Closely embedded Kre?n spaces and applications to Dirac operators

Motivated by energy space representation of Dirac operators, in the sense of K. Friedrichs, we recently introduced the notion of closely embedded Kre?n spaces. These spaces are associated to unbounded selfadjoint operators that play the role of kernel operators, in the sense of L. Schwartz, and they are special representations of induced Kre?n spaces. In this article we present a canonical representation of closely embedded Kre?n spaces in terms of a generalization of the notion of operator range and obtain a characterization of uniqueness. When applied to Dirac operators, the results differ according to a mass or a massless particle in a dramatic way: in the case of a particle with a nontrivial mass we obtain a dual of a Sobolev type space and we have uniqueness, while in the case of a massless particle we obtain a dual of a homogenous Sobolev type space and we lose uniqueness.     

Kernel methods and flexible inference for complex stochastic dynamics

Approximation theory suggests that series expansions and projections represent standard tools for random process applications from both numerical and statistical standpoints. Such instruments emphasize the role of both sparsity and smoothness for compression purposes, the decorrelation power achieved in the expansion coefficients space compared to the signal space, and the reproducing kernel property when some special conditions are met. We consider these three aspects central to the discussion in this paper, and attempt to analyze the characteristics of some known approximation instruments employed in a complex application domain such as financial market time series. Volatility models are often built ad hoc, parametrically and through very sophisticated methodologies. But they can hardly deal with stochastic processes with regard to non-Gaussianity, covariance non-stationarity or complex dependence without paying a big price in terms of either model mis-specification or computational efficiency. It is thus a good idea to look at other more flexible inference tools; hence the strategy of combining greedy approximation and space dimensionality reduction techniques, which are less dependent on distributional assumptions and more targeted to achieve computationally efficient performances. Advantages and limitations of their use will be evaluated by looking at algorithmic and model building strategies, and by reporting statistical diagnostics.     

Linearity and classification of ZpZp2-linear generalized Hadamard codes

The ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive codes are subgroups of ${\mathbb{Z}}_p^{{\alpha }_1}\times {\mathbb{Z}}_{p^2}^{{\alpha }_2}$, and can be seen as linear codes over ${\mathbb{Z}}_p$ when ${\alpha }_2=0$, ${\mathbb{Z}}_{p^2}$-additive codes when ${\alpha }_1=0$, or ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive codes when $p=2$. A ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear generalized Hadamard (GH) code is a GH code over ${\mathbb{Z}}_p$ which is the Gray map image of a ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive code. Recursive constructions of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive GH codes of type $({\alpha }_1,{\alpha }_2;t_1,t_2)$ with $t_1,t_2\ge 1$ are known. In this paper, we generalize some known results for ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear GH codes with $p=2$ to any $p\ge 3$ prime when ${\alpha }_1\ne 0$, and then we compare them with the ones obtained when ${\alpha }_1=0$. First, we show for which types the corresponding ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear GH codes are nonlinear over ${\mathbb{Z}}_p$. Then, for these codes, we compute the kernel and its dimension, which allow us to classify them completely. Moreover, by computing the rank of some of these codes, we show that, unlike ${\mathbb{Z}}_4$-linear Hadamard codes, the ${\mathbb{Z}}_{p^2}$-linear GH codes are not included in the family of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear GH codes with ${\alpha }_1\ne 0$ when $p\ge 3$ prime. Indeed, there are some families with infinite nonlinear ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear GH codes, where the codes are not equivalent to any ${\mathbb{Z}}_{p^s}$-linear GH code with $s\ge 2$.     

Recent studies on advance spectroscopic techniques for the identification of microorganisms: A review

The continuous development of resistance to antibiotic drugs by microorganisms causes high mortality and morbidity. Pathogens with distinct features and biochemical abilities make them destructive to human health. Therefore, early identification of the pathogen is of substantial importance for quick ailments and healthcare outcomes. Several phenotype methods are used for the identification and resistance determination but most of the conventional procedures are time-consuming, costly, and give qualitative results. Recently, great focus has been made on the utilization of advanced techniques for microbial identification. This review is focused on the research studies performed in the last five years for the identification of microorganisms particularly, bacteria using advanced spectroscopic techniques including mass spectrometry (MS), infrared (IR) spectroscopy, Raman spectroscopy (RS), and nuclear magnetic resonance (NMR) spectroscopy. Among all the techniques, MS techniques, particularly MALDI-TOF/MS have been widely utilized for microbial identification. A total of 44 bacteria i.e., 6 Staphylococcus spp., 3 Enterococcus spp., 6 Bacillus spp., 4 Streptococcus spp., 6 Salmonella spp., and one from each genus including Escherichia, Acinetobacter, Pseudomonas, Proteus, Clostridioides, Candida, Brucella, Burkholderia, Francisella, Yersinia, Moraxella, Vibrio, Shigella, Serratia, Citrobacter, and Haemophilus (spp.) were discussed in the review for their identification using the above-mentioned techniques. Among all the identified microorganisms, 21% of studies have been conducted for the identification of E. coli, 14% for S. aureus followed by 37% for other microorganisms.