On deflation and multiplicity structure

This paper presents two new constructions related to singular solutions of polynomial systems. The first is a new deflation method for an isolated singular root. This construction uses a single linear differential form defined from the Jacobian matrix of the input, and defines the deflated system by applying this differential form to the original system. The advantages of this new deflation is that it does not introduce new variables and the increase in the number of equations is linear in each iteration instead of the quadratic increase of previous methods. The second construction gives the coefficients of the so-called inverse system or dual basis, which defines the multiplicity structure at the singular root. We present a system of equations in the original variables plus a relatively small number of new variables that completely deflates the root in one step. We show that the isolated simple solutions of this new system correspond to roots of the original system with given multiplicity structure up to a given order. Both constructions are "exact" in that they permit one to treat all conjugate roots simultaneously and can be used in certification procedures for singular roots and their multiplicity structure with respect to an exact rational polynomial system.Comment: arXiv admin note: substantial text overlap with arXiv:1501.0508

The Effect of Fiscal Policy and Corruption Control Mechanisms on Firm Growth and Social Welfare: Theory and Evidence

The paper investigates the conflict that arises between the government, its bureaucrats and businesses in the tax collection process. We examine the effect of fiscal policy and corruption control mechanisms on the prevalence of tax evasion and corruption behaviour, and their impact on firm growth and social welfare. We first model a situation where bureaucrats are homogeneous and have complete negotiating power over the firms with which they interact. We show that in such a situation the government can set an optimal tax rate and put in place a corruption control mechanism involving detection of corrupt bureaucrats within the framework of a no-corruption equilibrium. However, when the public administration is composed of heterogeneous types of bureaucrats with the specific ability to impose red tape costs on firms, we show, like Acemoglu and Verdier (2000), that it might be best for the government to allow a certain level of corruption, given the cost of monitoring activities. We also show that the government could face lose-lose as well as win-win situations in the conduct of its fiscal policies. We then verify the predictions of the model using firm-level data collected from 243 businesses in Uganda. We test the effect of monitoring on bribe and tax payments. We also test the effect of tax rates and corruption control mechanisms on firm growth. We compare the effect of actual corruption (as measured by bribe payments) with the effect of government corruption expectations on firms’ growth.Corruption, Tax evasion, Tax administration, Firm growth

Tensor decomposition and homotopy continuation

A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness sets via numerical elimination theory, we develop computational methods for computing ranks and border ranks of tensors along with decompositions. More generally, we present our approach using joins of any collection of irreducible and nondegenerate projective varieties $X_1,\ldots,X_k\subset\mathbb{P}^N$ defined over $\mathbb{C}$. After computing ranks over $\mathbb{C}$, we also explore computing real ranks. Various examples are included to demonstrate this numerical algebraic geometric approach.Comment: We have added two examples: A Coppersmith-Winograd tensor, Matrix multiplication with zeros. (26 pages, 1 figure

Reaching Out: Advice and Information Networks within a Middle School English Language Development Program

English learners are one of the most rapidly growing student demographics in the United States. However, school systems have historically fallen short in providing English learners with adequate academic support. A review of the literature sheds light on academic tracking as a major factor in restricting access to the rigorous coursework English learners need to achieve at levels commensurate with their English-only speaking peers. Students tracked into the English language development pipeline have difficulty exiting. Those who do not reclassify and persist in this track experience lower levels of high school graduation and college completion. Several recent shifts in educational policy are seeking to address this problem. New language and curriculum standards, as well as detracking practices, have given rise to a more distributed approach to teaching English learners where all teachers have a responsibility to support this population. Through an explanatory sequential mixed methods design, this study examined the social networks a school has built around teaching English learners and how those networks impact the flow of social capital used to support this demographic of students. Research questions include: 1. Who do educators turn to for advice and information regarding the education of English learners? 2. How do social networks shape opportunities for educators to build social capital around teaching English learners? The first phase collected survey data in order to highlight advice and information seeking behaviors. After social network analysis, the results from phase one informed phase two. The second phase included interviews from salient actors to provide further depth into creating a rich description of the ELD networks at the research site and how they impact the English learner experience

The Nucleon Spin Polarizability at Order O(p4{\cal O}(p^4) in Chiral Perturbation Theory

We calculate the forward spin-dependent photon-nucleon Compton amplitude as a function of photon energy at the next-to-leading (${\cal O}(p^4)$) order in chiral perturbation theory, from which we extract the contribution to nucleon spin polarizability. The result shows a large correction to the leading order contribution.Comment: 7 pages, latex, 2 figures included as .eps file

Determinants for Use of Certified Maize Seed and the Relative Importance of Transaction Costs

The rising world prices for major tradable staples such as maize have been a concern for sub- Saharan countries such as Kenya which are maize deficit countries. Maize is a major staple food for over 80 percent of Kenya’s population. Kenya relies on maize for up to 40 percent of its dietary energy supply and is accordingly searching for ways to increase maize productivity. Maize productivity has been rising in the last decade manly as a result of the use of improved germplasm and fertilizer. However, the proportion of farmers using these technologies is low and the aggregate productivity in maize is low compared to other countries and its potential. Previous studies on input adoption have often assumed the existence of perfect supply and product markets, tending to ignore the important but significant role played by institutions as well as the role of transaction costs associated with market exchange. This study makes use of qualitative information from institutions and actors in seed input value chains as well as quantitative information collected from a sample of 150 farmers, in the Moist Transitional Maize Zones of Kenya. A two stage regression model was applied to analyze determinants of adoption and factors affecting degree of adoption of certified improved maize seed. The results show that as farmers adopt certified seeds, they incur higher transaction costs than non adopters, rural infrastructure, social capital such as membership in groups and trust play an important role in the decision of whether or not to use certified seed.Crop Production/Industries,

Improving the efficiency of Bayesian inverse reinforcement learning

Inverse reinforcement learning (IRL) is the task of learning the reward function of a Markov Decision Process (MDP) given knowledge of the transition function and a set of expert demonstrations. While many IRL algorithms exist, Bayesian IRL [1] provides a general and principled method of reward learning by casting the problem in the Bayesian inference framework. However, the algorithm as originally presented suffers from several inefficiencies that prohibit its use for even moderate problem sizes. This paper proposes modifications to the original Bayesian IRL algorithm to improve its efficiency and tractability in situations where the state space is large and the expert demonstrations span only a small portion of it. The key insight is that the inference task should be focused on states that are similar to those encountered by the expert, as opposed to making the naive assumption that the expert demonstrations contain enough information to accurately infer the reward function over the entire state space. A modified algorithm is presented and experimental results show substantially faster convergence while maintaining the solution quality of the original method.United States. Office of Naval Research (Science of Autonomy Program Contract N000140910625)

The Role of State Courts in Constraining Partisan Gerrymandering in Congressional Elections

Federal courts were once seen as the place for partisan gerrymandering challenges to be lodged, but after thirty-plus years of failing to find any redistricting plan to be a partisan gerrymander, even while holding partisan gerrymandering to be justiciable, the Supreme Court announced in Rucho v. Common Cause, 139 S.Ct. 2484, that partisan gerrymandering is not justiciable in federal courts. State courts are now seen as the only place where a remedy for egregious partisan gerrymandering might be sought (except, of course, for taking redistricting out of the hands of the state legislature and moving responsibility into a bipartisan or ostensibly non-partisan commission). Thus, we find that partisan gerrymandering claims, while almost entirely in federal courts in the 2010 and earlier rounds of redistricting, are now brought in state courts. We also expect that state courts would look to state constitutional provisions to evaluate partisan gerrymandering claims, especially language added in recent constitutional amendments that affected the procedures and criteria for redistricting. However, we also see some state courts creatively reevaluating older language in their state’s constitution to find a way to hold egregious gerrymanders in violation of that constitution. Moreover, we see various state court justices relying on a variety of statistical tests proposed by academic specialists, and/or examining the extent to which proposed maps satisfied traditional good government standards. Thus, they are implicitly challenging the Supreme Court’s view in Rucho that no manageable standard for egregious partisan gerrymandering existed