Refined upper bounds for the linear Diophantine problem of Frobenius

We study the Frobenius problem: given relatively prime positive integers a_1,...,a_d, find the largest value of t (the Frobenius number g(a_1,...,a_d)) such that m_1 a_1 + ... m_d a_d = t has no solution in nonnegative integers m_1,...,m_d. We introduce a method to compute upper bounds for g(a_1,a_2,a_3), which seem to grow considerably slower than previously known bounds. Our computations are based on a formula for the restricted partition function, which involves Dedekind-Rademacher sums, and the reciprocity law for these sums.Comment: 12 pages, 5 figure

Compound Poisson process with a Poisson subordinator

A compound Poisson process whose randomized time is an independent Poisson process is called compound Poisson process with Poisson subordinator. We provide its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps and normal jumps. Then for the iterated Poisson process we discuss some properties and provide convergence results to a Poisson process. The first-crossing-time problem for the iterated Poisson process is finally tackled in the cases of (i) a decreasing and constant boundary, where we provide some closed-form results, and (ii) a linearly increasing boundary, where we propose an iterative procedure to compute the first-crossing-time density and survival functions.Comment: 16 pages, 7 figure

A Standing Question: Mortgages, Assignment and Foreclosure

Banks are neither private attorneys general nor bounty hunters, armed with a roving commission to seek out defaulting homeowners and take away their homes in satisfaction of some other bank\u27s deed of trust. This Article examines the judicial treatment of mortgage assignments across various jurisdictions in the foreclosure context. Although some courts do permit debtors to challenge suspicious or problematic assignments, most have ignored such problems and denied standing to debtors attempting to assert assignment-based defenses. This is particularly surprising given the widespread and well-documented problems with foreclosure robo-litigation, including backdated documents, fraudulent notarizations, and unauthorized signatures. Despite the abuse of process by foreclosing entities, courts have permitted foreclosures to continue unabated and, in some instances, have even precluded the possibility of discovery to debtors seeking to ensure that title and assignments are legally valid. Judicial ambivalence about formal compliance by mortgage assignors and assignees in the foreclosure context is somewhat ironic given most courts\u27 routine enforcement of instruments against debtors who do not formally comply with all contractual terms. Current adjudicative approaches to mortgage assignment are seemingly disconnected from the devastating reality of the home mortgage crisis and its causes. Moreover, there are several rationales that would support a more robust enforcement of technical compliance with assignment procedures, including the need for procedural equity, title certainty, and public records integrity. Thus, as evidence exists that banks are still making many of the same problematic mistakes regarding transfer documentation, courts can perform an essential monitoring role as an important spur towards reform. Although it would not address all of the underlying causes of the housing crisis, an adjudicative approach that liberally permits challenges to mortgage assignments would encourage lenders and servicers to be more circumspect in their foreclosure processes