We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures. We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion. Then, we analyze some particular theories of interest (mostly various theories of fields of positive characteristic) and show that in all the cases considered the PAC property (...) is first order. (shrink)

We introduce and study the notions of a PAC-substructure of a stable structure, and a bounded substructure of an arbitrary substructure, generalizing [10]. We give precise definitions and equivalences, saying what it means for properties such as PAC to be first order, study some examples (such as differentially closed fields) in detail, relate the material to generic automorphisms, and generalize a "descent theorem" for pseudo-algebraically closed fields to the stable context. We also point out that the elementary invariants of pseudo-algebraically (...) closed fields from [6] are also valid for pseudo-differentially closed fields. (shrink)

The essays that follow grew out of a workshop held at the Center for Buddhist Studies, University of California, Berkeley, in March 2018, on the topic of conceptuality and non-conceptuality in Buddhist philosophy. Discussions at the workshop focused specifically on the tenability of the claim made by the two Buddhist epistemologists Dignāga and Dharmakīrti that perceptual cognitions are non-conceptual and yet also contribute to the contents of conceptual thought. The four contributions collected here present just a few of the resulting (...) reflections.This essay explores the Dharmakīrtian1 notion of a "concept" in the specific context of a "pseudo-perception", that is, cognitive events that... (shrink)

The Probably Approximately Correct (PAC) learning is a machine learning model introduced by Leslie Valiant in 1984. The PACi reducibility refers to the PAC reducibility independent of size and computation time. This reducibility in PAC learning resembles the reducibility in Turing computability. The ordering of concept classes under PAC reducibility is nonlinear, even when restricted to particular concrete examples.Due to the resemblance to Turing Reducibility, we suspected that there could be incomparable PACi and PAC degrees for the PACi and PAC (...) reducibilities as in Turing incomparable degrees. In 1957 Friedberg and in 1956 Muchnik independently solved the Post problem by constructing computably enumerable sets A and B of incomparable degrees using the priority construction method. We adapt this idea to PACi and PAC reducibilities and construct two effective concept classes C and D such that C is not reducible to D and vice versa. When considering PAC reducibility it was necessary to work on the size of an effective concept class, thus we use Kolmogorov complexity to obtain the size. The non-learnability of concept classes in the PAC learning model is explained by the existence of PAC incomparable degrees.Analogous to the Turing jump, we give a jump operation on effective concept classes for the zero jump. To define the zero jump operator for PACi degrees the join of all the effective concept classes is constructed and proved that it is a greatest element. There are many properties proven for existing degrees. Thus we can explore proving those properties to PACi and PAC degrees. But if we prove an embedding from those degrees to PACi and PAC degrees then those properties will be true for PACi and PAC degrees without explicitly proving them.Abstract prepared by Dodamgodage Gihnee M. Senadheera and taken directly from the thesisE-mail: [email protected]: https://www.proquest.com/docview/2717762461/abstract/ACD19F29A8774AF6PQ/1?accountid=13864. (shrink)

We compute that the index set of PAC-learnable concept classes is m-complete Σ30\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma^{0}_{3}}$$\end{document} within the set of indices for all concept classes of a reasonable form. All concept classes considered are computable enumerations of computable Π10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^{0}_{1}}$$\end{document} classes, in a sense made precise here. This family of concept classes is sufficient to cover all standard examples, and also has the property that PAC learnability (...) is equivalent to finite VC dimension. (shrink)

Computer scientists have provided a distinct justification of Occam’s Razor. Using the probably approximately correct framework, they provide a theorem that they claim demonstrates that we should favor simpler hypotheses. The argument relies on a philosophical interpretation of the theorem. I argue that the standard interpretation of the result in the literature is misguided and that a better reading does not, in fact, support Occam’s Razor at all. To this end, I state and prove a very similar theorem that, if (...) interpreted the same way, would justify the contradictory Anti-Occam’s Razor—the principle that we should favor more complex hypotheses. (shrink)

We generalize a well-known theorem binding the elementary equivalence relation on the level of PAC fields and the isomorphism type of their absolute Galois groups. Our results concern two cases: saturated PAC structures and nonsaturated PAC structures.

Denef and Loeser defined a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group. In addition, we define some maps between different Grothendieck rings of definable sets which provide additional information, not contained in the associated motive. In particular we infer that the map of Denef-Loeser is not injective.

Le présent article prend pour point de départ de sa réflexion sur les enjeux épistémologiques et méthodologiques des petits corpus un constat simple, à savoir que ceux-ci ont survécu à l’heure où l’essor de la linguistique de corpus et le développement du big data ont multiplié les ressources. Cette survie est, selon nous, la preuve d’une nécessité à la fois matérielle et logistique et d’une pertinence scientifique des petits corpus que nous explorons sous l’angle particulier de la langue orale et (...) du domaine spécifique de la phonologie. Nous montrons en quoi cette discipline entretient un rapport unique à l’objet langue et donc privilégié aux corpus et repensons la validité des corpus non pas en termes de taille, mais en termes de finalité. Nous défendons alors la méthodologie et les ambitions du programme PAC dont la base de données sur l’anglais oral contemporain repose exclusivement sur de petits corpus spécialisés constitués dans les différentes aires géographiques anglophones. Nous démontrons non seulement la pertinence individuelle de ces petits corpus pour l’étude quantitative et qualitative de phénomènes précis, comme le nivellement de la variété mancunienne dans le nord de l’Angleterre ; mais également l’apport concret de la comparabilité et de la jonction possible des analyses faites à partir de ces petits corpus pour la modélisation théorique des phénomènes de rhoticité et de r de sandhi dans les variétés d’anglais parlées à travers le monde. Ainsi, nous illustrons la manière dont les résultats obtenus grâce à l’étude en profondeur des petits corpus PAC permettent de penser la dynamique des systèmes de l’anglais ainsi que le poids sociolinguistique des identités, notamment nord-sud, dans la description et la compréhension de leurs caractéristiques phonético-phonologiques. Aussi, en défendant les petits corpus, nous défendons en creux la phonologie de corpus, et nous remettons en cause la supériorité automatique des grands corpus pour ce qui est de l’analyse en profondeur de la langue orale et de l’évolution des variétés d’anglais parlées à travers le monde. (shrink)

We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding absolute Galois groups of structures, so they can be interpreted in some monster model with an additional predicate. Third, we prove the “Weak Independence Theorem” for pseudo-algebraically closed (PAC) substructures of an ambient structure with no finite cover property (nfcp) and the property [Formula: see text]. Fourth, we describe Kim-dividing in (...) these PAC substructures and show several results related to the SOPnhierarchy. Fifth, we characterize the algebraic closure in PAC structures. (shrink)

BackgroundAcute myocardial infarction (AMI) research is challenging as it requires enrollment of acutely ill patients. Patients are generally in a suboptimal state for providing informed consent. Patients’ understanding to verbal assents have not been previously examined in AMI research. Patients Acceptance and Comprehension to Written and Verbal Consent (PAC–VC) compared patients’ understanding and attitudes to verbal and written consents in AMI RCTs.MethodsPAC–VC recruited patients from 3 AMI trials using both verbal N = 12 and written N = 6 consents. We (...) compared patients’ understanding using two survey questionnaires. The first questionnaire used open-ended questions with multiple choice answers. The second questionnaire used a 5-point Likert scale to measure patients understanding and attitudes to the consent process. Overall answers average scores were categorized into three groups: Adequate understanding (71–100) %, Partial understanding (41–70)% and Inadequate understanding (0–40)%.ResultsResponses showed patients with verbal assent had adequate understanding to most components of informed consent, close to those of written consent. Most patients did not read written information entirely and believed that it is not important to make a final decision. Patients favoured to have written information be part of the consent but not necessarily presented during the initial consent process. Patients felt less pressured in the verbal assent arm than those of written consent.ConclusionPatients had adequate understanding to most components of verbal assent and comparable to those of written consent. Utilizing verbal assents in the acute care setting should be further assessed in larger trials. (shrink)

"The Marriage Translation" argues that a Benjaminian account of translation can shed new light on the ways in which contemporary practices of queer alliance transform, and are transformed by, legal and political discourse. In a constellation of readings—of arguments for the French pacs, of Benjamin's essays on translation, and on Goethe's Elective Affinities, as well as of two photos of the illegal weddings celebrated in San Francisco and in Bègles, France—the article emphasizes the ways social contexts are sometimes reinvigorated and (...) at other times ignored by the demand for certain forms of legal recognition. (shrink)

Building on philosophy of science literature and two original studies, this paper argues for the necessity of incorporating all three portions of Wood''s (1991) theoretical model of corporate social performance (CSP) into its measurement. It begins by describing the two studies of an organizational phenomenon not commonly studied – internal fund drives to employees. Insights from these studies of corporate PAC and United Way campaigns are then used to illustrate how important it is to incorporate all three portions of Wood''s (...) model into the measurement of CSP to prevent drawing faulty conclusions. The paper concludes by providing a matrix for use in testing the validity of CSP measures. (shrink)

We produce a model of \ such that no outer model of \ has the same pure sets, answering a question asked privately by Eric Hall.

We prove that the theory of all sentences in the language of rings which are true in for almost all is decidable. Here is the field of all algebraic numbers; is the ring of all algebraic integers; is the absolute Galois group of ; for each , is the fixed field of σ1,…,σe in ; and the clause ‘almost all’ is used in the sense of the Haar measure of.

Bounded PAC substructures of models of stable theory T are generalizations of bounded PAC fields and bounded PAC beautiful pairs generalize Poizat's beautiful pairs. Both notions were introduced in the authors Ph.D. thesis. In this paper, we prove that under the assumption that the PAC property is first order for T, the theory of any bounded PAC structure is simple. Moreover, if the PAC property is first order for T and T does not have the finite cover property, then the (...) theory of any bounded PAC beautiful pair is simple. We, also, give a characterization of dividing in both cases. (shrink)

Offering wisdom for everyone, along with colorful original illustrations, arcade icon Pac-Man shares a collection of his most famous inspirational and feel-good quotes.

In this paper, we investigate a new model theoretical tree property (TP), called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and [Formula: see text]-ATP, and provide a criterion for theories to have not ATP (being NATP). Using these combinatorial observations, we find algebraic examples (...) of ATP and NATP, including pure groups, pure fields and valued fields. More precisely, we prove Mekler’s construction for groups, Chatzidakis’ style criterion for pseudo-algebraically closed (PAC) fields, and the AKE-style principle for valued fields preserving NATP. We give a construction of an antichain tree in the Skolem arithmetic and atomless Boolean algebras. (shrink)

Identifying biomarkers of developing mental disorder is crucial to improving early identification and treatment—a key strategy for reducing the burden of mental disorders. Cross-frequency coupling between two different frequencies of neural oscillations is one such promising measure, believed to reflect synchronization between local and global networks in the brain. Specifically, in adults phase–amplitude coupling (PAC) has been shown to be involved in a range of cognitive processes, including working and long-term memory, attention, language, and fluid intelligence. Evidence suggests that increased (...) PAC mediates both temporary and lasting improvements in working memory elicited by transcranial direct-current stimulation and reductions in depressive symptoms after transcranial magnetic stimulation. Moreover, research has shown that abnormal patterns of PAC are associated with depression and schizophrenia in adults. PAC is believed to be closely related to cortico-cortico white matter (WM) microstructure, which is well established in the literature as a structural mechanism underlying mental health. Some cognitive findings have been replicated in adolescents and abnormal patterns of PAC have also been linked to ADHD in young people. However, currently most research has focused on cross-sectional adult samples. Whereas initial hypotheses suggested that PAC was a state-based measure due to an early focus on cognitive, task-based research, current evidence suggests that PAC has both state-based and stable components. Future longitudinal research focusing on PAC throughout adolescent development could further our understanding of the relationship between mental health and cognition and facilitate the development of new methods for the identification and treatment of youth mental health. (shrink)

We study computable PAC (CPAC) learning as introduced by Agarwal et al. (2020). First, we consider the main open question of finding characterizations of proper and improper CPAC learning. We give a characterization of a closely related notion of strong CPAC learning, and provide a negative answer to the COLT open problem posed by Agarwal et al. (2021) whether all decidably representable VC classes are improperly CPAC learnable. Second, we consider undecidability of (computable) PAC learnability. We give a simple general (...) argument to exhibit such ndecidability, and initiate a study of the arithmetical complexity of learnability. We briefly discuss the relation to the undecidability result of Ben-David et al. (2019), that motivated the work of Agarwal et al. (shrink)

ABSTRACT Post-abortion care (PAC) research is increasingly being conducted in low- and middle-income countries (LMICs) to help reduce the high burden of unsafe abortion. This study aims to help address the evidence gap about ethical challenges that researchers in LMICs face when carrying out PAC research with adolescents. Employing an explorative qualitative approach, the study identified several ethics challenges encountered by PAC researchers in Zambia, including those associated with seeking ethics and regulatory approvals at institutional and national levels. Persistent stigma (...) around abortion and community perceptions that PAC studies encourage adolescents to seek abortion affected adolescents’ right to exercise their autonomy and to make decisions as well as exposed adolescents to social stigmatization risks. Challenges with recruitment was reported to result in abandoning of studies, thereby undermining development of PAC services that are more responsive to adolescent needs. Training needs identified included knowledge of best practices for conducting and disseminating PAC research. Strategies for addressing the ethical challenges included trust building and using less value-laden terminology when seeking permission and consent. It is essential to the future of PAC research in Zambia and globally that these important challenges be addressed through the development of comprehensive ethics guidance. (shrink)

We present a quantum model for the motion of N point particles, implying nonlocal (i.e., superluminal) influences of external fields on the trajectories, that is nonetheless fully relativistic. In contrast to other models that have been proposed, this one involves no additional space-time structure as would be provided by a (possibly dynamical) foliation of space-time. This is achieved through the interplay of opposite microcausal and macrocausal (i.e., thermodynamic) arrows of time. PACS numbers 03.65.Ud; 03.65.Ta; 03.30.+p..

Peace and conflict studies education has grown significantly in the last 30 years, mainly in Higher Education. This article critically analyzes the ways in which this field might be subject to poststructural critique, and posits Bourdieusian second-order reflexivity as a means of responding to these critiques. We propose here that theory-building within PACS education is often limited by the dominance of Galtung and Freire, and that, while the foundational ideas of positive and negative peace, structural and cultural violence, conscientization, reflexivity (...) and critical pedagogy are still relevant today, they nevertheless need to be combined in new ways with each other, and with Bourdieu’s notions of habitus and field, to adequately respond to poststructural critique. Thus, we call here for greater field-based reflexivity in twenty-first century PACS. (shrink)

Recent years have yielded many discussions on how to endow autonomous agents with the ability to make ethical decisions, and the need for explicit ethical reasoning and transparency is a persistent theme in this literature. We present a modular and transparent approach to equip autonomous agents with the ability to comply with ethical prescriptions, while still enacting pre-learned optimal behaviour. Our approach relies on a normative supervisor module, that integrates a theorem prover for defeasible deontic logic within the control loop (...) of a reinforcement learning agent. The supervisor operates as both an event recorder and an on-the-fly compliance checker w.r.t. an external norm base. We successfully evaluated our approach with several tests using variations of the game Pac-Man, subject to a variety of “ethical” constraints. (shrink)

Some opponents of abortion claim that fetuses are persons from the moment of conception. Following Berg (2017), let us call these individuals “Personhood-At-Conception” (or PAC), opponents of abortion. Berg argues that if fetuses are persons from the moment of conception, then miscarriage kills far more people than abortion. As such, PAC opponents of abortion face the following dilemma: They must “immediately” and “substantially” shift their attention, resources, etc., toward preventing miscarriage or they must admit that they do not actually believe (...) that fetuses are persons from the moment of conception (or, at least, they must recognize that they are not acting in ways that are consistent with this belief). In this essay, I show that Berg’s argument fails at each step. Specifically, after outlining her argument (in section 1), I consider the central claim of Berg’s style of argument: That “miscarriage…is much deadlier than abortion.” In section 2, I argue that this claim is false (when taken literally) and misleading otherwise. In section 3, I show that Berg’s style of argument is identical in structure to a criticism that is sometimes levied against the recent “Black Lives Matter” movement. In the latter context, the argument has been vehemently rejected. I argue that Berg’s style of argument should be rejected for the same reasons. Finally, in section 4, I show that Berg’s suggestion that opponents of abortion should divert “substantial” amounts of money and attention to miscarriage prevention faces two problems. First, these claims are made without any evidence regarding the actual priorities and spending habits of PAC opponents of abortion. If we are told “not enough is being done,” one wonders: What is being done and how much counts as “enough”? Berg gives no answer to either question. Second, even if it is true that opponents of abortion do not pay substantial attention to miscarriage prevention, Berg fails to notice that this may be for good reason. I conclude that PAC opponents of abortion do not face the dilemma that Berg presents. (shrink)

We study structures equipped with generic predicates and/or automorphisms, and show that in many cases we obtain simple theories. We also show that a bounded PAC field is simple. 1998 Published by Elsevier Science B.V. All rights reserved.

It is widely agreed that play and games contribute to the good life. One might naturally wonder how games in particular so contribute? Granted, games can be very good, what exactly is so good about them when they are good? Although a natural starting point, this question is perhaps naive. Games come in all shapes and sizes, and different games are often good in very different ways. Chess, Bridge, Bingo, Chutes and Ladders, Football, Spin the Bottle, Dungeons & Dragons, Pac-Man, (...) Minecraft and Charades are all games, and can all contribute to a good life, but each will characteristically enrich someone’s life in its own distinctive way. Some games facilitate socializing and sociability, other games improve physical fitness, some develop a sense of fair play and reciprocity, while others enhance concentration and analytic skills. Asking ‘what is good about games?’ and assuming a simple answer is as naïve as asking ‘what is good about fiction?’ or ‘what is good about sex?’ However, a less naïve question and philosophically interesting question is not hard to formulate. Plausibly, much of the heterogeneity of the value of games stems from the different kinds of instrumental value of different games. Perhaps we should therefore ask in what ways the activity of playing games is characteristically good for its own sake. Unfortunately, the philosophical literature on the non-instrumental value of playing games is sparse. One of the few sustained treatments of the topic can be found in an underappreciated exchange between Thomas Hurka and John Tasioulas. Interestingly, despite taking different views of what it is to play a game, they both make room for the non-instrumental value of play and achievement in game play and they both argue that these two goods stand in important an important explanatory relation. However, they take diametrically opposed views as to which of these good is more basic. Roughly, on Hurka’s view, the good of achievement is more basic, and it is because of the non-instrumental value of achievement that what Hurka calls “playing in a game,” which involves playing, is itself non-instrumentally good because of the non-instrumental value of achievement. The idea is that if something is non-instrumentally good then loving that thing is also non-instrumentally good, and that playing in a game involves loving the activity for its own sake. In this way, the value of achievement in a game grounds the value of playing in a game. Tasioulas takes exactly the opposite approach. He argues that there must be something independently good about playing a game which grounds the value of achievement in that game. On his view, the typical grounding good or “framing value” of games is play itself – what I am here calling “playing.” In this essay, I raise some objections to both Hurka’s view and Tasioulas’s view and develop a positive alternative conception of the non-instrumental value of games. I argue that while each view is insightful in its own way, neither gets things exactly right. For a start, the way in which they characterize the key concepts is problematic. Hurka’s definition of ‘play a game’, which he lifts from Suits, is problematic, while both of their characterizations of play are problematic for reasons I rehearse here. They also take an unduly narrow view of the possible role of achievement in game play, both effectively conflating achievement with “excellence.” I argue, by contrast, that there is an important sense in which even someone who is not very good at the game by any objective standard can still reap the goods of achievements in games. At the same time, those who play games typically do not do so for the sake of achievements as such, though they may do something which is in a sense tantamount to this, or so I shall argue. Finally, I argue that neither Hurka’s “achievement first” order of explanation nor Tasioulas’s “play first” order of explanation is fully correct. I argue for what I call a “variable priority” view. On this view, the value of play sometimes grounds the value of achievement in a game, while in other cases the independently grounded value of achievement in a game provides a further ground for the value of play, though even in that case play is independently good for its own sake. I begin by laying out Hurka’s and Tasioulas’s views. (shrink)

We define and investigate a family of local–global principles for fields involving both orderings and p-valuations. This family contains the PAC, PRC and PpC fields and exhausts the class of pseudo classically closed fields. We show that the fields satisfying such a local–global principle form an elementary class, admit diophantine definitions of holomorphy domains, and their orderings satisfy the strong approximation property.

In this paper, we perform a worst-case analysis of rule discovery based on generality and accuracy. A rule is defined as a probabilistic constraint of true assignment to the class attribute for corresponding examples. In data mining, a rule can be considered as representing an important class of discovered patterns. We accomplish the aforementioned objective by extending a preliminary version of PAC learning, which represents a worst-case analysis for classification. Our analysis consists of two cases: the case in which we (...) try to avoid finding a bad rule, and the case in which we try to avoid overlooking a good rule. Discussions on related work are also provided for PAC learning, multiple comparison, analysis of association rule discovery, and simultaneous reliability evaluation of a discovered rule. (shrink)

The analysis of Political Action Committee activities is dominated by the perspective that PAC contributions are a rational investment in political candidates; they yield valuable short-term payoffs. PACs buy access to officeholders and their votes on important legislation. Despite broad acceptance of this morally suspect theory, the evidence upon which it is based is weak. An alternative perspective — what we call the principled approach — both fits the evidence and rejects the morally repugnant interpretation of the relationship between business (...) and politics that permeates the rational investment perspective. (shrink)

Machiavelli advised us that people are so mean, small and selfish that they will only act under necessity, so the successful prince must force the population, through whatever means necessary, to follow his dictates. This book traces the influence of the Florentine thinker on American politics, from the Founders (c. 1770s) through today's rough-and-tumble political panorama. Machiavelli's ideas have been re-interpreted internationally as 'real-politik.' He proposed that the 'ends justify the means,' and that any manner of fraud, violence or corruption (...) must be utilized in attaining and retaining power. He maintained that the most powerful form of fraud was the appearance of religiosity and said that the successful prince must hold no art higher than that of war. In this disturbing, erudite and highly readable book, America is shown to be a surprising example of Machiavellian politics, utilizing all of the post-modern methods of information distribution and "legal" fraud and corruption. Lee Atwater, Karl Rove, George W. Bush, the Supreme Court's 'Citizens United v. the Federal Election Commission' (2010) and the Super PACs it spawned, the massive amounts of money ("power's master key"), the intermingling of the language of religion and war, and the 90% negative advertising of the 2012 Presidential campaign (channeling Machiavelli's dictum that the adversary must be "assassinated," though in contemporary America by character assassination) and even Barack Obama's Machiavellian machinations are looked at in light of the Renaissance political philosopher's ideas. The last section of the book offers a response to this with a specific, implementable program that will begin to devolve the power of American democracy back to the people. (shrink)

The objectives of this study were to develop a high-density chromosome bin map of homoeologous group 7 in hexaploid wheat, to identify gene distribution in these chromosomes, and to perform comparative studies of wheat with rice and barley. We mapped 2148 loci from 919 EST clones onto group 7 chromosomes of wheat. In the majority of cases the numbers of loci were significantly lower in the centromeric regions and tended to increase in the distal regions. The level of duplicated loci (...) in this group was 24% with most of these loci being localized toward the distal regions. One hundred nineteen EST probes that hybridized to three fragments and mapped to the three group 7 chromosomes were designated landmark probes and were used to construct a consensus homoeologous group 7 map. An additional 49 probes that mapped to 7AS, 7DS, and the ancestral translocated segment involving 7BS also were designated landmarks. Landmark probe orders and comparative maps of wheat, rice, and barley were produced on the basis of corresponding rice BAC/PAC and genetic markers that mapped on chromosomes 6 and 8 of rice. Identification of landmark ESTs and development of consensus maps may provide a framework of conserved coding regions predating the evolution of wheat genomes. (shrink)