A short note on the Frobenius norm of the commutator

This note mainly aims to improve the inequality, proposed by Böttcher and Wenzel, giving the upper bound of the Frobenius norm of the commutator of two particular matrices in ? ^n×n . We first propose a new upper bound on basis of the Böttcher and Wenzel’s inequality. Motivated by the method used, the inequality ‖XY ? XY‖ _F ² ≤ 2‖X‖ _F ² ‖Y‖ _F ² is finally improved into $$ \left\| {XY - YX} \right\|_F^2 \leqslant 2\left\| X \right\|_F^2 \left\| Y \right\|_F^2 - 2[tr(X^T Y)]^2 . $$ . In addition, a further improvement is made.     

Some classes of pre-Hilbert algebras with norm-one central idempotent

In this paper we first characterize the pre-Hilbert algebras with a norm-one central idempotent e such that ‖ex‖ = ‖x‖ for any x ∈ A. This generalizes a well-known theorem by Ingelstam asserting that every alternative pre-Hilbert algebra with a unit 1 such that ‖1‖ = 1 is isomorphic to ?, ?, ? or $\mathbb{O}$ . We also show that every power-associative pre-Hilbert algebra satisfying ‖x ²‖ = ‖x‖² for every element has a unique nonzero idempotent, which is a unit element. In fact, the same conclusion will be proved in a more general setting. As application we give some conditions characterizing when a real algebra A, which is a prehilbert space, is isomorphic to one of the Hilbert algebras ?, ?, ? or $\mathbb{O}$ .     

On bounds for solutions to nonlinear wave equations in Hilbert space with applications to nonlinear elastodynamics

For the nonlinear wave equationu _tt -Nu +G(t,u, u _t ) = ? in Hilbert space, with associated homogeneous initial data, we show how ana priori bound of the form ∫ ₀ ^T ∥G(τ,u, u _τ)∥² dτ ≤ κ ∫ ₀ ^T ∥?(τ)∥² dτ leads to upper and lower bounds for ∥u∥ in terms of ∥?∥. An application to nonlinear elastodynamics is presented.     

On the classification of conditionally integrable evolution systems in (1 + 1) dimensions

We generalize earlier results of Fokas and Liu and find all locally analytic (1 + 1)-dimensional evolution equations of order n that admit an N-shock-type solution with N ≤ n + 1. For this, we develop a refinement of the technique from our earlier work, where we completely characterized all (1+1)-dimensional evolution systems u _t = F (x, t, u, ?u/?x,..., ?ⁿ u/? x ⁿ) that are conditionally invariant under a given generalized (Lie-Bäcklund) vector field Q(x, t, u, ?u/?x,..., ?^k u/?x ^k)?/?u under the assumption that the system of ODEs Q = 0 is totally nondegenerate. Every such conditionally invariant evolution system admits a reduction to a system of ODEs in t, thus being a nonlinear counterpart to quasi-exactly solvable models in quantum mechanics.     

Wavelet characterization of growth spaces of harmonic functions

We consider the space h _ν ^∞ of harmonic functions in R ₊ ⁿ⁺¹ with finite norm ‖u‖ _ν = sup |u(x, t)|/v(t), where the weight ν satisfies the doubling condition. Boundary values of functions in h _ν ^∞ are characterized in terms of their smooth multiresolution approximations. The characterization yields the isomorphism of Banach spaces h _ν ^∞ ～ l ^∞. The results are also applied to obtain the law of the iterated logarithm for the oscillation of functions in h _ν ^∞ along vertical lines.     

The maximal Fejér operator on the spacesH p (R×···×R)

It is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from thed-dimensional Hardy spaceH _p (R×···×R) toL _p (R ^d ) (1/2<p<∞) and is of weak type (H ₁ ^?i ,L ₁) (i=1,…,d), where the Hardy spaceH ₁ ^?i is defined by a hybrid maximal function. As a consequence, we obtain that the Fejér means of a functionf ∈H ₁ ^?i ?L(logL) ^d?1 converge a.e. to the function in question. Moreover, we prove that the Fejér means are uniformly bounded onH _p (R×···×R) whenever 1/2<p<∞. Thus, in casef ∈H _p (R×···×R) the Fejér means converge tof inH _p (R×···×R) norm. The same results are proved for the conjugate Fejér means, too.     

Siegel’s Lemma and Sum-Distinct Sets

Let L(x)=a ₁ x ₁+a ₂ x ₂+???+a _n x _n , n≥2, be a linear form with integer coefficients a ₁,a ₂,…,a _n which are not all zero. A basic problem is to determine nonzero integer vectors x such that L(x)=0, and the maximum norm ‖x‖ is relatively small compared with the size of the coefficients a ₁,a ₂,…,a _n . The main result of this paper asserts that there exist linearly independent vectors x ₁,…,x _n?1∈? ⁿ such that L(x _i )=0, i=1,…,n?1, and $$\|{\mathbf{x}}_{1}\|\cdots\|{\mathbf{x}}_{n-1}\|<\frac{\|{\mathbf{a}}\|}{\sigma_{n}},$$ where a=(a ₁,a ₂,…,a _n ) and $$\sigma_{n}=\frac{2}{\pi}\int_{0}^{\infty}\left(\frac{\sin t}{t}\right)^{n}\,dt.$$ This result also implies a new lower bound on the greatest element of a sum-distinct set of positive integers (Erdös–Moser problem). The main tools are the Minkowski theorem on successive minima and the Busemann theorem from convex geometry.     

An example of asymmetry of convolution operators

Esistono un gruppo compatto non commutativoG ^～ ed un operatore di convoluzioneT tale che: perp∈[2,4] e perq∈[1,2),T∈L _p ^p (G ^～) eT?L _q ^q (G ^～).     

Interpolation and frames in certain Banach spaces of entire functions

The important class of generalized bases known as frames was first introduced by Duffin and Schaeffer in their study of nonharmonic Fourier series in L ² (?π, π) [4]. Here we consider more generally the classical Banach spacesE ^p(1 ≤ p ≤ ∞) consisting of all entire functions of exponential type at most π that belong to L^p (?∞, ∞) on the real axis. By virtue of the Paley-Wiener theorem, the Fourier transform establishes an isometric isomorphism between L ² (?π, π) andE ² . When p is finite, a sequence {λ _n} of complex numbers will be called aframe forE ^p provided the inequalities $$A\left\| f \right\|^p \leqslant \sum {\left| {f\left( {\lambda _\pi } \right)} \right|^p } \leqslant B\left\| f \right\|^p $$ hold for some positive constants A and B and all functions f inE ^p. We say that {λ _n} is aninterpolating sequence forE ^p if the set of all scalar sequences {f (λ _n)}, with f εE ^p, coincides with ?^p. If in addition {λ _n} is a set of uniqueness forE ^p, that is, if the relations f(λ _n)=0(?∞<n<∞), with f εE ^p, imply that f ≡0, then we call {λ _n} acomplete interpolating sequence. Plancherel and Pólya [7] showed that the integers form a complete interpolating sequence forE ^p whenever1<p<∞. In Section 2 we show that every complete interpolating sequence forE ^p(1<p<∞) remains stable under a very general set of displacements of its elements. In Section 3 we use this result to prove a far-reaching generalization of another classical interpolation theorem due to Ingham [6].     

Norm convergence of normalized iterates and the growth of Kœnigs maps

Let ? be an analytic function defined on the unit diskD, with ?(D)?D, ?(0)=0, and ?′(0)=λ≠0. Then by a classical result of G. K?nigs, the sequence of normalized iterates Φ _n /λ ⁿ converges uniformly on compact subsets ofD to a function σ analytic inD which satisfiesσ°φ=λσ. It is of interest in the study of composition operators to know if, whenever σ belongs to a Hardy spaceH _p , the sequence Φ _n /λ ⁿ converges to σ in the norm ofH _p . We show that this is indeed the case, generalizing a result of P. Bourdon obtained under the assumption that ? is univalent. When ? is inner, P. Bourdon and J. Shapiro have shown that σ does not belong to the Nevanlinna class, in particular it does not belong to anyH _p . It is natural to ask, how bad can the growth of σ be in this case? As a partial answer we show that σ always belongs to some Bergman spaceL _a ^p .     

On the expansion with respect to Steklov means of the second differences of functions

We shall prove that every function locally integrable in then-dimensional Euclidean spaceR ⁿ can be expanded into a series whose terms are the Steklov means of the second differences of the given function. In addition, the lengths of the edges of the cubes with respect to which averaging is taken form an infinite decreasing geometric progression. The series obtained in this way converge almost everywhere inR ⁿ . If the function expanded belongs to the Lebesgue spaceL _p on a compact set ofR ⁿ for some 1≤p<∞, then the expansion converges also in the norm of this space.     

Long time small solutions to nonlinear parabolic equations

A sharp result on global small solutions to the Cauchy problem $$u_t = \Delta u + f\left( {u,Du,D^2 u,u_t } \right)\left( {t > 0} \right),u\left( 0 \right) = u_0 $$ In Rⁿ is obtained under the the assumption thatf is C^1+r forr>2/n and ‖u ₀‖C²(R ⁿ ) +‖u ₀‖W ₁ ² (R ⁿ ) is small. This implies that the assumption thatf is smooth and ‖u ₀ ‖W ₁ ^k (R ⁿ )+‖u ₀‖W ₂ ^k (R ⁿ ) is small fork large enough, made in earlier work, is unnecessary.     

Difference Sets in {\mathbb{Z}}_4^m and {\mathbb{F}}_2^{2m}

Let R=GR(4,m) be the Galois ring of cardinality 4^m and let T be the Teichmüller system of R. For every map λ of T into { -1,+1} and for every permutation Π of T, we define a map φ _λ ^Π of Rinto { -1,+1} as follows: if x∈R and if x=a+2b is the 2-adic representation of x with x∈T and b∈T, then φ _λ ^Π (x)=λ(a)+2Tr(Π(a)b), where Tr is the trace function of R . For i=1 or i=-1, define D _i as the set of x in R such thatφ _λ ^Π =i. We prove the following results: 1) D _i is a Hadamard difference set of (R,+). 2) If φ is the Gray map of R into ${\mathbb{F}}_2^{2m}$ , then (D _i) is a difference set of ${\mathbb{F}}_2^{2m}$ . 3) The set of D _i and the set of φ(D _i) obtained for all maps λ and Π, both are one-to-one image of the set of binary Maiorana-McFarland difference sets in a simple way. We also prove that special multiplicative subgroups of R are difference sets of kind D _i in the additive group of R. Examples are given by means of morphisms and norm in R.     

On Dirichlet Type Problems of Polynomial Dirac Equations with Boundary Conditions

Let ${{\bf D}_{\bf x} := \sum_{i=1}^n \frac{\partial}{\partial x_i} e_i}$ be the Euclidean Dirac operator in ${\mathbb{R}^n}$ and let P(X) = a _m X ^m + . . . + a ₁ X +  a ₀ be a polynomial with real coefficients. Differential equations of the form P(D _x )u(x) = 0 are called homogeneous polynomial Dirac equations with real coefficients. In this paper we treat Dirichlet type problems of the a slightly less general form P(D _x )u(x) = f(x) (where the roots are exclusively real) with prescribed boundary conditions that avoid blow-ups inside the domain. We set up analytic representation formulas for the solutions in terms of hypercomplex integral operators and give exact formulas for the integral kernels in the particular cases dealing with spherical and concentric annular domains. The Maxwell and the Klein–Gordon equation are included as special subcases in this context.     

Duals of Hardy Spaces of B-valued Martingales for 0<r≤1

The dual of B-valued martingale Hardy space Hs(p)r(B) with small index 0 r ≤ 1,which is associated with the conditional p-variation of B-valued martingale,is characterized.In order to obtain the results,a new type of Campanato spaces for B-valued martingales is introduced and the classical technique of atomic decompositions is improved.Some results obtained here are connected closely with the p-uniform smoothness and q-uniform convexity of the underlying Banach space.     

On diversity and stability of unit bases for the Euclidean metric

We prove the existence of a family Ω(n) of 2 ^c (where c is the cardinality of the continuum) subgraphs of the unit distance graph (E ⁿ , 1) of the Euclidean space E ⁿ , n ≥ 2, such that (a) for each graph G ? Ω(n), any homomorphism of G to (E ⁿ , 1) is an isometry of E ⁿ ; moreover, for each subgraph G ₀ of the graph G obtained from G by deleting less than c vertices, less than c stars, and less than c edges (we call such a subgraph reduced), any homomorphism of G ₀ to (E ⁿ , 1) is an isometry (of the set of the vertices of G ₀); (b) each graph G ? Ω(n) cannot be homomorphically mapped to any other graph of the family Ω(n), and the same is true for each reduced subgraph of G.     

Conic volume ratio of the packing cone associate to a convex body

Given a convex body ${K\subset\mathbb{R}^n}$ (n??? 1) which contains o in its interior and ${{\bf u} \in S^{n-1}}$ , we introduce conic volume ratio r(K, u) of K in the direction of u by $$r(K, {\bf u})=\frac{vol(cone(K,{\bf u})\cap B_2^n)}{vol(B_2^n)},$$ where cone(K, u) is the packing cone of K in the direction of u. We prove that if K is an o-symmetric convex body in ${\mathbb{R}^n}$ and r(K, u) is a constant function of u, then K must be a Euclidean ball.     

Global existence for the Cauchy problem of the parabolic–parabolic Keller–Segel system on the plane

This paper is concerned with the Cauchy problem for the Keller–Segel system $$\left\{\begin{array}{l@{\quad}l}u_t = \nabla \cdot (\nabla u - u \nabla v) & \hbox{in } {\bf R}^{2} \times(0,\infty),\\v_t = \Delta v - \lambda v + u & \hbox{ in } {\bf R}^2 \times(0,\infty),\\u(x,0) = u_0 (x) \geq 0, \; v(x,0) = v_0 (x) \geq 0 & \hbox{ in} {\bf R}^2\end{array}\right.$$ with a constant λ ≥ 0, where ${(u_0, v_0) \in (L^1 ({\bf R}^2) \cap L^\infty ({\bf R}^2) ) \times (L^1 ({\bf R}^2) \cap H^1 ({\bf R}^2))}$ . Let $$m (u_0;{\bf R}^2) = \int\limits_{{\bf R}^2} u_0 (x) dx$$ . The same method as in [9] yields the existence of a blowup solution with m (u ₀; R ²) > 8π. On the other hand, it was recently shown in [7] that under additional hypotheses ${u_0 \log (1 + |x|^2) \in L^1 ({\bf R}^2)}$ and ${u_0 \log u_0 \in L^1 ({\bf R}^2)}$ , any solution with m(u ₀; R ²) < 8π exists globally in time. In[18], the extra assumptions were taken off, but the condition on mass was restricted to m (u ₀; R ²) < 4π. In this paper, we prove that any solution with m (u ₀; R ²) < 8π exists globally in time under no extra conditions. Furthermore the global existence of solutions is obtained under some condition on u ₀ also in the critical case m (u ₀; R ²) = 8π.     

Hankel operators on planar domains

Let Ω be a bounded domain in the plane whose boundary consists of a finite number of disjoint analytic simple closed curves LetA denote the space of analytic functions on Ω which are square integrable over Ω with respect to area measure and letP denote the orthogonal projection ofL ²(Ω,dA) ontoA. A functionb inA induces a Hankel operator (densely defined) onA by the ruleH _b (g)=(I?P)bg. This paper continues earlier investigations of the authors and others by determining conditions under whichH _b is bounded, compact, or lies in the Schatten-von Neumann idealS _p , 1<p<∞     

Existence and nonuniqueness of solutions to a functional-differential equation

We examine the functional-differential equation Δu(x) — div(u(H(x))f (x)) = 0 on a torus which is a generalization of the stationary Fokker-Planck equation. Under sufficiently general assumptions on the vector field f and the map H, we prove the existence of a nontrivial solution. In some cases the subspace of solutions is established to be multidimensional.