Boundary interpolation for slice hyperholomorphic Schur functions

A boundary Nevanlinna-Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers $\kappa_1, \ldots, \kappa_N$, quaternions $p_1, \ldots, p_N$ all of modulus $1$, so that the $2$-spheres determined by each point do not intersect and $p_u \neq 1$ for $u = 1,\ldots, N$, and quaternions $s_1, \ldots, s_N$, we wish to find a slice hyperholomorphic Schur function $s$ so that $$\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}} s(r p_u) = s_u\quad {\rm for} \quad u=1,\ldots, N,$$ and $$\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}}\frac{1-s(rp_u)\overline{s_u}}{1-r}\le\kappa_u,\quad {\rm for} \quad u=1,\ldots, N.$$ Our arguments relies on the theory of slice hyperholomorphic functions and reproducing kernel Hilbert spaces

Wiener algebra for the quaternions

We define and study the counterpart of the Wiener algebra in the quaternionic setting, both for the discrete and continuous case. We prove a Wiener-L\'evy type theorem and a factorization theorem. We give applications to Toeplitz and Wiener-Hopf operators

Hamiltonian purification

The problem of Hamiltonian purification introduced by Burgarth et al. [D. K. Burgarth et al., Nat. Commun. 5, 5173 (2014)] is formalized and discussed. Specifically, given a set of non-commuting Hamiltonians {h1, . . ., hm} operating on a d-dimensional quantum system Hd, the problem consists in identifying a set of commuting Hamiltonians {H1,...,Hm} operating on a larger dE-dimensional system H_{dE} which embeds H_d as a proper subspace, such that hj = PHjP with P being the projection which allows one to recover Hd from HdE . The notions of spanning-set purification and generator purification of an algebra are also introduced and optimal solutions for u(d) are provided.Comment: 13 pages, 3 figure

Cutaneous Head and Neck Squamous Cell Carcinoma with Regional Metastases: The Prognostic Importance of Soft Tissue Metastases and Extranodal Spread

Extranodal spread (ENS) is an established adverse prognostic factor in metastatic cutaneous squamous cell carcinoma (cSCC); however, the clinical significance of soft tissue metastases (STM) is unknown. The aim of this study was to evaluate the prognosis of patients with STM from head and neck cSCC, and to compare this with that of node metastases with and without ENS. Patients with cSCC metastatic to the parotid and/or neck treated by primary surgical resection between 1987 and 2007 were included. Metastatic nodes > 3 cm in size were an exclusion criterion. A Cox proportional hazard model was used to determine the effect of STM adjusting for other relevant prognostic factors. The population included 164 patients with a median follow-up of 26 months. There were 8 distant and 37 regional recurrences. There were 22 were cancer-specific deaths, and 29 patients died. STM was a significant predictor of reduced overall (hazard ratio 3.3; 95% confidence interval 1.6-6.4; P = 0.001) and disease-free survival (hazard ratio 2.4; 95% confidence interval 1.4-4.1; P = 0.001) when compared to patients with node disease with or without ENS. After adjusting for covariates, STM and number of involved nodes were significant independent predictors of overall and disease-free survival. In metastatic cSCC of the head and neck, the presence of STM is an independent predictor of reduced survival and is associated with a greater adverse effect than ENS alone

Prognostic value of the expression of C-Chemokine Receptor 6 and 7 and their ligands in non-metastatic breast cancer

<p>Abstract</p> <p>Background</p> <p>Chemokines and chemokine receptors are major actors of leukocytes trafficking and some have been shown to play an important role in cancer metastasis. Chemokines CCL19, CCL20 and CCL21 and their receptors CCR6 and CCR7, were assessed as potential biomarkers of metastatic dissemination in primary breast cancer.</p> <p>Methods</p> <p>Biomarker expression levels were evaluated using immunohistochemistry on paraffin-embedded tissue sections of breast cancer (n = 207).</p> <p>Results</p> <p>CCR6 was expressed by tumor cells in 35% of cases. CCR7 was expressed by spindle shaped stromal cells in 43% of cases but not by tumor cells in this series. CCL19 was the only chemokine found expressed in a significant number of breast cancers and was expressed by both tumor cells and dendritic cells (DC). CCR6, CCL19 and CCR7 expression correlated with histologic features of aggressive disease. CCR6 expression was associated with shorter relapse-free survival (RFS) in univariate and but not in multivariate analysis (p = 0.0316 and 0.055 respectively), and was not associated with shorter overall survival (OS). Expression of CCR7 was not significantly associated with shorter RFS or OS. The presence of CCL19-expressing DC was associated with shorter RFS in univariate and multivariate analysis (p = 0.042 and 0.020 respectively) but not with shorter OS.</p> <p>Conclusion</p> <p>These results suggest a contribution of CCR6 expression on tumor cells and CCL19-expressing DC in breast cancer dissemination. In our series, unlike what was previously published, CCR7 was exclusively expressed on stromal cells and was not associated with survival.</p

The Complete Genome Sequence of ‘Candidatus Liberibacter solanacearum’, the Bacterium Associated with Potato Zebra Chip Disease

Zebra Chip (ZC) is an emerging plant disease that causes aboveground decline of potato shoots and generally results in unusable tubers. This disease has led to multi-million dollar losses for growers in the central and western United States over the past decade and impacts the livelihood of potato farmers in Mexico and New Zealand. ZC is associated with ‘Candidatus Liberibacter solanacearum’, a fastidious alpha-proteobacterium that is transmitted by a phloem-feeding psyllid vector, Bactericera cockerelli Sulc. Research on this disease has been hampered by a lack of robust culture methods and paucity of genome sequence information for ‘Ca. L. solanacearum’. Here we present the sequence of the 1.26 Mbp metagenome of ‘Ca. L. solanacearum’, based on DNA isolated from potato psyllids. The coding inventory of the ‘Ca. L. solanacearum’ genome was analyzed and compared to related Rhizobiaceae to better understand ‘Ca. L. solanacearum’ physiology and identify potential targets to develop improved treatment strategies. This analysis revealed a number of unique transporters and pathways, all potentially contributing to ZC pathogenesis. Some of these factors may have been acquired through horizontal gene transfer. Taxonomically, ‘Ca. L. solanacearum’ is related to ‘Ca. L. asiaticus’, a suspected causative agent of citrus huanglongbing, yet many genome rearrangements and several gene gains/losses are evident when comparing these two Liberibacter. species. Relative to ‘Ca. L. asiaticus’, ‘Ca. L. solanacearum’ probably has reduced capacity for nucleic acid modification, increased amino acid and vitamin biosynthesis functionalities, and gained a high-affinity iron transport system characteristic of several pathogenic microbes

The spectral theorem for normal operators on a Clifford module

In this paper, using the recently discovered notion of the S-spectrum, we prove the spectral theorem for a bounded or unbounded normal operator on a Clifford module (i.e., a two-sided Hilbert module over a Clifford algebra based on units that all square to be - 1). Moreover, we establish the existence of a Borel functional calculus for bounded or unbounded normal operators on a Clifford module. Towards this end, we have developed many results on functional analysis, operator theory, integration theory and measure theory in a Clifford setting which may be of an independent interest. Our spectral theory is the natural spectral theory for the Dirac operator on manifolds in the non-self adjoint case. Moreover, our results provide a new notion of spectral theory and a Borel functional calculus for a class of n-tuples of commuting or non-commuting operators on a real or complex Hilbert space. Moreover, for a special class of n-tuples of operators on a Hilbert space our results provide a complementary functional calculus to the functional calculus of J. L. Taylor