On the local integrability condition for generalised translation-invariant systems

In this article we study finitely generated wavelet systems with arbitrary dilation sets. In 2002 Hernandez et al. gave a characterization of when such a system forms a Parseval frame, assuming that a certain hypothesis known as the local integrability condition (LIC) holds. We show that, under some mild regularity assumption on the wavelets, the LIC is solely a density condition on the dilation sets. Using this new interpretation of the LIC, we further discuss when the characterization result holds.

For arbitrary real b > 1 and for a large real parameter t let R (b, t) be the number of lattice points between the curve by = x (1 ≤x≤t) and the x-axis, the points on the x-axis being counted with the factor 1 / 2. Furthermore, let the lattice rest\( {\mit\Gamma} (b, t) \) be defined as the difference between R (b, t) and the area of the domain. The asymptotic behaviour of \( {\mit\Gamma} (b, t) \) for \( t\rightarrow \infty \) and also for b→ 1+ are investigated.

Higher moments for lattice point discrepancy of convex domains and annuli

Motivated by long-standing conjectures on the discretization of classical inequalities in the Geometry of Numbers, we investigate a new set of parameters, which we call \emph{packing minima}, associated to a convex body $K$ and a lattice $\Lambda$. These numbers interpolate between the successive minima of $K$ and the inverse of the successive minima of the polar body of $K$, and can be understood as packing counterparts to the covering minima of Kannan & Lov\'{a}sz (1988). As our main results, we prove sharp inequalities that relate the volume and the number of lattice points in $K$ to the sequence of packing minima. Moreover, we extend classical transference bounds and discuss a natural class of examples in detail.

Rank-1 lattice rules based on a weighted star discrepancy with weights of a product form have been previously constructed under the assumption that the number of points is prime. Here, we extend these results to the non-prime case. We show that if the weights are summable, there exist lattice rules whose weighted star discrepancy is O(n −1+δ ), for any δ > 0, with the implied constant independent of the dimension and the number of lattice points, but dependent on δ and the weights. Then we show that the generating vector of such a rule can be constructed using a component-by-component (CBC) technique. The cost of the CBC construction is analysed in the final part of the paper.

We consider a random Gaussian ensemble of Laplace eigenfunctions on the 3D torus, and investigate the 1-dimensional Hausdorff measure (‘length’) of nodal intersections against a smooth 2-dimensional toral sub-manifold (‘surface’). A prior result of ours prescribed the expected length, universally proportional to the area of the reference surface, times the wavenumber, independent of the geometry. In this paper, for surfaces contained in a plane, we give an upper bound for the nodal intersection length variance, depending on the arithmetic properties of the plane. The bound is established via estimates on the number of lattice points in specific regions of the sphere.

On the number of lattice points in convex symmetric bodies and their duals

A new estimate is obtained for the residue R in the asymptotics of the number of integer points in a ball of radius a. The estimate has the form R ≪ a 17/14 + ɛ .

In their survey article on cyclic Steiner Quadruple Systems SQS(v) M. J. Grannel and T. S. Griggs advanced the conjecture (cf. [8, p. 412]) that their necessary condition for the existence of S-cyclic SQS(v) (cf. [7, p. 51]) is also sufficient. Some years prior to that E. Kohler [10] used a graph theoretical method to construct S-cyclic SQS(v). This method was extended in [17]-[20] and eventually used to reduce the conjecture of Grannel and Griggs to a number theoretic claim (cf. also [21], research problem 146). The main purpose of the present paper is to attack this claim. For the long intervals we have to distinguish four cases. The proof of cases I–III can be accomplished by a thorough study of how the multiples of a certain set belonging to the first column of a certain matrix (the elements of which are essentially the vertices of a graph corresponding to SQS(2p)) are distributed over the columns. The proof is by contradiction. Case IV is most difficult to treat and could only be dealt with by very deep lying means. We have to use an asymptotic formula on the number of lattice points (x,y) with xy ≡ 1 mod p (we speak of 1-points) in a rectangle and this formula shows that the 1-points are equidistributed. But even so our claim could not be proved for all intervals of admissible length. Intervals [a,b] with ^p—_m+1 <a <b < ^p−_m for some m and 3√4 p <b−a <p ^11−15 could not be covered. In the last section we discuss some conclusions which would follow from the non-existence of complete intervals.

Analogs of classical results on accumulated spectrograms, the sum of spectrograms of eigenfunctions of localization operators, are established for Gabor multipliers. We show that the lattice ℓ1 distance between the accumulated spectrogram and the indicator function of the Gabor multiplier mask is bounded by the number of lattice points near the boundary of the mask and that this bound is sharp in general. The methods developed for the proofs are also used to show that the Weyl-Heisenberg ensemble restricted to a lattice is hyperuniform when the Gabor frame is tight.

We present a general framework for computing various optimal embeddings of undirected and directed connected graphs in two and multi-dimensional integer lattices in time sub-exponential either in the minimum number n of lattice points used by such optimal embeddings or in the budget upper bound b on the number of lattice points that may be used in an embedding. The sub-exponential upper bounds in the two dimensional case and d-dimensional case are respectively of the form 2\(^{O(\sqrt{ln}log n)}\), 2\(^{O(\sqrt{lb}log b)}\) and 2\(^{O(dl^{1/d_n (d-1)/d}{\rm log} n)}\), 2\(^{O(dl^{1/d_b (d-1)/d}{\rm log} b)}\), where l stands for the degree of the allowed overlap. For the problem of minimum total edge length planar or multi-dimensional embedding or layout of a graph and the problem of an optimal protein folding in the so called HP model we obtain the upper bounds in terms of n. Note that in case of protein folding n is also the size of the input. The list of problems for which we can derive the upper bounds in terms of b includes among other things:

We consider a discrete analogue of Hardy‐Littlewood maximal operator on the generalized Lebesque space of sequences defined on ℤ. It is known a necessary and sufficient condition which guarantees an existence of a real number p &gt; 1 such that the norms in the space and in the classical space are equivalent. Of course, this condition immediately implies the boundedness of maximal operator on and, moreover, lim⁡|n|→∞⁡pn = p. We construct two examples of sequences {pn} satisfying lim⁡|n|→∞⁡pn = p in this paper. In the first example the maximal operator is unbounded on and the sequence {pn} from the second example does not satisfy but the maximal operator is bounded. Moreover, it is known a sufficient integral condition to a behavior of a function p(x) at infinity which guarantees the boundedness of the maximal operator on Lp(⋅)(ℝn). As a main result of this paper we construct a function p(x) which does not satisfy this integral condition nevertheless the maximal operator is bounded.

We study the two boundary value problems with an integral boundary condition for a degenerate equation of even order. They are reduced to corresponding problems with local boundary conditions, but for an integral-differential equation . We prove the solvability of these auxiliary problems, using the method of consecutive approximations.

In the first part of this paper, we deal with the unique solvability of multidimensional backward stochastic differential equations (BSDEs) with a $p$-integrable terminal condition $(p>1)$ and a superlinear growth generator. We introduce a new local condition on the generator (assumption (H.4)) and then show that it ensures the existence and uniqueness, as well as the $L^p$-stability, of solutions. The assumptions that we impose on the generator are local in the three variables $y, z, \omega $, and therefore we also cover the BSDEs with stochastic Lipschitz coefficient. Our conditions on the generator go beyond all existing ones in the literature. For instance, the generator is not assumed uniformly continuous and therefore cannot satisfy the classical Osgood condition. Furthermore, it could be neither locally monotone in the $y$-variable nor locally Lipschitz in the $z$-variable. Although we are focused on multidimensional BSDEs, our results on uniqueness and stability are new even for one-dimensional BS...

Well-posedness of semilinear heat equations in L1