On boundary-value problems for generalized analytic and harmonic functions

The present paper is a natural continuation of our last articles on the Riemann, Hilbert, Dirichlet, Poincaré, and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic functions and the so-called A-harmonic functions with arbitrary boundary data that are measurable with respect to the logarithmic capacity. Here, we extend the corresponding results to generalized analytic functions h : D→C with sources g : ∂z-h = g ∈ Lp , p > 2, and to generalized harmonic functions U with sources G : ΔU =G ∈Lp , p > 2. Our approach is based on the geometric (functional-theoretic) interpretation of boundary values in comparison with the classical operator approach in PDE. Here, we will establish the corresponding existence theorems for the Poincaré problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations ΔU =G with arbitrary boundary data that are measurable with respect to the logarithmic capacity. A few mixed boundary-value problems are considered as well. These results can be also applied to semilinear equations of mathematical physics in anisotropic and inhomogeneous media.

Genericity and Universality for Operator Ideals

A bounded linear operator $U$ between Banach spaces is universal for the complement of some operator ideal $\mathfrak{J}$ if it is a member of the complement and it factors through every element of the complement of $\mathfrak{J}$. In the first part of this paper, we produce new universal operators for the complements of several ideals, and give examples of ideals whose complements do not admit such operators. In the second part of the paper, we use descriptive set theory to study operator ideals. After restricting attention to operators between separable Banach spaces, we call an operator ideal $\mathfrak{J}$ generic if whenever an operator $A$ has the property that every operator in $\mathfrak{J}$ factors through a restriction of $A$, then every operator between separable Banach spaces factors through a restriction of $A$. We prove that many classical operator ideals (such as strictly singular, weakly compact, Banach–Saks) are generic and give a sufficient condition, based on the complexity of the ideal, for when the complement does not admit a universal operator. Another result is a new proof of a theorem of M. Girardi and W. B. Johnson, which states that there is no universal operator for the complement of the ideal of completely continuous operators.

Spectral analysis of integro-differential operators applied in linear antenna modelling

The current on a linear strip or wire solves an equation governed by a linear integro-differential operator that is the composition of the Helmholtz operator and an integral operator with a logarithmically singular displacement kernel. Investigating the spectral behaviour of this classical operator, we first consider the composition of the second-order differentiation operator and the integral operator with logarithmic displacement kernel. Employing methods of an earlier work by J. B. Reade, in particular the Weyl–Courant minimax principle and properties of the Chebyshev polynomials of the first and second kind, we derive index-dependent bounds for the ordered sequence of eigenvalues of this operator and specify their ranges of validity. Additionally, we derive bounds for the eigenvalues of the integral operator with logarithmic kernel. With slight modification our result extends to kernels that are the sum of the logarithmic displacement kernel and a real displacement kernel whose second derivative is square integrable. Employing this extension, we derive bounds for the eigenvalues of the integro-differential operator of a linear strip with the complex kernel replaced by its real part. Finally, for specific geometry and frequency settings, we present numerical results for the eigenvalues of the considered operators using Ritz's methods with respect to finite bases.

A modified prefix operator well suited for area-efficient brick-based adder implementations

The implementation of integrated circuits becomes more and more difficult in the Ultra-Deep-Submicron regime due to sub-wavelength lithography issues. An approach called Brick-Based Design was recently proposed to eliminate the disadvantages of staying with the classical approach to layout design. Prefix adders are a core component in a wide variety of applications due to their high speed and regular topology. In this paper, a modified prefix operator for prefix adders is proposed which is well suited for brick-style layout implementation and, in addition, offers an increase in efficiency. The proposed operator makes it possible to use a mirror gate for the generation of both generate and propagate signals, which exhibits a forbidden input signal combination. This "forbidden state" causes an increase in power dissipation due to transient short circuit currents. The effect of the forbidden state was quantified as part of a comparison against the classical prefix operator, based on 64-bit Sklansky adders implemented in a 40-nm CMOS technology. The effects of the forbidden state were found to be well acceptable. The implementation of the adder based on the proposed prefix operator reduces the area by 29% while increasing the power by 13% compared to one based on the classical operator.

Embeddings for monotone functions and $B_p$ weights, between discrete and continuous

We characterize the embedding for general weighted Lebesgue spaces of monotone functions by using the analog embedding for discrete monotone sequences indexed over the integers. We then use these results to obtain the boundedness of the discrete Hardy operator and to study the connections with the Hardy classical operator in the continuous setting.

On the Boundedness of Classical Operators on Weighted Lorentz Spaces

Conditions on weights 𝑢(·), υ(·) are given so that a classical operator T sends the weighted Lorentz space Lrs (υd𝑥) into Lpq (υd𝑥). Here T is either a fractional maximal operator Mα or a fractional integral operator Iα or a Calderón–Zygmund operator. A characterization of this boundedness is obtained for Mα and Iα when the weights have some usual properties and max(r, s) ≤ min(p, q).

MATING-TYPE EFFECT ON CIS MUTATIONS LEADING TO CONSTITUTIVITY OF ORNITHINE TRANSAMINASE IN DIPLOID CELLS OF SACCHAROMYCES CEREVISIAE

ABSTRACT Cis-acting regulatory mutations have been isolated that affect L-ornithine transaminase (OTAse), an enzyme catalyzing the second step of arginine breakdown in yeast. These mutations lead to constitutive synthesis of OTAse at various levels. Two different types of mutations have been recovered, both of which are tightly linked to the structural gene (cargB) for this enzyme. One type behaves as a classical operator-constitutive mutation similar to the cargB+O-—l mutation previously described (DUBOIS et al. 1978) .—The second type is peculiar in two respects : the higher level of constitutive OTAse synthesis and the expression of constitutivity in diploid cells. These mutations are designated curgB+Oh. They behave as usual operator-constitutive mutations in diploid strains homozygous for mating type (a/a or α/α), but the constitutivity is strongly reduced in a/α diploid cells.