The smooth structure set of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We calculate \({\mathcal{S}^{{\it Diff}}(S^p \times S^q)}\), the smooth structure set of S ^p × S ^q , for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general \({\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}\) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of the forgetful map \({\mathcal{S}^{Diff}(S^{4j}\times S^{4k}) \rightarrow \mathcal{S}^{Top}(S^{4j}\times S^{4k})}\) is not in general a subgroup of the topological structure set.     

Difference sets with <Emphasis Type="Italic">n</Emphasis> = 5<Emphasis Type="Italic"> p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>

Let D be a (v, k, λ)-difference set in an abelian group G, and (v, 31) = 1. If n = 5p ^r with p a prime not dividing v and r a positive integer, then p is a multiplier of D. In the case 31|v, we get restrictions on the parameters of such difference sets D for which p may not be a multiplier.      

Free involutions on <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Topological free involutions on S ¹ × S ⁿ are classified up to conjugation. We prove that this is the same as classifying quotient manifolds up to homeomorphism. There are exactly four possible homotopy types of such quotients, and surgery theory is used to classify all manifolds within each homotopy type.     

Regular<Emphasis Type="Italic">n</Emphasis>-simplices in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with vertices in ℤ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper theI andII regularn-simplices are introduced. We prove that the sufficient and necessary conditions for existence of anI regularn-simplex in ℝ ⁿ are that ifn is even thenn = 4m(m + 1), and ifn is odd thenn = 4m + 1 with thatn + 1 can be expressed as a sum of two integral squares orn = 4m - 1, and that the sufficient and necessary condition for existence of aII regularn-simplex in ℝ ⁿ isn = 2m ² - 1 orn = 4m(m + 1)(m ∈ ℕ). The connection between regularn-simplex in ℝ ⁿ and combinational design is given.     

Cycles in <Emphasis Type="Italic">G</Emphasis>-orbits in <Emphasis Type="Italic">G</Emphasis><Superscript><Emphasis Type="Italic">ℂ</Emphasis></Superscript>-flag manifolds

There is a natural duality between orbits of a real form G of a complex semisimple group G on a homogeneous rational manifold Z=G /P and those of the complexification K of any of its maximal compact subgroups K: (,) is a dual pair if is a K-orbit. The cycle space C() is defined to be the connected component containing the identity of the interior of {g:g() is non-empty and compact}. Using methods which were recently developed for the case of open G-orbits, geometric properties of cycles are proved, and it is shown that C() is contained in a domain defined by incidence geometry. In the non-Hermitian case this is a key ingredient for proving that C() is a certain explicitly computable universal domain.Research of the first author partially supported by Schwerpunkt Global methods in complex geometry and SFB-237 of the Deutsche Forschungsgemeinschaft.The second author was supported by a stipend of the Deutsche Akademische Austauschdienst.     

<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">μ</Emphasis>) for vector-valued measure <Emphasis Type="Italic">μ</Emphasis>

Let μ be a measure from a σ-algebra of subsets of a set T into a sequentially complete Hausdorff topological vector space X. Assume that μ is convexly bounded, i.e., the convex hull of its range is bounded in X, and denote by L ¹(μ) the space of scalar valued functions on T which are integrable with respect to the vector measure μ. We study the inheritance of some properties from X to L ¹(μ). We show that the bounded multiplier property passes from X to L ¹(μ). Answering a 1972 problem of Erik Thomas, we show that for a rather large class of F-spaces X the non-containment of c ₀ passes from X to L ¹(μ).     

<Emphasis Type="Italic">C</Emphasis><Superscript>1+<Emphasis Type="Italic">α</Emphasis></Superscript>-Regularity for Two-Dimensional Almost-Minimal Sets in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We give a new proof and a partial generalization of Jean Taylor’s result (Ann. Math. (2) 103(3), 489–539, 1976) that says that Almgren almost-minimal sets of dimension 2 in ℝ³ are locally C ^1+α -equivalent to minimal cones. The proof is rather elementary, but uses a local separation result proved in Ann. Fac. Sci. Toulouse 18(1), 65–246, 2009 and an extension of Reifenberg’s parameterization theorem (David et al. in Geom. Funct. Anal. 18, 1168–1235, 2008). The key idea is still that if X is the cone over an arc of small Lipschitz graph in the unit sphere, but X is not contained in a disk, we can use the graph of a harmonic function to deform X and substantially diminish its area. The local separation result is used to reduce to unions of cones over arcs of Lipschitz graphs. A good part of the proof extends to minimal sets of dimension 2 in ℝ ⁿ , but in this setting our final regularity result on E may depend on the list of minimal cones obtained as blow-up limits of E at a point.     

Free completely <Emphasis Type="Italic">J</Emphasis><Superscript>(<Emphasis Type="Italic">ℓ</Emphasis>)</Superscript>-simple Semigroups

A semigroup is called completely J^(ι)-simple if it is isomorphic to some Rees matrix semigroup over a left cancellative monoid and each entry of whose sandwich matrix is in the group of units of the left cancellative monoid. It is proved that completely J^(ι)-simple semigroups form a quasivarity. Moreover, the construction of free completely J^(ι)-simple semigroups is given. It is found that a free completely J^(ι)-simple semigroup is just a free completely J ^*-simple semigroup and also a full subsemigroup of some completely simple semigroups.     

The fractional Hausdorff operators on the Hardy spaces <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

In this paper, we study the high-dimensional fractional Hausdorff operators and establish their boundedness on the real Hardy spaces H ^p (? ⁿ ) for 0 < p < 1.     

Square functions and <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> calculus on subspaces of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and on Hardy spaces

Let X be a (closed) subspace of L^p with 1≤p<∞, and let A be any sectorial operator on X. We consider associated square functions on X, of the form and we show that if A admits a bounded H^∞ functional calculus on X, then these square functions are equivalent to the original norm of X. Then we deduce a similar result when X=H¹(ℝ^N) is the usual Hardy space, for an appropriate choice of || ||_F. For example if N=1, the right choice is the sum for h ∈ H¹(ℝ), where H denotes the Hilbert transform.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript> norm estimates of the cost of the controllability for heat equations

This paper is concerned with the bound of the cost of approximate controllability and null controllability of heat equations, i.e., the minimal Lp norm and L∞ norm of a control needed to control the system approximately or a control needed to steer the state of the system to zero. The methods we use combine observability inequalities, energy estimates for heat equations and the dual theory.     

The <Emphasis Type="Italic">F</Emphasis><Superscript><Emphasis Type="Italic">ω</Emphasis></Superscript>-normalizers of finite groups

Given a nonempty set ω of primes and a nonempty formation F of finite groups, we define the F ^ω -normalizer in a finite group and study their properties (existence, invariance under certain homomorphisms, conjugacy, embedding, and so on) in the case that F is an ω-local formation. We so develop the results of Carter, Hawkes, and Shemetkov on the F-normalizers in groups.     

On blocks with <Emphasis Type="Italic">K</Emphasis>(<Emphasis Type="Italic">B</Emphasis>)−<Emphasis Type="Italic">L</Emphasis>(<Emphasis Type="Italic">B</Emphasis>)=1

We use the method of local representation and original method of Brauer to study the block with K(B)−L(B)=1, and get some properties on the defect group and the structure of this kind of blocks. Then, we show that K(B) conjecture holds for this kind of blocks.     

<Emphasis Type="Italic">C</Emphasis><Superscript>∞</Superscript> regularity of solutions of the Levi equation in<Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">2n+1</Emphasis></Superscript>

We study the regularity of the solutions of the Levi equation in ?²ⁿ⁺¹. It is a second order quasilinear equation whose characteristic matrix is positive semidefinite and has vanishing determinant at every point and for every functionu∈C ². We show that the operator associated to the equation can be represented as a sum of squares of non linear vector fields. Then, by using a freezing method, we prove theC ^∞ regularity of the solutions.     

On Lagrange interpolation for |<Emphasis Type="Italic">X</Emphasis>|<Superscript><Emphasis Type="Italic">α</Emphasis></Superscript> (0 < <Emphasis Type="Italic">α</Emphasis> < 1)

We study the optimal order of approximation for |x| ^α (0 < α < 1) by Lagrange interpolation polynomials based on Chebyshev nodes of the first kind. It is proved that the Jackson order of approximation is attained. Supported by the NSFC, 10601065.     

Optimum super-simple mixed covering arrays of type <Emphasis Type="Italic">a</Emphasis><Superscript>1</Superscript><Emphasis Type="Italic">b</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis>−1</Superscript>

A mixed covering array (MCA) of type (_v 1, v ₂,..., v _k ), denoted by MCA_λ (N; t, k, (v ₁, v ₂,..., v _k )), is an N × k array with entries in the i-th column from a set _{V i} of v _i symbols and has the property that each N × t sub-array covers all the t-tuples at least λ times, where 1 ≤ i ≤ k. An MCA _λ (N; t, k, (_v 1, v ₂,..., v _k )) is said to be super-simple, if each of its N × (t + 1) sub-arrays contains each (t + 1)-tuple at most once. Recently, it was proved by Tang, Yin and the author that an optimum super-simple MCA of type (a, b, b,..., b) is equivalent to a mixed detecting array (DTA) of type (a, b, b,..., b) with optimum size. Such DTAs can be used to generate test suites to identify and determine the interaction faults between the factors in a component-based system. In this paper, some combinatorial constructions of optimum super-simple MCAs of type (a, b, b,..., b) are provided. By employing these constructions, some optimum super-simple MCAs are then obtained. In particular, the spectrum across which optimum super-simple MCA₂(2b ²; 2, 4, (a, b, b, b))′s exist, is completely determined, where 2 ≤ a ≤ b.     

On the equation Δ<Emphasis Type="Italic">u</Emphasis> + <Emphasis Type="Italic">q</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)<Emphasis Type="Italic">u</Emphasis> = 0

Sufficient conditions for the blow-up of nontrivial generalized solutions of the interior Dirichlet problem with homogeneous boundary condition for the homogeneous elliptic-type equation Δu + q(x)u = 0, where either q(x) ≠ const or q(x) = const= λ > 0, are obtained. A priori upper bounds (Theorem 4 and Remark 6) for the exact constants in the well-known Sobolev and Steklov inequalities are established.     

Rigidity of Quasicrystallic and <Emphasis Type="Italic">Z</Emphasis><Superscript><Emphasis Type="Italic">γ</Emphasis></Superscript>-Circle Patterns

The uniqueness of the orthogonal Z ^γ -circle patterns as studied by Bobenko and Agafonov is shown, given the combinatorics and some boundary conditions. Furthermore we study (infinite) rhombic embeddings in the plane which are quasicrystallic, that is, they have only finitely many different edge directions. Bicoloring the vertices of the rhombi and adding circles with centers at vertices of one of the colors and radius equal to the edge length leads to isoradial quasicrystallic circle patterns. We prove for a large class of such circle patterns which cover the whole plane that they are uniquely determined up to affine transformations by the combinatorics and the intersection angles. Combining these two results, we obtain the rigidity of large classes of quasicrystallic Z ^γ -circle patterns.     

The <Emphasis Type="Italic">ψ</Emphasis>-associate <Emphasis Type="Italic">X</Emphasis><Superscript>#</Superscript> of a flat <Emphasis Type="Italic">X</Emphasis> in PG(<Emphasis Type="Italic">n</Emphasis>, 2)

For a given hypersurface ψ in PG(n, 2), with equation Q(x) = 0, where Q is a polynomial of (reduced) degree d > 1, a definition is given of the ψ-associate X ^# of a flat X in PG(n, 2). The definition involves the fully polarized form of the polynomial Q; in the cubic case d = 3 it reads: X ^# = {z ∈ PG(n, 2) | T(x, y, z) = 0 for all x, y ∈ X}, where T(x, y, z) denotes the alternating trilinear form obtained by completely polarizing Q. Some general results, valid for any degree d and projective dimension n, are first expounded. Thereafter several choices of ψ are visited, but for each choice just a few aspects are highlighted. Despite the partial nature of the survey quite a variety of behaviours of the ψ-associate are uncovered. Many of the choices of ψ which are considered are of cubic hypersurfaces in PG(5, 2). If ψ is the Segre variety it is shown that the 48 planes external to fall into eight pairs of ordered triplets {(P ₁, R ₁, S ₁), (P ₂, R ₂, S ₂)} such that and . Further those lines L of PG(5, 2) which are singular, satisfying that is L ^# = PG(5.2), are in this case shown to form a complete spread of 21 lines. Another result of note arises in the case where ψ is the underlying 35-set of a non-maximal partial spread Σ₅ of five planes in PG(5, 2), where it is shown that one plane is singled out by the property that every line is singular.      

On Analytic Campanato and <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>, <Emphasis Type="Italic">s</Emphasis>) Spaces

We investigate the relation between analytic Campanato spaces \(\mathcal {AL}_{p,s}\) and the spaces F(p, q, s), characterize the bounded and compact Riemann–Stieltjes operators from \(\mathcal {AL}_{p,s}\) to \(F(p,p-s-1,s)\). We also describe the corona theorem and the interpolating sequences for the class \(F(p,p-2,s)\), which is the Möbius invariant subspace of the analytic Besov type spaces \(B_p(s)\).