The paper is devoted to the development of the octonion Fourier transform (OFT) theory initiated in 2011 in articles by Hahn and Snopek. It is also a continuation and generalization of earlier work by Błaszczyk and Snopek, where they proved few essential properties of the OFT of real-valued functions, e.g. symmetry properties. The results of this article focus on proving that the OFT is well-defined for octonion-valued functions and almost all well-known properties of classical (complex) Fourier transform (e.g. argument scaling, modulation and shift theorems) have their direct equivalents in octonion setup. Those theorems, illustrated with some examples, lead to the generalization of another result presented in earlier work, i.e. Parseval and Plancherel Theorems, important from the signal and system processing point of view. Moreover, results presented in this paper associate the OFT with 3-D LTI systems of linear PDEs with constant coefficients. Properties of the OFT in context of signal-domain operations such as derivation and convolution of Rvalued functions will be stated. There are known results for the QFT, but they use the notion of other hypercomplex algebra, i.e. double-complex numbers. Considerations presented here require defining other higher-order hypercomplex structure, i.e. quadruple-complex numbers. This hypercomplex generalization of the Fourier transformation provides an excellent tool for the analysis of 3-D LTI systems.

No abstract

Background and Aims: Gastric varices (GVs) occur in 20% of patients with portal hypertension. GVs are associated with a 65% risk of bleeding over the course of 2 years and have a mortality rate of up to 20%. The standard treatment for GVs is obliteration with cyanoacrylate (CYA). This study presents our experience with combined therapy (vascular coils and CYA) under endoscopic ultrasound (EUS) guidance. Methods: 16 patients (9 male and 7 female) were included into our study. Etiology of portal hypertension included: portal vein thrombosis (PVT) (31.0%), isolated splenic vein thrombosis (SVT) (25.0%), alcoholic cirrhosis (12.5%), hepatitis C cirrhosis (19.0%), and alcoholic cirrhosis with PVT (12.5%). Varices type GOV-2 were diagnosed in 8 patients, type IGV-1 and IGV-2 in 6 and 2 patients, respectively. Indications for treatment were based on endoscopic and endosonographic evaluations of GVs. Inclusion and exclusion criteria were also specified. Technique depended on the size of varices (different size of coils + CYA additionally). The results were based on the achievement of technical success, therapeutic effects, and number of adverse events. Average follow-up period was 327 days. Results: From January to August 2017, 16 patients were treated with EUS-guided obliteration of GVs using vascular coils only or coils with CYA injections. 6 (37.5%) and 10 (62.5%) patients underwent primary and secondary prophylaxis for hemorrhage, respectively. Technical success was achieved in 15 patients (94.0%). Mean numbers of implanted coils and CYA volume during one procedure were 1.7 and 2 mL, respectively. Therapeutic success was achieved in all patients treated with the combination. There were no serious complications such as embolization or death due to the procedure. Three patients (19.0%) had transient abdominal pain and two (12.5%) had transient fever. 1 patient had clinical symptoms of gastrointestinal bleeding. Conclusions: Based on our retrospective research we have concluded, that EUS-guided implantation of intravascular coils combined with cyanoacrylate injections is an effective method of treatment with an acceptable number of complications.

No abstract

The article concerns compressed sensing methods in the quaternion algebra. We prove that it is possible to uniquely reconstruct -by 1 norm minimization -a sparse quaternion signal from a limited number of its real linear measurements, provided the measurement matrix satisfies so-called restricted isometry property with a sufficiently small constant. We also provide error estimates for the reconstruction of a non-sparse quaternion signal in the noisy and noiseless cases. arXiv:1605.07985v1 [math.FA]