Solving Non-homogeneous Nested Recursions Using Trees

The solutions to certain nested recursions, such as Conolly's C(n) = C(n-C(n-1))+C(n-1-C(n-2)), with initial conditions C(1)=1, C(2)=2, have a well-established combinatorial interpretation in terms of counting leaves in an infinite binary tree. This tree-based interpretation, which has a natural generalization to a k-term nested recursion of this type, only applies to homogeneous recursions, and only solves each recursion for one set of initial conditions determined by the tree. In this paper, we extend the tree-based interpretation to solve a non-homogeneous version of the k-term recursion that includes a constant term. To do so we introduce a tree-grafting methodology that inserts copies of a finite tree into the infinite k-ary tree associated with the solution of the corresponding homogeneous k-term recursion. This technique can also be used to solve the given non-homogeneous recursion with various sets of initial conditions.Comment: 14 page

Piling On? An Empirical Study of Parallel Derivative Suits

Using a sample of all companies named as defendants in securities class actions between July 1, 2005 and December 31, 2008, we study parallel suits relying on state corporate law arising out of the same allegations as the securities class actions. We test several ways that parallel suits may add value to a securities class action. Most parallel suits target cases involving obvious indicia of wrongdoing. Moreover, we find that although a modest percentage of parallel suits are filed first, over 80 percent are filed after a securities class action (termed “follow-on” parallel suits). We find that parallel suits and, in particular, follow-on parallel suits sometimes target individual officers not already named as defendants in the securities class action. Suing more officers, however, does not positively correlate with an increase in settlement incidence, monetary recovery amounts, or attorney fees. Parallel suits sometimes result in settlements when the corresponding class action is dismissed; however, only rarely do the parallel suit settlements provide monetary recovery for investors. We find that follow-on parallel suits often result in nonmonetary, corporate governance settlements, particularly for frequent-filing plaintiffs’ attorneys. Corporate governance settlements correlate with significantly lower attorney hours and attorney fees for the plaintiffs’ attorneys. We conclude that such settlements are used to justify fees in cases in which there is no monetary recovery

The Business of Securities Class Action Lawyering

Plaintiffs’ lawyers in the United States play a key role in combating corporate fraud. Shareholders who lose money as a result of fraud can file securities class actions to recover their losses, but most shareholders do not have enough money at stake to justify overseeing the cases filed on their behalf. As a result, plaintiffs’ lawyers control these cases, deciding which cases to file and how to litigate them. Recognizing the agency costs inherent in this model, the legal system relies on lead plaintiffs and judges to monitor these lawyers and protect the best interests of absent class members. Yet there is remarkably little data on the business of securities class action lawyers, leaving lead plaintiffs and judges to oversee this area without the tools to understand how it works.This Article looks inside the black box of securities class action lawyering to explore the business behind these cases. Our study includes hand-collected data on all securities fraud class actions against public corporations filed between 2005 and 2018, a total of nearly 2500 cases. We find that the business of securities class action lawyering is far more complex than prior scholarship has recognized. Contrary to conventional wisdom, there are not two tiers of plaintiffs’ law firms; instead, there are multiple tiers of firms, each with its own client base, litigation patterns, and revenue model. Our study gives lead plaintiffs and judges the data and tools they need to understand these tiers and to compare the performance of the law firms within them. We also examine how these law firms are compensated, finding that judges’ fee awards fail to account for the difficulty of cases or the risk of non-recovery in any systematic way. These fees are crucial to ensuring that law firms pursue the right cases on behalf of shareholders, so we suggest ways that judges can use data to improve fee awards. As we will see, the path to reforming securities class actions starts with understanding the business behind them

Paying for Performance? Attorneys’ Fees in Securities Fraud Class Actions

This Article studies whether plaintiffs\u27 lawyers matter in securities class actions. We use inverse propensity score weighting (IPW) to compare the results in cases led by top-tier firms against those brought by lower-tier firms. This technique addresses case selection effects by using all of the cases led by a top-tier firm and then weighting the cases led by lower-tier firms based on how similar these cases are to the cases led by top-tier firms. We do find that top-tier lawyers obtain better outcomes for shareholders in a subset of securities class actions, specifically the cases against the larger (although not the very largest) companies. Outside of these cases, we find that most of the difference in the results obtained by top-tier and lower-tier firms disappears when we balance observable characteristics using the IPW technique. Although the top-tier firms do not get better results in most cases, they do invest more hours and money into their cases

Coalitions among Plaintiffs’ Attorneys in Securities Class Actions

This paper examines contests among plaintiffs’ firms to become lead counsel in securities fraud class actions. We study lead plaintiff appointments in all federal securities class actions involving a disclosure claim from 2005 to 2016. We find that law firms’ decisions to combine correspond primarily with the existence of relationships—either with clients or between firms—rather than case characteristics. In the case of initial motions, the presence of a relationship between attorney firms and certain institutional investor movants for lead plaintiff corresponds with motions that propose multiple lead counsel. We also find that although the typical class action has several lead plaintiff motions, many of these motions will either withdraw or combine before the judge chooses the lead plaintiff and lead counsel. When motions combine, the number of selected lead counsel typically increases. The existence of prior relationships between law firms corresponds with the decision to withdraw or combine. Finally, we find that the correspondence of prior relationships between law firms on decisions to withdraw or combine does not increase with our proxies for the importance of synergies between law firms

The Business of Securities Class Action Lawyering

Plaintiffs’ lawyers in the United States play a key role in combating corporate fraud. Shareholders who lose money as a result of fraud can file securities class actions to recover their losses, but most shareholders do not have enough money at stake to justify overseeing the cases filed on their behalf. As a result, plaintiffs’ lawyers control these cases, deciding which cases to file and how to litigate them. Recognizing the agency costs inherent in this model, the legal system relies on lead plaintiffs and judges to monitor these lawyers and protect the best interests of absent class members. Yet there is remarkably little data on the business of securities class action lawyers, leaving lead plaintiffs and judges to oversee this area without the tools to understand how it works. This Article looks inside the black box of securities class action lawyering to explore the business behind these cases. Our study includes hand-collected data on all securities fraud class actions against public corporations filed between 2005 and 2018, a total of nearly 2500 cases. We find that the business of securities class action lawyering is far more complex than prior scholarship has recognized. Contrary to conventional wisdom, there are not two tiers of plaintiffs’ law firms; instead, there are multiple tiers of firms, each with its own client base, litigation patterns, and revenue model. Our study gives lead plaintiffs and judges the data and tools they need to understand these tiers and to compare the performance of the law firms within them. We also examine how these law firms are compensated, finding that judges’ fee awards fail to account for the difficulty of cases or the risk of non-recovery in any systematic way. These fees are crucial to ensuring that law firms pursue the right cases on behalf of shareholders, so we suggest ways that judges can use data to improve fee awards. As we will see, the path to reforming securities class actions starts with understanding the business behind them

Does Prehabilitation Prior to Ulnar Collateral Ligament Surgery Affect Return to Sport Rate or Time in Baseball Players with Partial UCL Tears?

Those who suffer a partial thickness ulnar collateral ligament (UCL) tear often undergo a period of nonoperative management including physical therapy rehabilitation. This treatment is aimed at optimizing range of motion (ROM) and strengthening the supporting structures around the elbow to help offload the UCL. Unfortunately, some of these patients fail nonoperative management and require surgical intervention. This creates a unique set of patients who essentially underwent “prehabilitation” prior to their UCL surgery. Prehabilitation is considered a period of structured physical therapy rehabilitation aimed at strengthening structures surrounding an injured tendon or ligament, to allow for dissipation of stress away from the repaired structure after surgery. Prehabilitation has been studied extensively and implemented into the clinical practice of anterior cruciate ligament (ACL) rehabilitation, and is being studied for other injuries as well. However, the efficacy of prehabilitation for UCL surgical patients with partial thickness UCL tears has not been evaluated. This chart review: - determines if baseball players with partial UCL tears who completed at least 4 weeks of prehabilitation prior to surgery (Prehab) had better return to play (RTP) rates and quicker return to sport (RTS) time than players who attempted 0-3 weeks of physical therapy prior to UCL surgery (No Prehab) - compares revision, reoperation, and patient reported outcomes between Prehab and No Prehab player

Dissipative preparation of W states in trapped ion systems

We present protocols for dissipative entanglement of three trapped-ion qubits and discuss a scheme that uses sympathetic cooling as the dissipation mechanism. This scheme relies on tailored destructive interference to generate any one of six entangled W states in a three-ion qubit space. Using a beryllium-magnesium ion crystal as an example system, we theoretically investigate the protocol's performance and the effects of likely error sources, including thermal secular motion of the ion crystal, calibration imperfections, and spontaneous photon scattering. We estimate that a fidelity of $\sim$ 98 % may be achieved in typical trapped ion experiments with $\sim$ 1 ms interaction time. These protocols avoid timescale hierarchies for faster preparation of entangled states

Helping Relationships and Genetic Propensities: A Combinatoric Study of DRD2, Mentoring, and Educational Continuation

Abstract From conception to death, helping relationships promote positive development and enable people to surmount challenges in their lives. Is it the case that the negative consequences of a genetic propensity for risky behaviors can be attenuated by helping relationships (a G × E)? But is it also the case that people with such a genetic propensity are less likely to have helping relationships compared to people without such a propensity (a rGE)? We illustrate this complex pattern of gene–environment interplay by drawing on the National Longitudinal Study of Adolescent Health and a combinatoric analytic strategy. We focus on a gene associated with dopamine receptor type 2 (DRD2 TaqIA), student–mentor relationships, and educational continuation beyond secondary school. Results reveal that, for both white and black males, DRD2 A1+ (A1A1 and A1A2 genotypes) is associated with a decreased likelihood of school continuation compared to their counterparts with DRD2 A1–; mentors who are teachers compensate for this negative association (a G × E); and youth with DRD2 A1+ are less likely to have a mentor who is a teacher than their counterparts with DRD2 A1– (a rGE)