter-dependent boundary conditions are considered. Theorems analogous to those of both Hochstadt and Gelfand and Levitan are proved.

In particular, let ly = (1/r)(−(py′)′+qy), l˜y = (1/r˜)(−(p˜y′)′+q˜y),

formula here

where det Δ = δ > 0, c ≠ 0, det [sum ] > 0, t ≠ 0 and (cs + dr − au − tb)2 < 4(cr − ta)(ds − ub). Denote by (l; α; Δ) the eigenvalue problem ly = λy with boundary conditions y(0)cosα+y′(0)sinα = 0 and (aλ+b)y(1) = (cλ+d)(py′)(1). Define (l˜; α; Δ) as above but with l replaced by l˜. Let wn denote the eigenfunction of (l; α; Δ) having eigenvalue λn and initial conditions wn(0) = sin α and pw′n(0) = −cos α and let γn = −awn(1)+cpw′n(1). Define w˜n and γ˜n similarly.

As sample results, it is proved that if (l; α; Δ) and (l˜; α; Δ) have the same spectrum, and (l; α; Σ) and (l˜; α; Σ) have the same spectrum or ∫10[mid ]wn[mid ]2rdt+([mid ]γn[mid ]2/δ) = ∫10[mid ]w˜n[mid ]2r˜dt+([mid ]γ˜n[mid ]2/δ) for all n, then q/r = q˜/r˜.

A proper vertex-colouring of a graph is acyclic if there are no 2-coloured cycles. It is known that every planar graph is acyclically 5-colourable, and that there are planar graphs with acyclic chromatic number χa = 5 and girth g = 4. It is proved here that a planar graph satisfies χa [les ] 4 if g [ges ] 5 and χa [les ] 3 if g [ges ] 7.

Two rings A and B are said to be derived Morita equivalent if the derived categories Db(Mod A) and Db(Mod B) are equivalent. If A and B are derived Morita equivalent algebras over a field k, then there is a complex of bimodules T such that the functor T[otimes ]LA−[ratio ]Db(Mod A) → Db(Mod B) is an equivalence. The complex T is called a tilting complex.

When B = A the isomorphism classes of tilting complexes T form the derived Picard group DPic(A). This group acts naturally on the Grothendieck group Ko(A).

It is proved that when the algebra A is either local or commutative, then any derived Morita equivalent algebra B is actually Morita equivalent. This enables one to compute DPic(A) in these cases.

Assume that A is noetherian. Dualizing complexes over A are complexes of bimodules which generalize the commutative definition. It is proved that the group DPic(A) classifies the set of isomorphism classes of dualizing complexes. This classification is used to deduce properties of rigid dualizing complexes.

Finally finite k-algebras are considered. For the algebra A of upper triangular 2×2 matrices over k, it is proved that t3 = s, where t, s ∈ DPic(A) are the classes of A*:= Homk(A, k) and A[1] respectively. In the appendix, by Elena Kreines, this result is generalized to upper triangular n×n matrices, and it is shown that the relation tn+1 = sn−1 holds.

We solve the Kato square root problem for second order elliptic systems in divergence form under mixed boundary conditions on Lipschitz domains. This answers a question posed by Lions in 1962. To do this we develop a general theory of quadratic estimates and functional calculi for complex perturbations of Dirac-type operators on Lipschitz domains.

In [9], Shelah introduced a class of first order theories, which he called simple, properly containing the class of stable theories. Here we prove for simple theories, (i) the equivalence of forking and dividing, (ii) the symmetry and transivity of forking.

Let T be a self-affine tile in ℝn defined by an integral expanding matrix A and a digit set D. The paper gives a necessary and sufficient condition for the connectedness of T. The condition can be checked algebraically via the characteristic polynomial of A. Through the use of this, it is shown that in ℝ2, for any integral expanding matrix A, there exists a digit set D such that the corresponding tile T is connected. This answers a question of Bandt and Gelbrich. Some partial results for the higher-dimensional cases are also given.

The points which converge to $\infty$ under iteration of the maps $z\mapsto\lambda\exp(z)$ for $\lambda \in \mathbb{C} \backslash \{0\}$ are investigated. A complete classification of such ‘escaping points’ is given: they are organized in the form of differentiable curves called rays which are diffeomorphic to open intervals, together with the endpoints of certain (but not all) of these rays. Every escaping point is either on a ray or the endpoint (landing point) of a ray. This answers a special case of a question of Eremenko. The combinatorics of occurring rays, and which of them land at escaping points, are described exactly. It turns out that this answer does not depend on the parameter $\lambda$.

It is also shown that the union of all the rays has Hausdorff dimension 1, while the endpoints alone have Hausdorff dimension 2. This generalizes results of Karpińska for specific choices of $\lambda$.

The paper considers the iterated function systems of similitudes which satisfy a separation condition weaker than the open set condition, in that it allows overlaps in the iteration. Such systems include the well-known Bernoulli convolutions associated with the PV numbers, and the contractive similitudes associated with integral matrices. The latter appears frequently in wavelet analysis and the theory of tilings. One of the basic questions is studied: the absolute continuity and singularity of the self-similar measures generated by such systems. Various conditions to determine the dichotomy are given.

The paper investigates the partial regularity of the minimizers for quadratic functionals whose integrands have VMO coefficients in principal part and nonlinear terms that are Carathéodory functions. We use some majorizations for the functional, rather than the well known Euler equation associated to it.

This paper concerns a wide class of singular perturbation problems arising from such diverse fields as phase transitions, chemotaxis, pattern formation, population dynamics and chemical reaction theory. The corresponding elliptic equations in a bounded domain without any symmetry assumptions are studied. It is assumed that the mean curvature of the boundary has M isolated, non-degenerate critical points. Then it is shown that for any positive integer M[les ]M there exists a stationary solution with M local peaks which are attained on the boundary and which lie close to these critical points. The method is based on Lyapunov–Schmidt reduction.

Let $F(n,r,k)$ denote the maximum possible number of distinct edge-colorings of a simple graph on $n$ vertices with $r$ colors which contain no monochromatic copy of $K_k$. It is shown that for every fixed $k$ and all $n\,{>}\,n_0(k)$, $F(n,2,k)\,{=}\,2^{t_{k-1}(n)}$ and $F(n,3,k)\,{=}\,3^{t_{k-1}(n)}$, where $t_{k-1}(n)$ is the maximum possible number of edges of a graph on $n$ vertices with no $K_k$ (determined by Turán's theorem). The case $r\,{=}\,2$ settles an old conjecture of Erdős and Rothschild, which was also independently raised later by Yuster. On the other hand, for every fixed $r\,{>}\,3$ and $k\,{>}\,2$, the function $F(n,r,k)$ is exponentially bigger than $r^{t_{k-1}(n)}.$ The proofs are based on Szemerédi's regularity lemma together with some additional tools in extremal graph theory, and provide one of the rare examples of a precise result proved by applying this lemma.

Operator-valued Fourier multiplier theorems are used to establish maximal regularity results for an integro-differential equation with infinite delay in Banach spaces. Results are obtained under general conditions for periodic solutions in the vector-valued Lebesgue and Besov spaces. The latter scale includes in particular the Hölder spaces $C^{\alpha},\,0\,{<}\, \alpha \,{<}\, 1 .$

The paper is motivated by an analogy between cluster algebras and Kac–Moody algebras: both theories share the same classification of finite type objects by familiar Cartan–Killing types. However, the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac–Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to skew-symmetrizable matrices. We study an interplay between the two classes of matrices, in particular, establishing a new criterion for deciding whether a given skew-symmetrizable matrix gives rise to a cluster algebra of finite type.

A numerical quasi-isometry invariant $R(\Gamma)$ of a finitely generated group $\Gamma$ is defined whose values parametrize the difference between $\Gamma$ being uniformly embeddable in a Hilbert space and $C^{*}_{r}(\Gamma)$ being exact.

The exact representation of symmetric polynomials on Banach spaces with symmetric basis and also on separable rearrangement-invariant function spaces over [0, 1] and [0, ∞) is given. As a consequence of this representation it is obtained that, among these spaces, [lscr ]2n, L2n[0, 1], L2n[0, ∞) and L2n[0, ∞)∩L2m[0, ∞) where n, m are both integers are the only spaces that admit separating polynomials.

For an artin algebra $\Lambda$, cotorsion pairs are studied for the category ${\rm mod}\Lambda$ of finitely presented $\Lambda$-modules and for the category ${\rm Mod}\Lambda$ of all $\Lambda$-modules. It is shown that every cotorsion pair for ${\rm mod}\Lambda$ induces a cotorsion pair for ${\rm Mod}\Lambda$. This has some interesting applications, even for the category of finitely presented modules. Another theme of the paper is the interplay between cotorsion and torsion pairs. This leads to a conjecture which is an analogue of the telescope conjecture in stable homotopy theory.