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IMR Press / RCM / Volume 23 / Issue 2 / DOI: 10.31083/j.rcm2302076
Open Access Review
Prognostic value of tricuspid regurgitation
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1 Department of Biomedical Sciences, Humanitas University, 20090 Pieve Emanuele-Milan, Italy
2 Cardio Center, IRCCS Humanitas Research Hospital, 20089 Rozzano-Milan, Italy
3 Division of Cardiology, Montefiore Medical Center, Bronx, NY 10467-2401, USA
*Correspondence: pierpasquale.leone@gmail.com (Pier Pasquale Leone); antonio.mangieri@gmail.com (Antonio Mangieri)
Rev. Cardiovasc. Med. 2022, 23(2), 76; https://doi.org/10.31083/j.rcm2302076
Submitted: 26 December 2021 | Revised: 24 January 2022 | Accepted: 27 January 2022 | Published: 22 February 2022
(This article belongs to the Special Issue Present and future of tricuspid regurgitation)
This is an open access article under the CC BY 4.0 license.
Abstract

Tricuspid regurgitation (TR) has a considerable prevalence in the overall population, that further increases in selected categories of patients. Three morphologic types of TR prevail, namely primary, secondary and atrial TR, mostly, but not always, occurring in different subsets of patients. Recent evidences demonstrate a negative impact of TR on outcomes, irrespective of etiology and even when less than severe in grading. Unfortunately, current surgical standards are void of strong prospective evidence of positive impact on clinical outcomes. While on one hand recent advances in diagnosis and risk stratification of patients with TR are shedding light onto the population that may benefit from intervention and its appropriate timing, on the other hand the arrival on stage of percutaneous treatment options is widening even more the therapeutic options for such population. In this review we will address and discuss the available evidence on the prognostic impact of TR in different clinical contexts encountered in practice.

Keywords
tricuspid regurgitation
valvular heart disease
right ventricle
heart failure

Progressive accrual of insights into the prevalence, pathophysiology, natural history and clinical relevance of tricuspid regurgitation (TR) has been witnessed in recent years [1, 2, 3, 4], and treatment of TR gradually shifted from a conservative to a more interventional therapeutic and preventive approach. Advances in echocardiographic techniques have been accomplices for such radical changes in management. Three different types of TR can be distinguished, that is primary, secondary and atrial TR (Fig. 1). While primary TR is due to an abnormality of the tricuspid valve (TV) apparatus, TR may be caused by dilation of TV annulus, right ventricular (RV) remodeling and leaflet tethering secondary to left heart disease and pulmonary hypertension (secondary TR), or caused by atrial fibrillation (AF) and right atrium (RA) remodeling (atrial TR) [4, 5, 6]. Secondary TR is the most common manifestation of TV disease, and has traditionally been considered a benign disease manageable with diuretic therapy and by addressing left heart disease. On the other hand, growing evidence has shown deleterious outcomes of untreated TR irrespective of left or right ventricular function and pulmonary hypertension in the surgical, percutaneous, and heart failure populations studied [7, 8]. In light of this evidence and the mortality risk associated with surgical intervention in candidate patients at high risk, novel transcatheter devices have been developed to address this unmet clinical need. This review will address the available evidence concerning the prognostic impact of TR in the different contexts with a particular focus on secondary TR (Table 1, Ref. [2, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]).

Fig. 1.

Classification of types of tricuspid regurgitation and related mechanisms. The three main types of TR are associated with different mechanisms of disease, which may overlap. CIED, cardiovascular implantable electronic device; LV, left ventricle; RA, right atrium; RV, right ventricle.

Table 1.Selected evidence on outcomes in patients with tricuspid regurgitation in different clinical settings.
 Year, author N Population Follow-up (years) Main results Mitral valve disease 2003, Henein et al. [9] 42 MVR for rheumatic disease 5 50% vs. 100% survival in patients with severe vs. mild TR 2004, Ruel et al. [10] 708 MVR 5 Increased risk of mortality with persistent TR (HR 2.17) 2009, Di Mauro et al. [11] 165 Mitral valve surgery for secondary MR 5 Increased risk of mortality with moderate or more vs. mild or less TR (HR 3.1) 2014, Ohno et al. [12] 146 TEER for secondary MR 1 Moderate/severe TR predicted death and re-hospitalization for heart failure (HR 2.67) 2020, Hahn et al. [13] 599 Medical therapy alone or medical therapy and TEER for secondary MR 2 Increased risk of death and hospitalization for heart failure if moderate or more TR treated with medical therapy alone (HR 1.74) 2020, Essayagh et al. [14] 5083 Primary MR 10 Increased adjusted risk of mortality with moderate (HR 1.40) and severe TR (HR 2.10) LV dysfunction 2002, Koelling et al. [15] 1,421 Left ventricular ejection fraction 35% or less 1 Increased adjusted risk of poor outcome with severe TR (HR 1.55) 2012, Agricola et al. [16] 373 Heart failure, left ventricular ejection fraction $<$50% and mild or more secondary MR 3 Moderate to severe functional TR independently predicted heart failure (HR 1.4) and mortality (HR 1.6) 2013, Neuhold et al. [17] 576 Congestive systolic heart failure 6 Increased risk of death/heart transplantation/left ventricular-assist device implantation in patients with TR (HR 2.07) 2019, Benfari et al. [8] 11,507 Heart failure stage B–C, left ventricular ejection fraction $<$50% 4 Increased adjusted risk of mortality with mild (HR 1.09), moderate (HR 1.21) and severe TR (HR 1.57) vs. trivial TR 2019, Bartko et al. [18] 382 Heart failure and reduced ejection fraction 5 0.9% and 1.3% increased risk of mortality per 1 mm${}^{2}$ increase in EROA and 1 mL increase in regurgitant volume, respectively Aortic valve disease 2021, Tomii et al. [19] 2,008 TAVI 1 Excess risk of mortality with severe (90%) and massive TR (117%) after TAVI Other settings 2004, Messika-Zeitoun et al. [20] 60 TR due to flail leaflets 10 Excess risk of mortality (39%) with severe TR when compared to matched United States population 2014, Hoke et al. [21] 239 CIED-related TR 5 Increased risk of mortality and heart failure-related events with significant TR (HR 1.69) 2014, Topilsky et al. [22] 353 Atrial TR 6 Increased risk of mortality with qualitatively and quantitatively measured severe TR (HR 1.78 and 2.67, respectively) 2019, Topilsky et al. [2] 417 Moderate or more TR of various etiologies (8.1% atrial TR) Up to 15 Increased risk of mortality with atrial TR (HR 1.68) 2020, Chorin et al. [23] 33,305 TR of various etiologies 3 Increased risk of mortality with moderate (1.15) and severe TR (1.43) vs. no or minimal TR EROA, effective regurgitant orifice area; HR, hazard ratio; MR, mitral regurgitation; MVR, mitral valve replacement TEER, transcatheter edge-to-edge repair; TR, tricuspid regurgitation.
1. Natural history of tricuspid regurgitation

TR is a common echocardiographic finding, reported in up to 70 to 90% of adults [1] and clinically relevant TR is found in approximately 4% of subjects aged 75 years [2]. Historically, the management of TR by the medical community has been to treat it medically, driven by the belief that it would resolve upon resolution of the underlying cause [7, 24, 25, 26]. Nonetheless, recent evidence from several studies suggests not only that significant TR impacts negatively patient prognosis (Fig. 2) [7, 15, 22], but also that it does so more and more for every increase in grade of severity [23, 27]. This might be well explained by the fact that chronic RV volume overload due to self-perpetuating significant TR may result in progressive RV dilation and remodeling, which in turn might then lead to progressive irreversible RV myocardial damage and symptomatic right heart failure [28]. In advanced stages, low cardiac output might not only be due to the relevant amount of regurgitant flow at expense of forward flow, but also due to the inability to proportionally increase cardiac output to metabolic needs related to increased left heart pressures caused by diastolic ventricular interaction and pericardial restraint [29, 30]. In the end, such natural history suggests a non-marginal role of TR in disease, especially upon the onset of the vicious cycle encompassing RV myocardial damage and remodeling.

Fig. 2.

The role of tricuspid regurgitation in disease. Incremental impact on survival is witnessed with worsening of the grade of TR. Mechanisms of disease and therapeutic considerations according to severity of TR are also depicted. RHF, right heart failure; RV, right ventricle; TR, tricuspid regurgitation.

2. Innocent bystander or covert foe—does severity of regurgitation matter?

Of note, whether TR predicts mortality independently from left-sided valve disease, pulmonary hypertension, RV dysfunction and other comorbidities has only recently been investigated in a more punctual and thorough manner [7, 15, 31]. Not surprisingly, the severity of disease does play a central role:

- Severe TR is associated with increased risk of mortality when compared with patients with no TR in retrospective registries, with reported overall mortality rates at 1-year of up to almost 50% vs. 8%, respectively [7, 23]. Of note, the association of severe TR with increased overall mortality persists when adjusting also for RV function, with more than 40% increase in events as compared with patients with no TR [23].

- Moderate TR also yields a 15% increase in mortality when compared with no or minimal TR [23]. Patients with moderate or severe primary or secondary TR from a comprehensive, although heterogeneous, large-scale meta-analysis including more than 30,000 patients [27] had double the risk of overall mortality when compared with no or minimal TR. Such results might allow inference on the generalizability of the dreadful prognostic impact of TR in different settings, irrespective of the underlying etiology [2, 7, 22, 32, 33, 34].

- Mild TR has been linked to increased mortality as well [27]. This might be related to the progressive nature of TR, and the fact that TR progression is paralleled by worse outcomes [35, 36, 37]. Also, the rapidity of significant TR development impacts all-cause mortality [33, 36]. Presence of pacemaker or defibrillator leads, surgical left-sided valvular treatment, elevated pulmonary pressures, increase of RV sphericity, tricuspid annular dilatation and increased TV tenting area predict fast development of significant TR [35], and could reveal useful for early identification of patients at risk and intervention before onset of irreversible right ventricular dysfunction.

3. Tricuspid regurgitation in mitral valve disease

TR is frequently present in patients with primary and secondary mitral valve disease, and may occur also late after mitral valve intervention [15, 38, 39, 40, 41, 42]. For this reason, aggressive correction with tricuspid annuloplasty at the time of mitral valve surgery is actually encouraged.

3.1 Tricuspid regurgitation in rheumatic primary mitral valve disease

More than 30% of patients with mitral stenosis have at least moderate TR [43, 44], which is associated with higher New York Heart Association (NYHA) functional class than mild or no TR at long term follow-up [44]. Also, patients with pre-procedural severe TR among those undergoing balloon mitral valvotomy not only present more advanced mitral valve disease, higher pulmonary vascular resistance and smaller increase in mitral valve area after valvotomy, but also they have poorer outcome in terms of overall survival, heart failure episodes and need for repeat intervention or mitral valve replacement (MVR) [45]. The incidence of moderate or severe late TR in rheumatic patients is up to 37% and 68%, respectively. In these cases, TR is often diagnosed years after MVR, ten on average [46, 47]. The pathophysiological basis for such condition is poorly defined [48], nonetheless significant TR is associated with higher NYHA functional class III or IV, heart failure episodes and all-cause death at 5 years [9, 10].

3.2 Tricuspid regurgitation in non-rheumatic primary mitral valve disease

TR is common among patients undergoing mitral valve surgery for primary mitral regurgitation (MR) [2, 14]. When severe, TR is often addressed during the index procedure, but the operative management of mild or moderate TR is widely debated. If not corrected at the time of left-sided cardiac surgery, TR may progress in a quarter of patients, especially in presence of annular dilation of 40 mm (or 21 mm/m${}^{2}$) or more in diameter, leaflet tethering, transvalvular leads, RV dysfunction, pulmonary hypertension and atrial fibrillation [49], and finally jeopardize late survival and functional outcomes [50, 51]. It is important to underline that patients with severe TR after mitral valve surgery undergoing isolated TV surgery usually have high perioperative mortality (10%) and poor late survival [52, 53, 54, 55], so that TV repair at index procedure may limit disease progression and eventually provide long-term clinical benefit [11].

3.3 Tricuspid regurgitation in secondary mitral valve disease

When focusing on patients with secondary MR undergoing mitral valve surgery, the impact of at least moderate functional TR on outcomes is substantial. Indeed, affected patients presented one half the chance to be alive at 5 years when compared with patients with less than moderate TR [11].

Thus, The prognostic role of tricuspid regurgitation (TR) associated with organic left-sided valvular heart disease is well known.

More recent reports addressed the role of TR in patients undergoing transcatheter edge-to-edge repair (TEER) treatment of secondary MR. Although TEER leads to improvement in MR, TR and NYHA functional class irrespective of baseline TR, NYHA functional class $>$2 was more common in patients with moderate/severe TR, suggesting that significant TR limits mid-term benefits of TEER in patients affected by secondary MR. Of note, moderate/severe TR independently predicted death and re-hospitalization for heart failure at 1 year [12]. In a recent analysis [13], patients with at least moderate TR presented worse clinical and echocardiographic characteristics than those with no or mild TR, and had higher rate of the composite of death or hospitalization for heart failure at 2 years when treated with medical therapy alone, while this was not the case following TEER with MitraClip.

Finally, although TR after mitral valve surgery has mostly been reported in patients with primary mitral valve disease, it may present also in patients after surgical repair of MR due to dilated ischemic or non-ischemic cardiomyopathy [56, 57, 58].

4. Tricuspid regurgitation in aortic valve disease

Secondary TR is often found in conjunction with aortic stenosis, especially when low-flow, low-gradient, and it is associated with increased risk of mortality [59]. The reported incidence of TR in patients candidate to surgical aortic valve replacement or transcatheter aortic valve implantation (TAVI) is of 11% [60]. TR does not improve up to 4 years after intervention in almost half of patients undergoing aortic valve replacement [61], and post-operative progression is observed in a quarter of patients [60]. In a recent analysis conducted on patients undergoing TAVI, more than 50% of patients experienced TR reduction after the procedure, whereas in little less than 10% of cases TR worsened [19]. Such evidence may lead to speculate that, in order to avoid late TR, a concomitant TV procedure might be considered in selected patients.

The impact of TR in patients undergoing TAVI has also been recently addressed: severe and massive TR post-TAVI is associated with a 2-fold increased risk of 1-year all-cause death, while this is not the case for baseline TR severity [19].

5. Tricuspid regurgitation in left ventricle systolic dysfunction

Notwithstanding the pathophysiological mechanisms described above by which TR can negatively affect patient outcomes, its independent association to poor prognosis in the subset of patients with left ventricle (LV) systolic dysfunction has been subject of debate as well. In this population, functional TR has been consistently associated with more dyspnea, older age, renal dysfunction, lower cardiac output, lower LV ejection fraction, MR, atrial fibrillation, and pulmonary hypertension [58]. Nonetheless, evidence of TR as a prognostic interventional target rather than mere indicator of worse prognosis in patients with heart failure has been accruing [62].

Early reports revealed a strong impact of TR on clinical outcome [15], and this is true also when focusing on patients with severe heart failure secondary to idiopathic-dilated cardiomyopathy or to ischemic cardiomyopathy [42]. When compared to trivial TR, different degrees of TR jeopardize outcomes to different levels in patients with secondary TR and heart failure with reduced LV ejection fraction. In particular, 9%, 21% and 57% increase in mortality is described at long-term follow-up for patients with mild, moderate and severe TR, independently of LV systolic dysfunction and pulmonary hypertension [8].

5.1 Does severity of heart failure matter?

Of note, the prognostic role of TR might depend upon the degree of heart failure. Indeed, when stratifying patients according to mildly or moderately vs. severely impaired LV function, TR is significantly associated to the risk of death/heart transplantation/LV assist device implantation in the former group, not in the latter [17]. Also, when stratifying patients according to N-terminal-pro-B-type natriuretic peptide concentrations, TR is related with outcome in patients with concentrations below the median, not those with values above it [63]. The impact of TR in advanced heart failure might thus be limited, and might be considered merely a marker of advanced disease, and, as such, not an attractive therapeutic target. On the other hand, treatment of severe TR might have a favorable impact on outcome in patients with less severe HF.

5.2 Does quantification of tricuspid regurgitation severity matter?

Earlier studies addressing significance of TR in chronic heart failure used qualitative measures of TR grading, but assessment of secondary TR severity by quantitative measures might be useful to unravel its natural history [18]. Quantitative metrics of TR severity were indeed found to be consistently associated with mortality; namely, an increase in risk of mortality of 0.9% and 1.3% were observed per 1 mm${}^{2}$ increase in effective regurgitant orifice area (EROA) and per 1 mL increase in regurgitant volume, respectively. Severe functional TR (defined as EROA $>$40 mm${}^{2}$) yields an increased risk of mortality and cardiac events of 80% and 120%, respectively [32].

Even more striking, thresholds for moderate TR as per current guidelines, that is $\geq$20 mm${}^{2}$ of EROA, $>$5 mm of vena contracta and $>$20 mL of regurgitant volume, are actually associated with increased mortality [63]. Moderate to severe functional TR increased the risk of heart failure and overall mortality also in patients with at least mild secondary MR. In particular, patients with no, mild, moderate and severe functional TR had a survival free of all-cause mortality at 6 years of 69%, 67%, 51% and 40%, respectively [16].

Overall, the evidence supports an important prognostic role for TR in patients with LV systolic dysfunction and calls for a paradigm shift in evaluation and therapeutic approach of patients with LV systolic dysfunction and secondary TR. Interventions aiming at TR treatment should be investigated also in this population, and the definition of the appropriate timing for intervention will be of paramount importance. In this setting, awareness not only of the dynamic nature of TR, but also of the risk of worsening pulmonary hypertension, right ventricular remodeling and mortality associated with non-severe TR progression is key [36, 64].

6. Tricuspid regurgitation in other settings

As already introduced, additional morphologic types of TR can be recognized, including primary and atrial TR [65], and it is important to recognize that although they often have a separate course of disease, they might coexist in the settings described above as well [66]. As already introduced, the main cause of TR is mitral or aortic valve disease, in more than 50% of TR cases [1]. On the other hand, a primary etiology is present in 7.4% of TR patients, with cardiovascular implantable electronic device (CIED)-related TR being the most common cause (66%), while atrial TR is present in 17% of non-primary cases [1, 65].

6.1 Primary tricuspid regurgitation

Patients with severe primary TR due to any congenital or organic cause, irrespective of LV function, left valvular function or pulmonary pressure, demonstrate a trend towards worse prognosis than those with no or trace TR [23].

The mechanism behind CIED-related TR, the most common cause of primary TR, is usually impingement of leaflet, subvalvular apparatus or papillary muscles, but leaflet perforation, fibrosis, thrombosis and endocarditis may occur as well. In this subgroup of patients, worse long-term survival and increased risk of heart failure-related events are reported when patients with significant TR are compared to those without significant TR, especially when focusing on patients with baseline LV ejection fraction $<$40% [21].

TV leaflet flail is often due to trauma, and leads to TR of mostly severe grade. Excess mortality is present in patients with TR due to flail leaflets, and excess tricuspid-related events accrue also in those asymptomatic at presentation. Moreover, severe dilation of right-sided chambers is an independent predictor of poor outcomes in such patients, so that early intervention before irreversible right heart chamber remodeling and dysfunction should be considered [20].

6.2 Atrial tricuspid regurgitation

Atrial TR is defined as such when holosystolic and functional, with no likely pulmonary hypertension, no overt cause and no previous valve surgery [27]. It is important to note that outcomes do differ when analyzing TR patients according to etiology. Indeed, atrial TR, even though associated with excess mortality, was observed to have less impact on yearly mortality, atrial fibrillation or heart failure hospitalization rates when compared to secondary or even primary disease [2]. Nonetheless, the reported 10-year survival rate of patients with an effective regurgitant orifice $\geq$40 mm${}^{2}$ is 38%, and severe TR predicts higher mortality when graded both qualitatively and quantitatively, regardless of RV size or function, pulmonary pressures or relevant comorbidities [27, 36]. The surgical benefit in patients with significant atrial TR was also investigated, and a trend towards benefit in survival rate was observed when comparing patients undergoing TV surgery and patients managed medically [67].

7. Reappraisal of adequate indication and timing for treatment of tricuspid regurgitation

The proposed algorithm in the latest American College of Cardiology/American Heart Association guidelines [68] supports TV surgery with Class I indication only at the time of left-sided valve surgery. On the other hand, only Class IIa and IIb indications exist for treatment of TR in other clinical settings, and these are based upon evaluation not only of TR severity and etiology, but also of symptoms, severity of pulmonary hypertension and RV size and function. The Class I indication to proceed to surgery for severe symptomatic isolated TR in absence of severe RV dysfunction and Class IIa indication for surgery in severe secondary TR either symptomatic or progressive RV dilation reported in the latest European Society of Cardiology guidelines might stem from the recent evidence of improved outcomes after isolated TV surgery with careful patient selection and current perioperative management [69].

In addition, the latest European guidelines report transcatheter treatment of symptomatic severe secondary TR in inoperable patients with a Class IIb indication (Table 2) [70]. Thus, upon re-evaluation of correct indication and timing of TR intervention, different factors need to be taken into account.

Table 2.2020 AHA/ACC and 2021 ESC/EACTS Guidelines for the Management of TR.
 Class of recommendation Level of evidence AHA/ACC 2020 Guidelines Primary TR Surgery for severe TR at the time of left-sided valve surgery I B Surgery for symptoms caused by severe TR not responsive to medical therapy IIa B Surgery for asymptomatic severe TR and progressive right ventricular dilatation and dysfunction IIb C Secondary TR Surgery for severe TR at the time of left-sided valve surgery I B Surgery for progressive TR at the time of left-sided valve surgery with either a dilated annulus ($\geq$40 mm or $\geq$21 mm/m${}^{2}$) or evidence of previous right heart failure IIa B Surgery for symptoms caused by severe TR attributable to annular dilation (in the absence of pulmonary hypertension or left-sided dysfunction) not responsive to medical therapy IIa B Surgery for persistent symptoms caused by isolated severe TR after previous left-sided valve surgery in absence of severe pulmonary hypertension or right ventricular dysfunction IIb B ESC/EACTS 2021 guidelines Primary TR Surgery for severe TR at the time of left-sided valve surgery I C Surgery for severe symptomatic isolated TR without severe right ventricular dysfunction I C Surgery for moderate TR at the time of left-sided valve surgery IIa C Surgery for asymptomatic or mildly symptomatic isolated severe TR and right ventricular dilatation, if appropriate for surgery IIa C Secondary TR Surgery for severe TR at the time of left-sided valve surgery I B Surgery for mild or moderate TR with dilated annulus ($\geq$40 mm or $\geq$21 mm/m${}^{2}$) at the time of left-sided valve surgery IIa B Surgery for severe TR with or without previous left-sided valve surgery with symptoms or progressive right ventricular dilatation, in absence of severe right or left ventricular dysfunction and severe pulmonary vascular disease/hypertension IIa B Transcatheter treatment for symptomatic severe TR in inoperable patients at a Heart Valve Centre with expertise in the treatment of tricuspid valve disease IIb C

Despite the evidence that quantitative grading of TR is a strong independent predictor of clinical outcomes, even superior to qualitative assessment [22], only one single study reported results according to such grading in a recent meta-analysis of 70 studies [27]. Given the possible underestimation of severity of disease with biplanar measurement of vena contracta [71], focus moved upon the utility of proximal isovelocity hemispheric surface area (PISA) for risk stratification [22]. Nonetheless, it is important to recognize that PISA might be underestimating TR severity as well [71, 72, 73], which might explain the sustained increase in mortality associated with EROA $>$20 mm${}^{2}$ and regurgitant volume $>$20 mL [18]. An additional issue is the extended grading scheme of TR [74] that was recently proposed, which addresses the unmet need of classifying patients currently undergoing percutaneous treatment in clinical trials, who often suffer from a degree of TR greater than merely severe, be it for the long symptoms tolerance or for the missed acknowledgment of the disease. In this context, many studies investigated the role of using additional cutoffs for vena contracta width and EROA, and consistently showed worse prognosis in patients classified as having either massive or torrential TR [72, 73, 74, 75].

7.2 The right ventricle and pulmonary artery coupling

Quantification of RV size and function [75, 76, 77], of severity of pulmonary hypertension and of the relationship between RV and pulmonary artery [78, 79] should be integral to the evaluation of TR. Ventriculo-arterial coupling represents the ability of the RV to balance its contractility with the afterload due to the pulmonary arterial vascular bed. Its reference method of assessment is pressure-volume loop analysis, but several non-invasive surrogated combining indices of RV function or chamber size with a metric of load have been proposed [78, 80]. Recently, abnormal ventriculo-arterial coupling, defined as RV free wall longitudinal strain/RV systolic pressure $\leq$0.5%/mmHg, predicted all-cause mortality in patients with severe secondary MR enrolled in the Cardiovascular Outcomes Assessment of the MitraClip Percutaneous Therapy for Heart Failure Patients With Functional Mitral Regurgitation (COAPT) trial [81]. Right ventricular systolic pressure is also often measured non-invasively by estimation via TR velocity, but some limitations need to be ascertained, especially when encountering patients with higher degrees of severity of TR. Indeed, discordance of pulmonary hypertension estimation via invasive and non-invasive measurement was a marker of more advanced valvular and ventricular disease, and should be taken into account when predicting procedural success and long-term outcomes [82]. It is thus important to keep in mind that invasive measurements yielding often underestimated hemodynamic information, including the type and eventual reversibility of pulmonary hypertension, might shed light onto the optimal TR population to be treated.

7.3 Implications for intervention

The first major implication for intervention is that TR should be treated during index left-sided valve surgery, especially if the latter is of rheumatic origin. Given the wide variation in adoption of concomitant TV and mitral valve surgery [8, 49, 83, 84], evidence not only of the absence of increased risk-adjusted mortality irrespective of the grade of TR, but actually of the lower risk of cardiac-related mortality and improved echocardiographic TR outcomes after concomitant TV repair at the time of left-sided valve surgery is of particular importance [50, 51, 83].

Similarly, prevention of long-term progression of functional TR by adopting a more aggressive strategy involving concomitant TR treatment upon mitral valve surgery should be acknowledged, especially when considering optimal timing for intervention [70, 85].

The second major implication for intervention is that, notwithstanding the potential for positively affecting the natural history of TR, isolated TV surgery has long been paralleled by high risk of mortality. Unfortunately, it has been difficult to generate prospective randomized evidence to demonstrate whether isolated TV surgery for TR has a positive prognostic effect, mainly due to a vicious cycle deriving from high reported mortality (9–10%), in-hospital complications (31%), poor late survival and no significant improvement in functional capacity after TV surgery [9, 10, 47, 48], yielding reluctancy of cardiologists to refer patients to surgery unless extremely symptomatic. Indeed, given the good response of TR to diuretic therapy, referral usually occurred late in the course of the disease concomitantly with onset of RV dysfunction. Earlier referral of TV surgery before RV function compromise might yield better outcomes [53], and, although not thoroughly investigated yet, recent studies reported promising results [69, 86, 87].

Finally, pre-operative clinical and echocardiographic features affect outcomes more than etiology or mechanism of TR [88], so that scoring systems to predict mortality after TV surgery might reveal useful to properly select patients to intervene upon and guide clinical decision-making [85]. Taking into consideration the lessons learned in the surgical context [51, 86, 87], the introduction of transcatheter tricuspid valve interventions might offer patients a less invasive and safer procedure [6]. Appropriate patient selection and timing of intervention will be of paramount importance in order to impact positively the history of disease and avoid futility in patients at a too advanced disease stage.

8. Conclusions

There is a growing body of evidence today that significant TR is not merely a bystander in the different disease processes in which it is involved, but rather is an active player in the pathophysiology and evolution of disease, impacting negatively on outcomes. Moreover, recent advances in diagnostic evaluation and risk stratification will allow us to more appropriately identify the adequate population to treat and the proper timing of intervention. Thus, we believe that the synergistic effect of medical, surgical and, more recently, percutaneous treatment of TR has the potential to positively affect clinical outcomes in this large population of patients, and testing of this concept in properly designed trials is eagerly awaited.

Abbreviations

CIED, cardiovascular implantable electronic device; EROA, effective regurgitant orifice area; HR, hazard ratio; LV, left ventricular; MR, mitral regurgitation; MVR, mitral valve replacement; NYHA, New York Heart Association; PISA, proximal isovelocity surface area; RA, right atrium; RV, right ventricle; TAVI, transcatheter aortic valve implantation; TEER, transcatheter edge-to-edge repair; TR, tricuspid regurgitation; TV, tricuspid valve.

Author contributions

PPL and AM contributed to conception of the review. PPL, AM and MC contributed to design of the article. PPL wrote the first draft of the manuscript. MC wrote sections of the manuscript. DR, MP, LM, BP, FL, GS, DP, BR, AC, AL contributed to editorial changes and manuscript revision. PPL, MC, DR, MP, LM, BP, FL, GS, DP, BR, AC, AL and AM read and approved the submitted version.

Ethics approval and consent to participate

Not applicable.

Acknowledgment

We would like to express our gratitude to all those who helped us during the writing of this manuscript.

Funding

This research received no external funding.

Conflict of interest

The authors declare no conflict of interest. Azeem Latib and Antonio Mangieri are serving as Guest editors of this journal. We declare that Azeem Latib and Antonio Mangieri had no involvement in the peer review of this article and have no access to information regarding its peer review. Full responsibility for the editorial process for this article was delegated to Alexander Lauten.

Publisher’s Note: IMR Press stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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# The Wronskian of vector valued functions vs. the Wronskian of real valued functions.

When discussing, for example, two solutions $\phi_1(t)$ and $\phi_2(t)$ of a second order homogeneous ode the Wronskian $$W[\phi_1,\phi_2](t)=\begin{vmatrix}\phi_1(t)&\phi_2(t)\\\phi_1'(t)&\phi_2'(t)\end{vmatrix}$$ of the solutions might be mentioned. Likewise, when discussing two solutions $$\phi_1(t)=\begin{bmatrix}x_1(t)\\y_1(t)\end{bmatrix}\qquad\phi_2(t)=\begin{bmatrix}x_2(t)\\y_2(t)\end{bmatrix}$$ of a system of two first order linear ode's the Wronskian $$W[\phi_1,\phi_2](t)=\begin{vmatrix}x_1(t)&x_2(t)\\y_1(t)&y_2(t)\end{vmatrix}$$ of the solutions might be mentioned. Now, in the first case the Wronskian is an array of derivatives, while in the second, it is an array of vector components.

As in both cases the Wronskian provides information about linear independence of the solutions, it would appear that the two "versions" (array of derivatives vs. array of components) are not really different but I am missing the connection between the two. Can anyone provide some insight into that connection?

1. Consider vector functions $$y_1(t),\ldots,y_k(t)$$ with components $$y_j(t)=(x_j(t),x_j'(t),\ldots,x_j^{(k-1)}(t)).$$ Then the system of functions $\{y_1,\ldots,y_k\}$ is linearly dependent on interval $I$ if and only if the system of functions $\{x_1,\ldots,x_k\}$ is linearly dependent on $I$.
2. Consider an $k$-th order linear ODE with a fundamental system of solutions $\{x_1,\ldots,x_k\}$ and reduce it to $k$ first order equations with the fundamental system of solutions $\{y_1,\ldots,y_k\}$.

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# Limit of a sequence

by KLscilevothma
Tags: limit, sequence
 P: 321 It isn't a homework problem but I think I better post it here instead of Mathematics forum, since it belongs to "exam help". Prove that for any positive real numbers a and b, lim [(an+b)1/n-1] = 0 n->inf I don't need to use things like |a-b|
PF Patron
Emeritus
P: 5,539
 Originally posted by KL Kam Prove that for any positive real numbers a and b, lim [(an+b)1/n-1] = 0 n->inf
This one just screams "L'Hopital!"

First, rearrange it to:

lim(an+b)1/n=1
n-->∞

Then take the natural log of both sides to get:

lim ln(an+b)/n=0
n-->∞

This goes to ∞/∞, which is an indeterminate form and ripe for L'Hopital's rule.
 P: 321 LOL, thanks Tom and L'hopital lim ln(an+b)/n n->[oo] = lim a/(an+b) n->[oo] =0
P: 321

## Limit of a sequence

Oh sorry, I forgot to mention
(an+b)1/n-1
is a sequence, not a function. I think L'hopital's rule applies to differentiable functions only.

Perhaps I better rephase the question a bit.
A sequence {an} is defined by (an+b)1/n-1
Prove that
lim (an+b)1/n-1 = 0
n->inf
(a and b are real numbers and n is a positive integer)
 PF Patron Sci Advisor Thanks Emeritus P: 38,429 It is true that L'hopital's rule applies to functions rather than sequences. However, IF we can convert a sequence an to a function f(x) (we can't if the sequence involves things like n! or "floor" or "ceiling" that can't be written simply as a continuous function), then f(x)-> L, an-> L. The other way doesn't necessarily work- the function might not have a limit, depending on how it is defined for non-integer values.
P: 321
 Originally posted by HallsofIvy However, IF we can convert a sequence an to a function f(x) (we can't if the sequence involves things like n! or "floor" or "ceiling" that can't be written simply as a continuous function), then f(x)-> L, an-> L. The other way doesn't necessarily work- the function might not have a limit, depending on how it is defined for non-integer values.
So we can treat a sequence as a function if it is an "elementary" one like the one I posted, and can apply L'hopital's rule, is it correct?

 Related Discussions Calculus & Beyond Homework 3 Calculus & Beyond Homework 6 Calculus 1 Introductory Physics Homework 3 Calculus 28


text:
## Book Store

Currently only available for.
CBSE Gujarat Board Haryana Board

## Previous Year Papers

Download the PDF Question Papers Free for off line practice and view the Solutions online.
Currently only available for.
Class 10 Class 12

The distance travelled by a particle starting from rest and moving with an acceleration 4/3 ms-2 in the third second is

• 6 m

• 4 m

• 10/3

• 19/3

C.

10/3

Distance travelled by  the particle in nth second

Where u is initial speed and a is an acceleration of the particle.

Here, n = 3, u =0, a = 4/3 m/s2

Distance travelled by  the particle in nth second

Where u is initial speed and a is an acceleration of the particle.

Here, n = 3, u =0, a = 4/3 m/s2

521 Views

What is a vector quantity?

A physical quantity that requires direction along with magnitude, for its complete specification is called a vector quantity.
835 Views

What are the basic characteristics that a quantity must possess so that it may be a vector quantity?

A quantity must possess the direction and must follow the vector axioms. Any quantity that follows the vector axioms are classified as vectors.

814 Views

Give three examples of vector quantities.

Force, impulse and momentum.
865 Views

What is a scalar quantity?

A physical quantity that requires only magnitude for its complete specification is called a scalar quantity.
1212 Views

Give three examples of scalar quantities.

Mass, temperature and energy
769 Views


text:
## Aug 10, 2012

### Longest Common Subsequence in Python (version 3.0)

This is my new implementation of the Longest Common Subsequence algorithm, and it's far better than my two previous attempts. This algorithm has less memory usage and is faster than the others because it just compares the common elements.

This algorithm has 3 main variables: the first index of a and b (ia and ib) in wich the comparisson must begin, and the number of elements to compare in each phase (s).

• Phase 1: The number of elements to compare is equal to 1, the last index of a with the first index of b, and for every iteration the number of elements to compare increases by one. then ia starts with the last index of a and decreases by one in every iteration, while ib is always 0.
• Phase 2: The number of elements to compare becomes constant and equal to the size of the shortest sequence. if ia is not equal to the first element of a, then decrease ia by one until it becomes equal to 0, and stays constant. Then begin to increase ib by one.
• Phase 3: For every iteration, decreases by one the number of elements to compare. ia is always equal to the first element of a, while increases ib by one.
The number of iterations is equal to:
$$l = len(a) + len(b) - 1$$
#!/usr/bin/env python
# -*- coding: utf-8 -*-
#
# Longest Common Subsequence, 3 phases comparisson algorithm.
# Copyright (C) 2012  Gonzalo Exequiel Pedone
#
# This program is free software: you can redistribute it and/or modify
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with This program. If not, see <http://www.gnu.org/licenses/>.
#
# Email   : hipersayan DOT x AT gmail DOT com
# Web-Site: http://hipersayanx.blogspot.com/

def lcs(a=[], b=[]):
ja = -1
jb = -1
n = 0

if a == [] or b == []:
return ja, jb, n

l = len(a) + len(b) - 1
ia = len(a) - 1
ib = 0
s = 1

for k in range(l):
nCur = 0

for r in range(s):
if a[ia + r] == b[ib + r]:
nCur += 1

if nCur > n:
ja = ia + r - nCur + 1
jb = ib + r - nCur + 1
n = nCur
else:
nCur = 0

if k < min(len(a), len(b)) - 1:
ia -= 1
s += 1
elif k > l - min(len(a), len(b)) - 1:
ib += 1
s -= 1
elif ia > 0:
ia -= 1
else:
ib += 1

return ja, jb, n

a = 'Andrea always wears black cat to all places she go.'
b = 'She likes her black cat because it is very mischievous and playful.'

ia, ib, n = lcs(list(a), list(b))

print([a[ia: ia + n]])
print([b[ib: ib + n]])

#### 1 comment:

1. Hi! Great post! I wonder if there is any way to fit this lcs v3.0 into the alignSequences function (http://hipersayanx.blogspot.com.co/2012/08/multiple-sequences-alignment-in-python.html).


text:
# Math Help - Convergence, Absolute Convergenge and Divergence

1. ## Convergence, Absolute Convergenge and Divergence

Determine whether each of the following absolutely converges, converges but not absolutely, or diverges:

2. Originally Posted by My Little Pony
Determine whether each of the following absolutely converges, converges but not absolutely, or diverges:
Do you have any thoughts on any of these?

3. I think I got number 1.

From the limit test, we end up getting:

(-7)^n as n approaches infinity. Since this does not approach a number, it is divergent?

4. Also number 4 fails the test for divergence, because as n approaches infinity, the limit is 0. Therefore, it is convergent?


text:
# Conflict between amsmath and pdfcomment

I use both amsmath and pdfcomment in one document. This leads to the references produced by (\ref{}) to appear like (zref@1) instead of numbers like (1). Without amsmath or pdfcomment this problem does not show up.

A minimal (not) working example:

\documentclass{article}
\usepackage{amsmath}
\usepackage{hyperref}
\usepackage{pdfcomment}

\begin{document}

$$\label{eq:1} E = m \pdftooltip{c}{speed of light}^2$$

See eq. (\ref{eq:1}). % <- this produces (zref@1) instead of (1)

\end{document}

Compile it with pdflatex.

Any suggestions are very welcome.

-

The problem is not really to do with pdfcomment, except that one is more likely to use a \pdftooltip inside an equation than, say, a \section command. But if you were so foolish as to try:

$$\label{eq:1} x^2 + y^2 = z^2 \section{Pythagoras}$$

then (assuming it actually compiled, which I doubt) you would find exactly the same problem.

It is due to the fact that when amsmath processes an equation environment, including the equation one itself (since it redefines it), then it does some Complicated Stuff on the equation which involves processing it a few times over (this causes Issues with beamer and tikz as well, I believe). One consequence of this is that if you put a \label command in an equation environment then the label is not saved right away, rather the equation is processed at least once before the label is saved to the .aux file. This is presumably to deal with things like align where more than one piece of the whole equation contents can be labelled (I'm guessing here).

Now the \label command figures out what information to save by using the value of \@currentlabel. This is a neat piece of work, and it's what ensures things like \section{Introduction}\label{sec:intro} Just Works(TM). But this does mean that you need to ensure that between when the counter is incremented and the \label command is processed then the the \@currentlabel doesn't get redefined.

Here's where amsmaths wizardry works against it. If there is a command inside the equation which increments a counter in such a way that the \@currentlabel is redefined then it is that which gets saved rather than the equation label. My silly example with \section is one such command, but that's unlikely to appear in an equation environment. Another is \pdftooltip and that, as we see, is a little more likely.

To verify that what I'm saying is true, we can simulate the \section command:

\documentclass{article}
\usepackage{amsmath}

\renewcommand{\thesection}{\alph{section}}

\begin{document}

\section{Introduction}
\label{sec:intro}

$$\label{eq:1} E = m c^2 \refstepcounter{section}$$

See eq.\ \ref{eq:1} in Section~\ref{sec:intro}.

\end{document}

This produces:

See eq. b in Section a.

If we comment out the \refstepcounter line, we get:

See eq. 1 in Section a.

As another experiment, we can remove amsmath and get the problem with the following code:

\documentclass{article}
\usepackage{pdfcomment}

\renewcommand{\thesection}{\alph{section}}

\begin{document}

\section{Introduction}
\label{sec:intro}

\pdftooltip{hello}{world}

\label{sec:introb}

Sections~\ref{sec:intro} and~\ref{sec:introb} are the same.

\end{document}

In this case, the answer is simple: make sure you issue the \label before the \pdftooltip. But for the amsmath situation, things aren't so easy because everything is internal.

Looking in the source of pdfcomment I can see that \pdftooltip uses \refstepcounter which is the magic command which resets the \@currentlabel. My guess is that it only needs to use it internally, and afterwards it doesn't need to have \@currentlabel set to anything particular. So the best solution would be to patch the pdfcomment so that it saves the current value of \@currentlabel when invoked and restores it again afterwards.

However, I don't recommend patching packages yourself. So you should contact the maintainer of the package and ask him to do so. In the meantime, there is a way to get around the problem. This is simply to ensure that the correct \@currentlabel is in force at the end of the equation. So we need to call \refstepcounter but without actually incrementing the counter. Here's a simple macro that does that:

\makeatletter
\def\refsetcounter#1{%
\protected@edef\@currentlabel
{\csname p@#1\endcsname\csname the#1\endcsname}%
}
\makeatother

Call it as \refsetcounter{equation}. It has to be placed at the end of the equation environment (or at least, after the \pdftooltip).

Now, it would be nice to have this done automatically. This would involve patching the amsmath commands. Normally, I'd be wary about this as well, but I feel a little more on sure ground with the amsmath package than the pdfcomment one so I'm prepared to have a go!

My experiments show that it is actually only the equation command that has this problem. Somehow, things like align, gather, and multline work just fine. So we only need to ensure that our new command, \refsetcounter{equation} is run at the end of the equation environment. Here's the code for that:

\makeatletter
\def\refsetcounter#1{%
\protected@edef\@currentlabel
{\csname p@#1\endcsname\csname the#1\endcsname}%
}
\let\orig@endequation=\endequation
\def\endequation{%
\refsetcounter{equation}%
\orig@endequation}
\makeatother

With this in the preamble, your code produces:

See eq. (1).

As it should.

-
 Thanks a lot. It has solved my problem. – amorua Sep 13 '11 at 19:13 Andrew, this was a very interesting read, though I confess, that I'm not latex literate enough to understand your code. It seems that the problem here is almost identical in nature to a problem that I posted here to do with markup in figure captions: tex.stackexchange.com/q/42960/1269 . I tried your temporary solution using \refsetcounter{figure} in figure captions, and also at the end of the figure environment, but this didn't work. I'm guessing that different variables are involved. – aghsmith Feb 8 '12 at 17:36 @aghsmith I've had a look at your question and it certainly looks similar enough. However, the figure environment doesn't work in the same way as the equation one so even if the problems are the same then the solutions may be different. – Andrew Stacey Feb 8 '12 at 19:18


text:
## 305 – Friedman’s theorem

February 4, 2012

I am sketching here a proof of Harvey Friedman‘s result mentioned in lecture. The proof will not be needed for the rest of the course, but it is a nice argument that you may enjoy reading.

Recall that a sequence $x_0 x_1 \dots x_n$ has property $*$ iff there are no integers $i such that the sequence $x_i x_{i+1}\dots x_{2i}$ is a subsequence of $x_j x_{j+1}\dots x_{2j}$.

Here, we are using the notion of subsequence where terms must appear in order but are not required to be consecutive: A sequence $a_1 a_2\dots a_k$ is a subsequence of $b_1 b_2\dots b_l$ iff there are indices $1\le i_1 such that

$b_{i_1}=a_1,b_{i_2}=a_2,\dots,b_{i_k}=a_k.$

Theorem (H. Friedman, 2001). For any positive integer $k$ there is a longest finite sequence $x_1 x_2\dots x_n$ in $k$ letters with property $*$.

(Of course, once we know that the theorem is true, we can proceed as in lecture and define $n(k)$ as the length of this longest sequence.)

## 502 – König’s lemma (2)

August 26, 2009

We continue with the example of domino systems.

Remark 1 There is no algorithm that determines whether a given ${D}$ can tile the plane or not.

Of course, for specific systems ${D,}$ we usually can tell by ad hoc methods which one is the case. What the remark above indicates is that there is no uniform way of doing this.

## 502 – König’s lemma

August 24, 2009

(The material on mathematical logic is covered in the textbook starting with Chapter 5; however, for the first few lectures, I will be providing some required background topics and will not be following the book. There are several references that you may find useful. For example,

• Kaye, Richard. The mathematics of logic, Cambridge University Press (2007).

I am also making use of a set of notes originally developed by Alexander Kechris for the course Math 6c at Caltech.)

## 580 -III. Partition calculus

March 21, 2009

1. Introduction

Partition calculus is the area of set theory that deals with Ramsey theory; it is devoted to Ramsey’s theorem and its infinite and infinitary generalizations. This means both strengthenings of Ramsey’s theorem for sets of natural numbers (like the Carlson-Simpson or the Galvin-Prikry theorems characterizing the completely Ramsey sets in terms of the Baire property) and for larger cardinalities (like the ${\mbox{Erd\H os}}$-Rado theorem), as well as variations in which the homogeneous sets are required to possess additional structure (like the Baumgartner-Hajnal theorem).

Ramsey theory is a vast area and by necessity we won’t be able to cover (even summarily) all of it. There are many excellent references, depending on your particular interests. Here are but a few:

• Paul ${\mbox{Erd\H os},}$ András Hajnal, Attila Máté, Richard Rado, Combinatorial set theory: partition relations for cardinals, North-Holland, (1984).
• Ronald Graham, Bruce Rothschild, Joel Spencer, Ramsey theory, John Wiley & Sons, second edn., (1990).
• Neil Hindman, Dona Strauss, Algebra in the Stone-${\mbox{\bf \v Cech}}$ compactification, De Gruyter, (1998).
• Stevo ${\mbox{Todor\v cevi\'c},}$ High-dimensional Ramsey theory and Banach space geometry, in Ramsey methods in Analysis, Spiros Argyros, Stevo ${\mbox{Todor\v cevi\'c},}$ Birkhäuser (2005), 121–257.
• András Hajnal, Jean Larson, Partition relations, in Handbook of set theory, Matthew Foreman, Akihiro Kanamori, eds., to appear.

I taught a course on Ramsey theory at Caltech a couple of years ago, and expect to post notes from it at some point. Here we will concentrate on infinitary combinatorics, but I will briefly mention a few finitary results.

## 580 -II. Cardinal arithmetic

February 5, 2009

Homework problem 5. (${\sf ZF}$). Show that if $\omega\preceq{\mathcal P}(X)$ then in fact ${\mathcal P}(\omega)\preceq{\mathcal P}(X)$.

Question. If $X$ is Dedekind-finite but ${\mathcal P}(X)$ is Dedekind-infinite, does it follow that there is an infinite Dedekind-finite set $Y$ such that ${\mathcal P}(Y)\preceq X$?

From now on, until we cover determinacy, we assume the axiom of choice unless stated otherwise.


text:
# Famous Finite Sets [closed]

What are the most famous (or most beautiful, IYO) finite sets in mathematics? I'm especially looking for 'large' sets that contain more than $2^{10} \approx 1000$ but fewer than $2^{20} \approx 1{,}000{,}000$ elements.

I'll start the ball rolling with the five platonic solids. (Unfortunately not large.)

-

## closed as too localized by Grumpy Parsnip, Pete L. Clark, Ilya, J. M., Asaf KaragilaOct 20 '11 at 11:31

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

6 775 235 700 (en.wikipedia.org/wiki/World_population) is a bit bigger then 2^20, but still it is a finite set of such cardinality where all of us are elements. – Ilya Oct 19 '11 at 15:12
@Gortaur: "in mathematics." – Qiaochu Yuan Oct 19 '11 at 15:13
I can't wait until Asaf sees this question... – Willie Wong Oct 19 '11 at 15:19
I don't understand the motivation behind this question. – Grumpy Parsnip Oct 19 '11 at 15:43
This is not actually a question in the sense of this site. It is a call for a discussion, or referendum, on people's tastes and preferences. I have voted to close. – Pete L. Clark Oct 19 '11 at 16:58

The sporadic groups? In particular they are finite sets... quite a few are too big to fit into your range, but the smallest (Mathieu groups) would do the trick.

-

As is well known, every finite (natural) number can be associated with a finite set of that cardinality. So in particular, the cardinality of a famous or special finite set must be a famous or special number. Here's a list of all the special numbers less than or equal to 9999 and contains quite a few items between $2^{10}$ and $2^{20}$.

Oh, what, you actually want the sets, and not just their cardinality, because there is more than one way of realising a set of a given cardinality? (Me grumbles something about bijective maps and isomorphisms of sets.) Fine:

• 1132 is the number of 3-valent trees with 15 vertices
• 1144 is the number of non-invertible knots with 12 crossings.
• 1165 is the number of conjugacy classes in the automorphism group of the 12 dimensional hypercube.
• 1205 is the number of fullerenes with 58 carbon atoms
• 1294 is the number of 4 dimensional polytopes with 8 vertices.
• 1378 is the number of symmetric idempotent 6×6 matrices over GF(2).
• 1411 is the number of quasi-groups of order 5.
• ...
• 3240 is the number of 3×3×3 Rubik's cube positions that require exactly 3 moves to solve.
• 3286 is the number of stable patterns with 16 cells in Conway's game of Life.
• ...
• 4535 is the number of unlabeled topologies with 7 elements.
• ...

As beauty is in the eye of the beholder, I'm sure there are mathematicians out there who think each of the above numbers ought to be better known.

-


text:
# The necessity of using tangent space as the vectors in general relativity

I’ve recently learnt what manifolds are to prepare myself for a course in GR. My relevant mathematical background is linear algebra (abstract, proof-ish course) and multivariable/vector calculus (course mostly focussed on computations). I use Knuth and Renteln as my main references.

The idea of an n-dimensional manifold is introduced as a combination of open sets whose union forms the manifold. Each such open set must have a continuous 1-to-1 map to an open set in n-dimensional Euclidean space; that is: each point within these open sets can be described as an n-tuple, just like vectors and points in ”regular” space can. Please correct me if I’m wrong here, I really want to get my definitions right.

Subsequently the tangent space is introduced as a vector space for each point on the manifold, whose elements are differential operators. I recognise that that this vector space is very much usable to describe vector fields on the manifold.

But as I read the definition of manifolds, I intuitively expected the displacement between points to be a good definition for vectors on my manifold. If our manifold consists of multiple charts, this would not be possible, since the space would not be closed: if we added the displacement between points to itself long enough we would eventually “exit” the open set in which the tuples are defined, which is by definition not possible. I’m still interested in knowing why in general relativity we use tangent spaces instead of conventional classic coordinate tuples.

So my first question is: if a manifold is describable by a single chart, can we define a vector space simply by taking displacements between points on that chart as n-tuple vectors? For instance, take the manifold $$\mathbf{r} = (x,y,z)^T = (f_1(u,v), f_2(u,v), f_3(u,v))^T$$: a 2D-manifold embedded in 3D space. If the functions $$f_i$$ are well behaved so that the manifold is smooth and has no sharp edges/crossings/etc, wouldn’t $$(u,v)$$ tuples form a nice vector space? You could add them, scale them, and fulfil all the properties of a vector space.

My second question is more to manage my own expectations for later: is this single-coordinate chart situation one that occurs at all in G.R.? If each/most manifold(s) in G.R. naturally requires two or more charts, then it would be senseless to take these displacement tuples as vectors. The reason I still ask is because I intuitively expected that a single chart should encompass all of space; I would be very surprised if I was working out a physics problem and I couldn’t describe my worldline in the same coordinate system everywhere.

So my first question is: if a manifold is describable by a single chart, can we define a vector space simply by taking displacements between points on that chart as n-tuple vectors?

Not in general. Consider the real line as a manifold $$M$$, equipped with the chart $$(\mathbb R, x)$$ where the chart map $$x$$ is defined by

$$x:M \rightarrow \mathbb R$$ $$p \mapsto x(p) = \tan^{-1}(p)$$

I cannot define a vector space from the coordinates of points on the manifold, because the range of the chart map is only $$x(\mathbb R)=(-\pi/2,\pi/2)$$.

You might say that this is simply a bad choice of chart (should that matter?), but note that if the Riemann curvature tensor of the manifold does not vanish everywhere, then it is not possible to construct a globally Euclidean chart.

is this single-coordinate chart situation one that occurs at all in G.R.? [...]I would be very surprised if I was working out a physics problem and I couldn’t describe my worldline in the same coordinate system everywhere.

Polar coordinates have singular behavior at the coordinate origin; spherical coordinates have singular behavior at the origin as well as the poles. Even manifolds as mundane as $$\mathbb R^n$$ generically require more than one chart if you use non-cartesian coordinates.

It can be shown that if a manifold possesses intrinsic curvature then it cannot be described by a single, globally Euclidean chart. In other words, while it is possible to find very special cases in which you could find a single chart of the kind you want, it is only possible when the space possesses no curvature.

Essentially, you could (on rare occasions) artificially shoehorn a vector space structure into your spacetime if you want, but it would be generally pointless. Don't bother.

As far as I’m aware vectors do not need to follow any transformation requirements (though it’s nice if they do) but only a set of eight definitions. (comment on other answer)

This is the mathematical definition of a vector space. The vectors that puppetsock is referring to are tangent vectors to a manifold; those objects (or rather, their components in a given chart), which are the ones you'll be dealing with in GR, do possess certain transformation properties which follow from the requirement that tangent vectors be chart-independent geometrical objects.

• “It can...no curvature.” This is exactly the confirmation I needed! It’s not that I don’t understand tangent space, it’s just that I found it so silly to work with if the displacement between points sufficed as well. Now I know that it doesn’t (is a proof of this a standard part of diff geometry courses?), so thanks a lot! One last tiny question: in my OP I wrote that I figured in a multiple-chart manifold we could never use coordinate displacements as vectors since eventually you’d “walk out” of the open space on which those coordinates are defined. Was that correct (if perhaps a tad loose)? Jul 29, 2019 at 19:06
• @Heatherfield That's the general idea - or at least a component of it. Loosely speaking, a manifold admits a global chart which you could equip with a vector space structure if and only if the manifold is a smooth deformation of $\mathbb R^n$ for some $n$. There are many manifolds for which this is not the case - any compact manifold, like a circle or sphere, would serve as an example. More importantly, your vector space structure would only make sense for that chart - if you wanted to switch to different coordinates (polar, cylindrical, etc) the structure would not carry over. Jul 29, 2019 at 19:44
• "if the Riemann curvature tensor of the manifold does not vanish everywhere, then it is not possible to construct a globally Euclidean chart." What exactly do you mean by a "globally Euclidean chart"? It is certainly possible to have spaces diffeomorphic to $\Bbb R^n$ (i.e. manifolds that can be completely covered by a single chart) which nevertheless have non-trivial curvature.
– Danu
Jul 30, 2019 at 15:30
• @Danu By globally Euclidean chart, I mean a chart whose range is $\mathbb R^n$ and in which the connection coefficients all vanish. It's a stronger statement than was asked for, but I interpreted subtext of the question to be "why can't we just equip spacetime with an affine structure like we do in nonrelativistic physics?" in which case I felt that things like the connection would be more relevant. Jul 30, 2019 at 16:27
• OK. When in your terminology a manifolds admits a "globally Euclidean chart", the usual terminology is to say that it is isometric to Euclidean space. As you note, this really isn't the notion that the OP asked about, so it's potentially confusing to bring it up here. For instance, many Riemannian manifolds do not admit any "Euclidean chart" (i.e. local isometry to $\Bbb R^n$) even locally, though every smooth manifold has many charts (local diffeomorphisms to $\Bbb R^n$).
– Danu
Jul 30, 2019 at 20:36

You can define the tangent space in sort of the way you have in mind, but it's not enough to make the displacements small enough to keep from crossing into a different chart. You need to make them infinitesimal in size, and we also want a definition that's coordinate-independent. For a mathematically rigorous treatment in this style, see Nowik and Katz, "Differential geometry via infinitesimal displacements," https://arxiv.org/abs/1405.0984 .

• Thanks, I’ll save it for later when I feel more prepared to tackle a subject using nonstandard analysis (since I barely understand usual nonstandard analysis as is). Jul 29, 2019 at 19:42

Vectors must transform in a specific manner under a coordinate transform. It's not clear that your $${u,v}$$ necessarily forms a vector.

• As far as I’m aware vectors do not need to follow any transformation requirements (though it’s nice if they do) but only a set of eight definitions. A copy of the list is available on the relevant wiki page: en.m.wikipedia.org/wiki/Vector_space. Does this mean that I’m not looking to just define vectors on my manifold, but I’m looking to define a special kind? That’d seem strange to me, while I know some textbooks define tangent vectors in terms of their transformation properties, the resources I use derive these transformation properties as a handy consequence, not as an intended one. Jul 29, 2019 at 18:32


text:
# How do I describe a DFA?

Consider the alphabet E = ${[abc] : a, b, c \in 0,1,...,9)}$ Example [234], [567], [897] are symbols of the alphabet. For a string $w \in$ let n(w) denote the number represented by $w$: Example for symbol [345], n([345]) = 345.

1) Describe a DFA for the language of strings of the form $[x_0y_0z_0][x_1y_1z_1] [x_ny_nz_n]$ such that

$n(x_n... x_1x_0) + n(y_n... y_1y_0) = n(z_n... z_1z_0)$

This language corresponds of reading the numbers from right to left and position by position; this is how we add numbers by hand.Example $[819][606][213]$ is in the language because 268 + 101 =369.

How can we describe something like this? $M = (Q, E, d, q_0, F)$

where Q : ?

E = ${[abc] : a, b, c \in 0,1,...,9)}$

d: ?

$q_0 =$

F: ?

• Normally, what you would do is just "draw the DFA", as you put it. It is not always practical to do so - for instance, it may be too big. In these cases, what you do is describe how would you draw it, like an algorithm. – André Souza Lemos Apr 6 '16 at 22:50
• hmmm so in general we can use the M = (Q, E, d , q_0, F) to describe it? – Darkflame Apr 6 '16 at 22:53
• Try to understand why your teacher (I suppose you are a student) has given you the exercise. Use your creativity. – André Souza Lemos Apr 6 '16 at 23:16
• So you basically ask what does it mean to describe DFA? why the long story attached to it? describing a DFA, is either (1) writing the formal 5-tuple, or (2) drawing the "state machine", or (3) describing in words what the DFA does. In "tests", way (1) and (2) are usually accepted, but its up to your professor. Consider editing the question to reflect what you really want to ask. – Ran G. Apr 7 '16 at 0:34
• Then talk to your teacher and/or get a book. Seriously. The internet is not a good place to ask people if you don't even know what the task is. (And yes, plenty of people know how to start and still post here -- because then they can ask a real question.) Your question is similar to a person asking "how to solve this problem in Java?" when they don't know any Java. – Raphael Apr 7 '16 at 20:08

Notice that it's not unreasonable to slightly extend this problem to an arbitrary number of triples. For example, the single triplet [123] is a reasonable candidate to be in your language, since $1+2=3$. Similarly, [280][135] is another reasonable candidate, since $12+38=50$. With that in mind, I'll extend the language to all strings of an arbitrary number (including zero) of triples that satisfy the summation requirement. I'll also decide that we'll accept a sequence of triplets only if they represent a sum without a final carry, so we wouldn't accept the string [000][000][190] since $100+900=1000$ and there's a final carry out (though [000][000][190][001] would be accepted).

The fundamental idea is a very common one: "remember" the current state of the problem by using one state in the DFA for each possible configuration. Here's what we might do:

1. Label the start state $start$. We haven't read any input yet. This will also be the unique final state.
2. Let state $t$ remember that we've seen the first [, so we have the move $\delta(start, [)=t$.
3. Next, construct ten states $p_0, p_1, \dotsc,p_9$ to remember the first digit. We'll have the moves $\delta(t,i)=p_i$, i.e., from $t$ we'll be in state $p_i$ after having seen input $i$.
4. Now make twenty more states, $q_0, \dotsc,q_9$ and $c_0,\dotsc,c_9$. The $q$'s will record the sum of the first two inputs if there's no carry and the $c$'s will record the sum if there is a carry. In other words $$\delta(p_i,j) =\begin{cases} q_{i+j} &\text{if } i+j\le 9\\ c_{(i+j)\bmod{10}} &\text{if } i+j\ge 10 \end{cases}$$
5. Then make two states, $a,b$ to note whether the third input is the sum of the first two and wehther or not we've generated a carry. From state $q_k$ (sum is $k$ with no carry) we'll pass to state $a$ if the third input is the digit $k$. Similarly, we'll have $\delta(c_k,k)=b$ meaning that we've just seen the appropriate third digit and we had a carry.
6. Now we've read the first triplet and we're expecting to see a ]. In state $a$ we know we've completed the first triplet and don't have to adjust any subsequent triplet, so $\delta(a,]) = start$, i.e., we loop back to the start and accept if there's no further input. On the other hand, if we're in state $b$ then we have a correct digit sum, but we also have a carry, so we can't yet accept when we see a ], instead we need to read another triplet. We'll process the ], read a [ and duplicate the whole process again, this time adjusting for the carry. We'll then have $\delta(b, ])=u$ and $\delta(u,[)=v$.
7. Starting anew from state $v$, we construct 34 new states exactly as we did in steps (1) - (6). The only difference is that the transitions from the new states corresponding to the original $p$ states (call them $p'_i$) in step (4) we'll compensate for the previous carry in and have $$\delta(p_i,j) =\begin{cases} q_{i+j+1} &\text{if } i+j+1\le 9\\ c_{(i+j+1)\bmod{10}} &\text{if } i+j+1\ge 10 \end{cases}$$ The final difference with these new states is that if we get to the new state, $b'$ corresponding to the original $b$, then on seeing ], we go back to state $start'$ and start again.

In high-level terms, we have two nearly-identical functions to check whether a triplet $t$ is valid:

• $N(t)$ which checks whether $t$ is valid, assuming no carry in. This will return 0 or 1, depending on whether or not $t$ generated a carry.
• $C(t)$ which checks whether $t$ is valid, assuming a carry in. This will return 0 or 1, depending on whether or not $t$ generated a carry.

Then, we would have the following pseudocode

c = 0
while there's a triplet t to process
get t
if c = 0
c = N(t)
else
c = C(t)
end loop
if c = 0
accept
else
reject


text:
# Help with plotting in matlab

• MATLAB
Okay so I am really new to matlab, I mean REALLY new. I am trying to write a program for one of my classes but I am having some trouble. I keep getting this error message:

??? Error using ==> plot
Invalid first data argument

Error in ==> guessTraj at 89
plot('b-')

And here is my code:

Code:
function [output]= guessTraj(target_dis)

target_dis=input('distance :');

error = sqrt(target_dis);

real = isreal(error);

if real == 0 & isnumeric(target)== 0

output = 4

disp('Invalid Distance')

end

angle = input('Please enter the angle in degrees:');

velocity = input('Please enter the velocity the ball is thrown:');

vel_error = sqrt(velocity);

vel_check = isreal(vel_error);

if vel_check == 0

disp('Invalid Velocity')

end

VY = velocity * sin(radians) ;

VX = velocity * cos(radians) ;

t = (2*velocity)/9.81 ;

x_dis = VX * t ;

y_dis = VY * t ;

H1 = target_dis + 2.5 ;

H2 = target_dis - 2.5 ;

x = 0:x_dis/50:x_dis;

plot(x,x_dis,target_dis,0,'mo')

xlabel ('Distance the ball Traveled')

ylabel ('Height of the ball')

legend ('Trajectory of Ball', 'Location', 'SouthEastOutside')

hold on

timeVector = 0:t/50:t;

plot('b-')

if x_dis <= H1 && x_dis >= H2

title('The ball hit the target')

output = 1;

disp('1. The ball hit the target!!!!!')

elseif x_dis < H1

title('The ball was underthrown')

under = target_dis-x_dis;

output = 2;

disp('2. The ball was underthrown')

elseif x_dis > H2

title('The ball was overthrown')

over = x_dis-target_dis;

output = 3;

disp('3. The ball was overthrown')

end

I'm not sure what I'm doing wrong. I want to plot the path of the object... What am I doing wrong?

Thanks so much!!
Code:


text:
## Birational Symmetries, Hirota Bilinear Forms and Special Solutions of the Garnier Systems in 2-variables

J. Math. Sci. Univ. Tokyo
Vol. 10 (2003), No. 2, Page 355--371.

Tsuda, Teruhisa
Birational Symmetries, Hirota Bilinear Forms and Special Solutions of the Garnier Systems in 2-variables
[Full Article (PDF)] [MathSciNet Review (HTML)] [MathSciNet Review (PDF)]

Abstract:
Hirota bilinear forms of the Garnier system in 2-vari\-ables, $G(1,1,1,1,1)$, are given. By using Hirota bilinear forms we construct new birational symmetries of $G(1,1,1,1,1)$. We obtain special solutions of the Garnier system in $n$-variables, which are described in terms of solutions of the Garnier system in $(n-1)$-variables. We investigate also algebraic solutions for $n=2$.

Mathematics Subject Classification (2000): Primary 34M55; Secondary 33E17
Mathematical Reviews Number: MR1987136

Received: 2002-05-27


text:
Usage
• 42 views

# Under-Dense Laser-Plasma Interactions in Relativistic Optical Vortices

• Author / Creator
Longman, Andrew M
• In the ever-evolving field of high-intensity laser interactions with matter, many approaches have been taken to control and enhance laser-plasma interactions. An alternative, and potentially beneficial approach to further controlling laser-plasma interactions comes from spatial structuring of the laser phase. In this thesis, we explore high intensity ($>1\\times10^{18}Wcm^{-2}$) structured light that carries orbital angular momentum (OAM) via a helical wavefront, also known as an optical vortex (OV). These beams have the unique property of carrying unprecedented amounts of angular momentum, even when linearly polarized. Coupling of this OAM to a plasma is of great interest for the generation of strong axial magnetic fields, particle guiding, enhanced radiation, and gaining an additional control parameter in the laser plasma interactions. We introduce the concept of an off-axis spiral phase mirror for the mode conversion of ultra-fast high-intensity laser systems, and demonstrate the highest intensity OAM optical vortices to date. Diffraction models of the high-intensity OV's are developed analytically and then approximated using Laguerre-Gaussian basis functions for both symmetric and asymmetric modes. The interaction of high-intensity OV's with single electrons and plasma is explored analytically, numerically and experimentally. We show that linearly polarized OAM coupling to single electrons and plasma occurs from nonlinear mechanisms, and can be used to accelerate particles and enhance the laser-plasma interactions. We explore both numerically with particle in cell (PIC) simulations and experimentally new modes of relativistic self-focussing, wakefield acceleration, betatron radiation, and magnetic field generation in the presence of an OAM pulse. We find that angular momentum is indeed coupled to the electrons and that the critical energy of the emitted betatron spectrum is increased with an increase in beam OAM. Additionally, we demonstrate in realistic full scale 3D PIC simulations kilo-Tesla magnetic field generation from the inverse Faraday effect and characterize the spatial extent and temporal duration of these fields.

• Subjects / Keywords
Spring 2021
• Type of Item
Thesis
• Degree
Doctor of Philosophy
• DOI
https://doi.org/10.7939/r3-9hzf-1449