Bounding differences in Jager Pairs

Symmetrical subdivisions in the space of Jager Pairs for continued fractions-like expansions will provide us with bounds on their difference. Results will also apply to the classical regular and backwards continued fractions expansions, which are realized as special cases

Solving for best linear approximates

Our goal is to finally settle the persistent problem in Diophantine Approximation of finding best inhomogeneous linear approximates. Classical results from the theory of continued fractions solve the special homogeneous case in the form of a complete sequence of normal approximates. Real expansions that allow the notion of normality to percolate into the inhomogeneous setting will provide us with the general solution

Symmetry in the sequence of approximation coefficients

Let $\{a_n\}_1^\infty$ and $\{\theta_n\}_0^\infty$ be the sequences of partial quotients and approximation coefficients for the continued fraction expansion of an irrational number. We will provide a function $f$ such that $a_{n+1} = f(\theta_{n\pm1},\theta_n)$. In tandem with a formula due to Dajani and Kraaikamp, we will write $\theta_{n \pm 1}$ as a function of $(\theta_{n \mp 1}, \theta_n)$, revealing an elegant symmetry in this classical sequence and allowing for its recovery from a pair of consecutive terms

Logarithmically larger deletion codes of all distances

The deletion distance between two binary words $u,v \in \{0,1\}^n$ is the smallest $k$ such that $u$ and $v$ share a common subsequence of length $n-k$. A set $C$ of binary words of length $n$ is called a $k$-deletion code if every pair of distinct words in $C$ has deletion distance greater than $k$. In 1965, Levenshtein initiated the study of deletion codes by showing that, for $k\ge 1$ fixed and $n$ going to infinity, a $k$-deletion code $C\subseteq \{0,1\}^n$ of maximum size satisfies $\Omega_k(2^n/n^{2k}) \leq |C| \leq O_k( 2^n/n^k)$. We make the first asymptotic improvement to these bounds by showing that there exist $k$-deletion codes with size at least $\Omega_k(2^n \log n/n^{2k})$. Our proof is inspired by Jiang and Vardy's improvement to the classical Gilbert--Varshamov bounds. We also establish several related results on the number of longest common subsequences and shortest common supersequences of a pair of words with given length and deletion distance

Antrochoanal Polyp: Updated Clinical Approach, Histology Characteristics, Diagnosis and Treatment

Antrochoanal polyp (ACP) is a benign unilateral polyp, originating from the maxillary sinus and expanding through the accessory or natural ostia into the nasal cavity and choanae. It has a 2: 1 male predominance and is more common in children and young adults. The exact pathophysiology is unclear, and it is thought to have less of the inflammatory reactions as opposed to typical bilateral nasal polyps which are commonly seen in diffused chronic rhinosinusitis. The presenting symptoms of ACP are unilateral nasal obstruction and rhinitis. Epistaxis, pain, and foul-smelling secretions are not typically seen and point towards a different etiology. Diagnosis is mainly clinical via endoscopic examination and supported by Computed tomography (CT) imaging. In CT images the three components of the polyp can be identified; an intramaxillary portion, intranasal and choanal components. Treatment is surgical, where Endoscopic sinus surgery (ESS) is the main technique used with other assisting approaches to reach the more challenging anterior and inferior areas of the maxillary sinus. Successful resection depends on complete removal of the intramaxillary component of the polyp to avoid polyp regrowth. The typical histologic characteristics are cyst formation, fibrosis and squamous metaplasia that are significantly more common in ACP than diffused nasal polyps

Digital phenotype of mood disorders: A conceptual and critical review

BackgroundMood disorders are commonly diagnosed and staged using clinical features that rely merely on subjective data. The concept of digital phenotyping is based on the idea that collecting real-time markers of human behavior allows us to determine the digital signature of a pathology. This strategy assumes that behaviors are quantifiable from data extracted and analyzed through digital sensors, wearable devices, or smartphones. That concept could bring a shift in the diagnosis of mood disorders, introducing for the first time additional examinations on psychiatric routine care.ObjectiveThe main objective of this review was to propose a conceptual and critical review of the literature regarding the theoretical and technical principles of the digital phenotypes applied to mood disorders.MethodsWe conducted a review of the literature by updating a previous article and querying the PubMed database between February 2017 and November 2021 on titles with relevant keywords regarding digital phenotyping, mood disorders and artificial intelligence.ResultsOut of 884 articles included for evaluation, 45 articles were taken into account and classified by data source (multimodal, actigraphy, ECG, smartphone use, voice analysis, or body temperature). For depressive episodes, the main finding is a decrease in terms of functional and biological parameters [decrease in activities and walking, decrease in the number of calls and SMS messages, decrease in temperature and heart rate variability (HRV)], while the manic phase produces the reverse phenomenon (increase in activities, number of calls and HRV).ConclusionThe various studies presented support the potential interest in digital phenotyping to computerize the clinical characteristics of mood disorders

Estrogen Antagonist and Development of Macular Hole

To describe the clinical and optical coherence tomography (OCT) features of a macular hole (MH) or its precursor lesion in patients treated with systemic antiestrogen agents. We reviewed the medical history of the patient, ophthalmic examination, and both fundus and OCT findings. Three female patients receiving antiestrogen therapy sought treatment for visual disturbance. All of the patients showed foveal cystic changes with outer retinal defect upon OCT. Visual improvement was achieved through surgery for the treatment of MH in two patients. Antiestrogen therapy may result in MH or its precursor lesion, in addition to perifoveal refractile deposits. OCT examination would be helpful for early detection in such cases