Incorrect conclusions drawn for plausible looking diagrams

In Mathematics is common to make a mistake and therefore a false conclusion arises. In each case it is important to recognize the mistake in order to avoid a similar one in the future. Geometric figures provide decisive help in order to have a strict mathematical proof, but also can easily lead to wrong conclusions without a mathematical proof. In this paper, several incorrect conclusions drawn for plausible looking diagrams are presented, motivated by a well-known faulty model for measuring the length of a segment. Similar models that lead to a contradiction are developed and a model that leads to the correct result is derived. The presented models prove the usefulness of paradoxes and can be implemented in a classroom in order to point out to students the significance of a strict mathematical proof as well as the construction of a correct mathematical model. The geometric nature of the problems provides the opportunity to use a dynamic geometric software.Comment: 16 pages, 15 figure

Eigenfunction Expansions for the Stokes Flow Operators in the Inverted Oblate Coordinate System

When studying axisymmetric particle fluid flows, a scalar function, ψ, is usually employed, which is called a stream function. It serves as a velocity potential and it can be used for the derivation of significant hydrodynamic quantities. The governing equation is a fourth-order partial differential equation; namely, E4ψ=0, where E2 is the Stokes irrotational operator and E4=E2∘E2 is the Stokes bistream operator. As it is already known, E2ψ=0 in some axisymmetric coordinate systems, such as the cylindrical, spherical, and spheroidal ones, separates variables, while in the inverted prolate spheroidal coordinate system, this equation accepts R-separable solutions, as it was shown recently by the authors. Notably, the kernel space of the operator E4 does not decompose in a similar way, since it accepts separable solutions in cylindrical and spherical system of coordinates, while E4ψ=0 semiseparates variables in the spheroidal coordinate systems and it R-semiseparates variables in the inverted prolate spheroidal coordinates. In addition to these results, we show in the present work that in the inverted oblate spheroidal coordinates, the equation E′2ψ=0 also R-separates variables and we derive the eigenfunctions of the Stokes operator in this particular coordinate system. Furthermore, we demonstrate that the equation E′4ψ=0  R-semiseparates variables. Since the generalized eigenfunctions of E′2 cannot be obtained in a closed form, we present a methodology through which we can derive the complete set of the generalized eigenfunctions of E′2 in the modified inverted oblate spheroidal coordinate system

Diagnostic Yield of Non-Invasive Testing in Patients with Anomalous Aortic Origin of Coronary Arteries : A Multicentric Experience

Background: Anomalous aortic origin of a coronary artery (AAOCA) is a congenital heart disease with a 0.3%-0.5% prevalence. Diagnosis is challenging due to nonspecific clinical presentation. Risk stratification and treatment are currently based on expert consensus and single-center case series. Methods: Demographical and clinical data of AAOCA patients from 17 tertiary-care centers were analyzed. Diagnostic imaging studies (Bidimensional echocardiography, coronary computed tomography angiography [CCTA] were collected. Clinical correlations with anomalous coronary course and origin were evaluated. Results: Data from 239 patients (42% males, mean age 15 y) affected by AAOCA were collected; 154 had AAOCA involving the right coronary artery (AAORCA), 62 the left (AAOLCA), 23 other anomalies. 211 (88%) presented with an inter-arterial course. Basal electrocardiogram (ECG) was abnormal in 37 (16%). AAOCA was detected by transthoracic echocardiography and CCTA in 53% and 92% of patients, respectively. Half of the patients reported cardiac symptoms (119/239; 50%), mostly during exercise in 121/178 (68%). An ischemic response was demonstrated in 37/106 (35%) and 16/31 (52%) of patients undergoing ECG stress test and stress-rest single positron emission cardiac tomography. Compared with AAORCA, patients with AAOLCA presented more frequently with syncope (18% vs. 5%, P = 0.002), in particular when associated with inter-arterial course (22% vs. 5%, P < 0.001). Conclusion: Diagnosis of AAOCA is a clinical challenge due to nonspecific clinical presentations and low sensitivity of first-line cardiac screening exams. Syncope seems to be strictly correlated to AAOLCA with inter-arterial course.Peer reviewe

Anomalous aortic origin of coronary arteries : Early results on clinical management from an international multicenter study

Background: Anomalous aortic origin of coronary arteries (AAOCA) is a rare abnormality, whose optimal management is still undefined. We describe early outcomes in patients treated with different management strategies. Methods: This is a retrospective clinicalmulticenter study including patients with AAOCA, undergoing or not surgical treatment. Patients with isolated high coronary take off and associated major congenital heart disease were excluded. Preoperative, intraoperative, anatomical and postoperative data were retrieved from a common database. Results: Among 217 patients, 156 underwent Surgical repair (median age 39 years, IQR: 15-53), while 61 were Medical (median age 15 years, IQR: 8-52), inwhomAAOCA was incidentally diagnosed during screening or clinical evaluations. Surgical patients were more often symptomatic when compared to medical ones (87.2% vs 44.3%, p b 0.001). Coronary unroofing was the most frequent procedure (56.4%). Operative mortality was 1.3% (2 patients with preoperative severe heart failure). At a median follow up of 18 months (range 0.1-23 years), 89.9% of survivors are in NYHA Conclusions: Surgery for AAOCA is safe andwith low morbidity. When compared to Medical patients, who remain on exercise restriction and medical therapy, surgical patients have a benefit in terms of symptoms and return to normal life. Since the long term-risk of sudden cardiac death is still unknown, we currently recommend accurate long term surveillance in all patients with AAOCA. (C) 2019 The Authors. Published by Elsevier B.V.Peer reviewe

Strokes flow problems with applications in hemodynamics

Aiming in the better understanding of the micro-reological behaviour of the human blood, a detailed mathematical analysis of two Stokes flow problems concerning the relative movement of a rigid inverted prolate spheroid and a viscous fluid is presented. In the first problem we study blood plasma flow past a red blood cell (RBC) and we modeled it as a creeping flow past an inverted prolate spheroid. Furthermore we imposed no slip condition, impenetrability of the RBC and that the fluid extends to infinity. We expressed the problem in the modified inverted prolate coordinate system and we used Kelvin’s inversion to translate the problem in an equivalent one, expressing the flow in a prolate spheroid. Using the concept of semiseparation in spheroidal coordinates, orthogonality arguments and the reduction of our solution to the one in the spherical coordinates, we derived the stream function in the interior of the prolate spheroid. Using Kelvin’s transformation we have the stream function that expresses the flow past the RBC. The procedure leads to a series solution in terms of Gegenbauer functions. By employing this solution the translation of a rigid inverted prolate spheroid through the blood plasma is also obtained, using the stream function that represents the unperturbed flow at infinity. These two models/problems also provide a new insight in the theoretical investigation of the flow behaviour of RBCs as we obtain analytical solutions while, as far to our knowledge, only numerical studies are available in the literature. The obtained analytical solutions can be used for assessing and verifying numerical methods and results. Furthermore these models can be applied to medical tests like the Erythrocyte Sedimentation Rate (ESR) and also in the modelling transport processes concerning the RBCs and the blood. Furthermore, by introducing for the first time here, the concept of R-semiseparation of variables, the eigenfunctions and the generalized eigenfunctions of the Stokes operator 2 E are derived, in the inverse modified prolate spheroidal system of coordinates. We proved that equation 2 E' ψ = 0 in the modified inverted prolate spheroid R-separates variables and we derived the eigenfunctions (i) Θ'n for i = 1, 2, 3, 4 and n ∈ N of the Stokes operator. We also proved that equation 4 E' ψ = 0 in the modified inverted prolate spheroid R-semiseparates variables and we presented a method to derive all the generalized eigenfunctions (i) n ' for i = 1, 2, 3, 4 and n ≥4 of the Stokes operator, since a closed formula can not be obtained.Στοχεύοντας στην καλύτερη κατανόηση της μικρορεολογικής συμπεριφοράς του ανθρώπινου αίματος, παρουσιάζεται μια λεπτομερής μαθηματική ανάλυση δύο προβλημάτων έρπουσας ροής για τη σχετική κίνηση ενός συμπαγούς αντίστροφου επιμήκους σφαιροειδούς σε ιξώδες ρευστό. Στο πρώτο πρόβλημα μελετήθηκε η ροή του πλάσματος του αίματος γύρω από το ερυθρό αιμοσφαίριο, η οποία μοντελοποιήθηκε σαν ροή Stokes γύρω από ένα αντίστροφο επίμηκες σφαιροειδές. Επιπλέον, υποτέθηκε συνθήκη μη ολίσθησης, αδιαπερατότητα του ερυθρού αιμοσφαιρίου και ότι το ρευστό εκτείνεται μέχρι το άπειρο. Εκφράστηκε το πρόβλημα στο τροποποιημένο αντίστροφο επίμηκες σφαιροειδές σύστημα συντεταγμένων και χρησιμοποιήθηκε η αντιστροφή Kelvin για να μετασχηματιστεί το πρόβλημα σε ένα ισοδύναμό του, που εκφράζει τη ροή στο εσωτερικό ενός επιμήκους σφαιροειδούς. Η μέθοδος του ημιχωρισμού των μεταβλητών, συνθήκες ορθογωνιότητας και αναγωγή της λύσης μας στην αντίστοιχη του σφαιρικού συστήματος συντεταγμένων ήταν τα εργαλεία που χρησιμοποιήθηκαν για να βρεθεί η συνάρτηση ροής στο εσωτερικό του επιμήκους σφαιροειδούς. Η αντιστροφή του θεμελιώδους χωρίου με τη χρήση του μετασχηματισμού Kelvin δίνει τη ζητούμενη συνάρτηση ροής που εκφράζει τη ροή γύρω από το ερυθρό αιμοσφαίριο. Η ζητούμενη συνάρτηση ροής εκφράζεται με τη βοήθεια μιας σειράς με όρους συναρτήσεις Gegenbauer. Εμπλέκοντας τη λύση αυτή, υπολογίστηκε η συνάρτηση ροής που εκφράζει τη μεταφορά του ερυθρού αιμοσφαιρίου μέσα στο πλάσμα, αφαιρώντας το κομμάτι της λύσης που εκφράζει τη ροή στο άπειρο. Επίσης, αυτά τα δύο μοντέλα/προβλήματα δίνουν μια νέα οπτική στη θεωρητική διερεύνηση της ροϊκής συμπεριφοράς των ερυθροκυττάρων, αφού βρέθηκαν αναλυτικές λύσεις, όταν (και στο επίπεδο που γνωρίζουμε) μόνο αριθμητικές μελέτες υπάρχουν στη βιβλιογραφία. Οι αναλυτικές λύσεις μπορούν να χρησιμοποιηθούν για την επιβεβαίωση των αριθμητικών μεθόδων και των αποτελεσμάτων που υπάρχουν. Επιπλέον, αυτά τα μοντέλα μπορούν να εφαρμοστούν σε ιατρικά τεστ, όπως είναι η καθίζηση των ερυθρών αιμοσφαιρίων, αλλά και για τη μοντελοποίηση διαδικασιών που αφορούν φαινόμενα μεταφοράς στο αίμα που εμπλέκουν το ερυθρό αιμοσφαίριο. Τέλος, παρουσιάστηκε εδώ για πρώτη φορά η ιδέα του R-ημιχωρισμού των μεταβλητών, οι ιδιοσυναρτήσεις και οι γενικευμένες ιδιοσυναρτήσεις του τελεστή Stokes στο τροποποιημένο αντίστροφο επίμηκες σφαιροειδές σύστημα συντεταγμένων. Αποδείχθηκε ότι η εξίσωση 2 E' ψ = 0 στο τροποποιημένο αντίστροφο επίμηκες σφαιροειδές σύστημα συντεταγμένων R-χωρίζει μεταβλητές και υπολογίστηκαν οι ιδιοσυναρτήσεις (i) Θ'n για i = 1, 2, 3, 4 και n ∈ N του τελεστή Stokes. Επίσης, αποδείχθηκε ότι η εξίσωση 4 E' ψ = 0 στο τροποποιημένο επίμηκες σφαιροειδές σύστημα συντεταγμένων R-ημιχωρίζει μεταβλητές και παρουσιάσθηκε μια μέθοδος για την εύρεση των γενικευμένων ιδιοσυναρτήσεων (i) 'n για i = 1, 2, 3, 4 και n ≥4 του τελεστή Stokes, αφού μια κλειστή μορφή τους δεν μπορεί να βρεθεί

On the Analytical Solution of the Kuwabara-Type Particle-in-Cell Model for the Non-Axisymmetric Spheroidal Stokes Flow via the Papkovich–Neuber Representation

Modern engineering technology often involves the physical application of heat and mass transfer. These processes are associated with the creeping motion of a relatively homogeneous swarm of small particles, where the spheroidal geometry represents the shape of the embedded particles within such aggregates. Here, the steady Stokes flow of an incompressible, viscous fluid through an assemblage of particles, at low Reynolds numbers, is studied by employing a particle-in-cell model. The mathematical formulation adopts the Kuwabara-type assumption, according to which each spheroidal particle is stationary and it is surrounded by a confocal spheroid that creates a fluid envelope, in which the Newtonian fluid moves with a constant velocity of arbitrary orientation. The boundary value problem in the fluid envelope is solved by imposing non-slip conditions on the surface of the spheroid, which is also considered as non-penetrable, while zero vorticity is assumed on the fictitious spheroidal boundary along with a uniform approaching velocity. The three-dimensional flow fields are calculated analytically for the first time, in the spheroidal geometry, by virtue of the Papkovich–Neuber representation. Through this, the velocity and the total pressure fields are provided in terms of a vector and the scalar spheroidal harmonic potentials, which enables the thorough study of the relevant physical characteristics of the flow fields. The newly obtained analytical expressions generalize to any direction with the existing results holding for the asymmetrical case, which were obtained with the aid of a stream function. These can be employed for the calculation of quantities of physical or engineering interest. Numerical implementation reveals the flow behavior within the fluid envelope for different geometrical cell characteristics and for the arbitrarily-assumed velocity field, thus reflecting the different flow/porous media situations. Sample calculations show the excellent agreement of the obtained results with those available for special geometrical cases. All of these findings demonstrate the usefulness of the proposed method and the powerfulness of the obtained analytical expansions

On the Analytical Solution of the Kuwabara-Type Particle-in-Cell Model for the Non-Axisymmetric Spheroidal Stokes Flow via the Papkovich&ndash;Neuber Representation

Modern engineering technology often involves the physical application of heat and mass transfer. These processes are associated with the creeping motion of a relatively homogeneous swarm of small particles, where the spheroidal geometry represents the shape of the embedded particles within such aggregates. Here, the steady Stokes flow of an incompressible, viscous fluid through an assemblage of particles, at low Reynolds numbers, is studied by employing a particle-in-cell model. The mathematical formulation adopts the Kuwabara-type assumption, according to which each spheroidal particle is stationary and it is surrounded by a confocal spheroid that creates a fluid envelope, in which the Newtonian fluid moves with a constant velocity of arbitrary orientation. The boundary value problem in the fluid envelope is solved by imposing non-slip conditions on the surface of the spheroid, which is also considered as non-penetrable, while zero vorticity is assumed on the fictitious spheroidal boundary along with a uniform approaching velocity. The three-dimensional flow fields are calculated analytically for the first time, in the spheroidal geometry, by virtue of the Papkovich&ndash;Neuber representation. Through this, the velocity and the total pressure fields are provided in terms of a vector and the scalar spheroidal harmonic potentials, which enables the thorough study of the relevant physical characteristics of the flow fields. The newly obtained analytical expressions generalize to any direction with the existing results holding for the asymmetrical case, which were obtained with the aid of a stream function. These can be employed for the calculation of quantities of physical or engineering interest. Numerical implementation reveals the flow behavior within the fluid envelope for different geometrical cell characteristics and for the arbitrarily-assumed velocity field, thus reflecting the different flow/porous media situations. Sample calculations show the excellent agreement of the obtained results with those available for special geometrical cases. All of these findings demonstrate the usefulness of the proposed method and the powerfulness of the obtained analytical expansions

Late management of the aortic root after repair of tetralogy of Fallot: A European multicentre study.

OBJECTIVES: We sought to determine the indications, type, and outcomes of reoperations on the aortic root after repair of tetralogy of Fallot (TOF). METHODS: Eleven centers belonging to the European Congenital Heart Surgeons Association contributed to the data collection process. We included 36 patients who underwent surgical procedures on the aortic root, including surgery on the aortic valve and ascending aorta, between January 1975 and December 2017. Original diagnoses included TOF-pulmonary stenosis (n = 18) and TOF-pulmonary atresia (n = 18). The main indications for reoperation were aortic insufficiency (n = 19, 53%), aortic insufficiency and dilatation of the ascending aorta (n = 10, 28%), aortic root dilatation (n = 4, 11%), and ascending aorta dilatation (n = 3, 8%). RESULTS: The median age at reoperation was 30.4 years (interquartile range 20.3-45.3 years), and mechanical aortic valve replacement was the most common procedure performed. Five patients died early after reoperation (14%), and larger ascending aorta diameters were associated with early mortality (P = .04). The median age at the last follow-up was 41.4 years (interquartile range 24.5-51.6 years). Late death occurred in five patients (5/31, 16%). Most survivors (15/26, 58%) were asymptomatic at the last clinical examination (New York Heart Association, NYHA class I). The remaining patients were NYHA class II (n = 7) and III (n = 3). The most common symptoms were fatigue (n = 5), dyspnea (n = 4), and exercise intolerance (n = 3). CONCLUSIONS: Reoperations on the aortic root are infrequent but may become necessary late after TOF repair. The main indications for reoperation are aortic insufficiency, either isolated or associated with a dilatation of the ascending aorta. The surgical risk at reoperation was high and the presence of ascending aorta dilation is related to higher mortality

Late management of the aortic root after repair of tetralogy of Fallot: A European multicentre study

OBJECTIVES: We sought to determine the indications, type, and outcomes of reoperations on the aortic root after repair of tetralogy of Fallot (TOF). METHODS: Eleven centers belonging to the European Congenital Heart Surgeons Association contributed to the data collection process. We included 36 patients who underwent surgical procedures on the aortic root, including surgery on the aortic valve and ascending aorta, between January 1975 and December 2017. Original diagnoses included TOF-pulmonary stenosis (n = 18) and TOF-pulmonary atresia (n = 18). The main indications for reoperation were aortic insufficiency (n = 19, 53%), aortic insufficiency and dilatation of the ascending aorta (n = 10, 28%), aortic root dilatation (n = 4, 11%), and ascending aorta dilatation (n = 3, 8%). RESULTS: The median age at reoperation was 30.4 years (interquartile range 20.3-45.3 years), and mechanical aortic valve replacement was the most common procedure performed. Five patients died early after reoperation (14%), and larger ascending aorta diameters were associated with early mortality (P = .04). The median age at the last follow-up was 41.4 years (interquartile range 24.5-51.6 years). Late death occurred in five patients (5/31, 16%). Most survivors (15/26, 58%) were asymptomatic at the last clinical examination (New York Heart Association, NYHA class I). The remaining patients were NYHA class II (n = 7) and III (n = 3). The most common symptoms were fatigue (n = 5), dyspnea (n = 4), and exercise intolerance (n = 3). CONCLUSIONS: Reoperations on the aortic root are infrequent but may become necessary late after TOF repair. The main indications for reoperation are aortic insufficiency, either isolated or associated with a dilatation of the ascending aorta. The surgical risk at reoperation was high and the presence of ascending aorta dilation is related to higher mortality.status: publishe

Anomalous aortic origin of coronary arteries: Early results on clinical management from an international multicenter study

BACKGROUND: Anomalous aortic origin of coronary arteries (AAOCA) is a rare abnormality, whose optimal management is still undefined. We describe early outcomes in patients treated with different management strategies. METHODS: This is a retrospective clinical multicenter study including patients with AAOCA, undergoing or not surgical treatment. Patients with isolated high coronary take off and associated major congenital heart disease were excluded. Preoperative, intraoperative, anatomical and postoperative data were retrieved from a common database. RESULTS: Among 217 patients, 156 underwent Surgical repair (median age 39 years, IQR: 15-53), while 61 were Medical (median age 15 years, IQR: 8-52), in whom AAOCA was incidentally diagnosed during screening or clinical evaluations. Surgical patients were more often symptomatic when compared to medical ones (87.2% vs 44.3%, p < 0.001). Coronary unroofing was the most frequent procedure (56.4%). Operative mortality was 1.3% (2 patients with preoperative severe heart failure). At a median follow up of 18 months (range 0.1-23 years), 89.9% of survivors are in NYHA ≤ II, while only 3 elderly surgical patients died late. Return to sport activity was significantly higher in Surgical patients (48.1% vs 18.2%, p < 0.001). CONCLUSIONS: Surgery for AAOCA is safe and with low morbidity. When compared to Medical patients, who remain on exercise restriction and medical therapy, surgical patients have a benefit in terms of symptoms and return to normal life. Since the long term-risk of sudden cardiac death is still unknown, we currently recommend accurate long term surveillance in all patients with AAOCA.status: publishe