A triangle model of criminality

This paper is concerned with a quantitative model describing the interaction of three sociological species, termed as owners, criminals and security guards, and denoted by X, Y and Z respectively. In our model, Y is a predator of the species X, and so is Z with respect to Y . Moreover, Z can also be thought of as a predator of X, since this last population is required to bear the costs of maintaining Z. We propose a system of three ordinary differential equations to account for the time evolution of X(t), Y (t) and Z(t) according to our previous assumptions. Out of the various parameters that appear in that system, we select two of them, denoted by H, and h, which are related with the efficiency of the security forces as a control parameter in our discussion. To begin with, we consider the case of large and constant owners population, which allows us to reduce (3)–(5) to a bidimensional system for Y (t) and Z(t). As a preliminary step, this situation is first discussed under the additional assumption that Y (t) + Z(t) is constant. A bifurcation study is then performed in terms of H and h, which shows the key role played by the rate of casualties in Y and Z, that results particularly in a possible onset of bistability. When the previous restriction is dropped, we observe the appearance of oscillatory behaviours in the full two-dimensional system. We finally provide a exploratory study of the complete model (3)–(5), where a number of bifurcations appear as parameter H changes, and the corresponding solutions behaviours are described

A Guide to Localized Frames and Applications to Galerkin-like Representations of Operators

This chapter offers a detailed survey on intrinsically localized frames and the corresponding matrix representation of operators. We re-investigate the properties of localized frames and the associated Banach spaces in full detail. We investigate the representation of operators using localized frames in a Galerkin-type scheme. We show how the boundedness and the invertibility of matrices and operators are linked and give some sufficient and necessary conditions for the boundedness of operators between the associated Banach spaces.Comment: 32 page

An optimally concentrated Gabor transform for localized time-frequency components

Gabor analysis is one of the most common instances of time-frequency signal analysis. Choosing a suitable window for the Gabor transform of a signal is often a challenge for practical applications, in particular in audio signal processing. Many time-frequency (TF) patterns of different shapes may be present in a signal and they can not all be sparsely represented in the same spectrogram. We propose several algorithms, which provide optimal windows for a user-selected TF pattern with respect to different concentration criteria. We base our optimization algorithm on $l^p$-norms as measure of TF spreading. For a given number of sampling points in the TF plane we also propose optimal lattices to be used with the obtained windows. We illustrate the potentiality of the method on selected numerical examples

A spoonful of L‐fucose—an efficient therapy for GFUS‐CDG, a new glycosylation disorder

Abstract Congenital disorders of glycosylation are a genetically and phenotypically heterogeneous family of diseases affecting the co‐ and posttranslational modification of proteins. Using exome sequencing, we detected biallelic variants in GFUS (NM_003313.4) c.[632G>A];[659C>T] (p.[Gly211Glu];[Ser220Leu]) in a patient presenting with global developmental delay, mild coarse facial features and faltering growth. GFUS encodes GDP‐L‐fucose synthase, the terminal enzyme in de novo synthesis of GDP‐L‐fucose, required for fucosylation of N‐ and O‐glycans. We found reduced GFUS protein and decreased GDP‐L‐fucose levels leading to a general hypofucosylation determined in patient's glycoproteins in serum, leukocytes, thrombocytes and fibroblasts. Complementation of patient fibroblasts with wild‐type GFUS cDNA restored fucosylation. Making use of the GDP‐L‐fucose salvage pathway, oral fucose supplementation normalized fucosylation of proteins within 4 weeks as measured in serum and leukocytes. During the follow‐up of 19 months, a moderate improvement of growth was seen, as well as a clear improvement of cognitive skills as measured by the Kaufmann ABC and the Nijmegen Pediatric CDG Rating Scale. In conclusion, GFUS‐CDG is a new glycosylation disorder for which oral L‐fucose supplementation is promising

Abstract composition laws and their modulation spaces

On classes of functions defined on R^2n we introduce abstract composition laws modelled after the pseudodifferential product of symbols. We attach to these composition laws modulation mappings and spaces with useful algebraic and topological properties.Comment: 19 page

Constructions of regular algebras Lpw(G)L_p^w(G)

Criterion of (Shilov) regularity for weighted algebras $L_1^w(G)$ on a locally compact abelian group $G$ is known by works of Beurling (1949) and Domar (1956). In the present paper this criterion is extended to translation invariant weighted algebras $L_p^w(G)$ with $p>1$. Regular algebras $L_p^w(G)$ are constructed on any sigma-compact abelian group $G$. It was proved earlier by the author that sigma-compactness is necessary (in the abelian case) for the existence of weighted algebras $L_p^w(G)$ with $p>1$.Comment: Submitted to Mat. Sborni