<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.     

<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">L</Emphasis>Σ(≤ <Emphasis Type="Italic">ω</Emphasis>)-spaces and spaces of continuous functions

We present a few results and problems related to spaces of continuous functions with the topology of pointwise convergence and the classes of LΣ(≤ ω)-spaces; in particular, we prove that every Gul’ko compact space of cardinality less or equal to $ \mathfrak{c} $ \mathfrak{c} is an LΣ(≤ ω)-space.     

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.     

An <Emphasis Type="Italic">L</Emphasis>-approach for packing (<Emphasis Type="Italic">ℓ</Emphasis>, <Emphasis Type="Italic">w</Emphasis>)-rectangles into rectangular and <Emphasis Type="Italic">L</Emphasis>-shaped pieces

This paper presents an approach using a recursive algorithm for packing (?, w)-rectangles into larger rectangular and L-shaped pieces. Such a problem has actual applications for non-guillotine cutting and pallet/container loading. Our motivation for developing the L-approach is based on the fact that it can solve difficult pallet loading instances. Indeed, it is able to solve all testing problems (more than 20 000 representatives of infinite equivalence classes of the literature), including the 18 hard instances unresolved by other heuristics. We conjecture that the L-approach always finds optimum packings of (?, w)-rectangles into rectangular pieces. Moreover, the approach may also be useful when dealing with cutting and packing problems involving L-shaped pieces.     

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.     

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.     

Bases of exponents in weighted spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Italic">−π, π</Emphasis>)

The system of exponents $ \left\{ {e^{i\lambda _n t} } \right\}_{n \in \mathbb{Z}} $ \left\{ {e^{i\lambda _n t} } \right\}_{n \in \mathbb{Z}} is considered. A sufficient condition for a Riesz-property basis in the weighted space L ^p (−π, π) is obtained.     

Ultracontractivity and embedding into <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript>

Given a self-adjoint semigroup e^−tA satisfying an ultracontractivity bound of the type , we find conditions on the sequence that imply that f is a bounded function. Sobolev’s classical embedding theorem says that, when A is the Laplace operator on , for some k > d/4 suffices to imply that f is bounded. In the cases we are interested in, the desired condition involves the whole sequence and depends on the behavior of the ultracontractivity function m. Research of A. Bendikov was supported by the Polish Goverment Scientific Research Fund, Grant 1 PO3 A 03129. Research of T. Coulhon was partially supported by the European Commission (IHP Network “Harmonic Analysis and Related Problems” 2002–2006, Contract HPRN-CT-2001-00273-HARP). Research of L. Saloff-Coste was partially supported by NSF grant DMS-0102126.     

Continuity properties of law-invariant (quasi-)convex risk functions on <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript>

We study continuity properties of law-invariant (quasi-)convex functions \({f:L^\infty(\Omega, \mathcal{F}, \mathbb{P}) \to (-\infty,\infty]}\) over a non-atomic probability space \({(\Omega, \mathcal{F}, \mathbb{P})}\). This is a supplementary note to Jouini et al. (Adv Math Econ 9:49–71, 2006).     

On the divergence of trigonometric Fourier series in classes <Emphasis Type="Italic">ϕ</Emphasis>(<Emphasis Type="Italic">L</Emphasis>) close to <Emphasis Type="Italic">L</Emphasis>

We show the unimprovability of a theorem on sufficient convergence conditions for the trigonometric Fourier series of a function in classes ?(L) in the case when the class ?(L) is “close” to L.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Poincaré inequality for general symmetric forms

Lp Poincare inequalities for general symmetric forms are established by new Cheeger＇s isoperimetric constants. Lp super-Poincare inequalities are introduced to describe the equivalent conditions for the Lp compact embedding, and the criteria via the new Cheeger＇s constants for those inequalities are presented. Finally, the concentration or the volume growth of measures for these inequalities are studied.     

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.     

Optimal <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Riemannian Gagliardo–Nirenberg inequalities

Let (M, g) be a compact Riemannian manifold of dimension n ≥  2 and 1 <  p ≤  2. In this work we prove the validity of the optimal Gagliardo–Nirenberg inequality

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.     

<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.     

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.      

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 numerical index of real <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(µ)-spaces

We give a lower bound for the numerical index of the real space L _p (µ) showing, in particular, that it is non-zero for p ≠ 2. In other words, it is shown that for every bounded linear operator T on the real space L _p (µ), one has

$\sup \left\{ {|\int {|x{|^{p - 1}}{\rm{sign}}(x)Tx d\mu |:x \in {L_p}\left( \mu \right), ||x|| = 1} } \right\} \ge {{{M_p}} \over {12{\rm{e}}}}||T||$

where \({M_p} = {\max _{t \in \left[ {0,1} \right]}}{{|{t^{p - 1}} - t|} \over {1 + {t^p}}} > 0\) for every p ≠ 2. It is also shown that for every bounded linear operator T on the real space L _p (µ), one has

$\sup \left\{ {\int {|x{|^{p - 1}}|Tx| d\mu :x \in {L_p}\left( \mu \right), ||x|| = 1} } \right\} \ge {1 \over {2{\rm{e}}}}||T||$

.

     

Fibers of the <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript> algebra and disintegration of measures

It is shown that Gelfand transforms of elements \({f\in L^{\infty} (\mu)}\) are almost constant at almost every fiber \({\Pi^{-1}(\{x\})}\) of the spectrum of L ^∞(μ) in the following sense: for each \({f\in L^{\infty} (\mu)}\) there is an open dense subset U = U(f) of this spectrum having full measure and such that the Gelfand transform of f is constant on the intersection \({\Pi^{-1}(\{x\})\cap U}\). As an application a new approach to disintegration of measures is presented, allowing one to drop the usually taken separability assumption.