Analysis of Dirac systems and computational algebra / Fabrizio Colombo .. let al.]. p. cm. -(Progress in mathematical physics ; v. 39) Inc1udes bibliographical references and index.

In this paper we prove a new Representation Formula for slice regular functions, which shows that the value of a slice regular function f at a point q = x + yI can be recovered by the values of f at the points q + yJ and q + yK for any choice of imaginary units I, J, K. This result allows us to extend the known properties of slice regular functions defined on balls centered on the real axis to a much larger class of domains, called axially symmetric domains. We show, in particular, that axially symmetric domains play, for slice regular functions, the role played by domains of holomorphy for holomorphic functions.

In this paper we offer a new definition of monogenicity for functions defined on R n+1 with values in the Clifford algebra R n following an idea inspired by the recent papers [6], [7]. This new class of monogenic functions contains the polynomials (and, more in general, power series) with coefficients in the Clifford algebra R n . We will prove a Cauchy integral formula as well as some of its consequences. Finally, we deal with the zeroes of some polynomials and power series.

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Abstract. In this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of S-spectrum. The proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of spectral analysis of quaternionic operators. The S-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the spectral theorem for quaternionic normal operators. The fact that the correct notion of spectrum for quaternionic operators was not previously known has been one of the main obstructions to fully understanding the spectral theorem in this setting. A prime motivation for studying the spectral theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti-self adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics.

Background: As the novel SARS-CoV-2 pandemic occurred, no specific treatment was yet available. Inflammatory response secondary to viral infection might be the driver of severe diseases. We report the safety and efficacy (in terms of overall survival and hospital discharge) of the anti-IL6 tocilizumab (TCZ) in subjects with COVID-19. Methods: This retrospective, single-center analysis included all the patients consecutively admitted to our Hospital with severe or critical COVID-19 who started TCZ treatment from March 13th to April 03rd, 2020. A 1:2 matching to patients not treated with TCZ was performed according to age, sex, severity of disease, P/F, Charlson Comorbidity Index and length of time between symptoms onset and hospital admittance. Descriptive statistics and non-parametric tests to compare the groups were applied. Kaplan Meier probability curves and Cox regression models for survival, hospital discharge and orotracheal intubation were used. Results: Seventy-four patients treated with TCZ were matched with 148 matched controls. They were mainly males (81.5%), Caucasian (82.0%) and with a median age of 59 years. The majority (69.8%) showed critical stage COVID-19 disease. TCZ use was associated with a better overall survival (HR 0.499 [95% CI 0.262-0.952], p = 0.035) compared to controls but with a longer hospital stay (HR 1.658 [95% CI 1.088-2.524], p = 0.019) mainly due to biochemical, respiratory and infectious adverse events. Discussion: TCZ use resulted potentially effective on COVID-19 in terms of overall survival. Caution is warranted given the potential occurrence of adverse events. Financial support: Some of the tocilizumab doses used in the subjects included in this analysis were provided by the "Multicenter study on the efficacy and tolerability of tocilizumab in the treatment of patients with COVID-19 pneumonia" (EudraCT Number: 2020-001110-38) supported by the Italian National Agency for Drugs (AIFA). No specific funding support was planned for study design, data collection and analysis and manuscript writing of this paper.

We study reproducing kernel Hilbert and Pontryagin spaces of slice\ud hyperholomorphic functions which are analogs of the Hilbert\ud spaces of analytic functions introduced by de Branges and Rovnyak.\ud In the first part of the paper we focus on the case of Hilbert\ud spaces, and introduce in particular a version of the Hardy space. Then we\ud define Blaschke factors and Blaschke products and we consider an\ud interpolation problem. In the second part of the paper we turn to the case of Pontryagin spaces. We first prove some results from the theory of Pontryagin spaces in the quaternionic setting and, in particular, a theorem of Shmulyan on densely defined contractive linear relations. We then study realizations of generalized\ud Schur functions and of generalized Carath\'eodory functions