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Root-theoretic Young diagrams and Schubert calculus II
Dominic Searles
Mathematics , 2013,
Abstract: We continue the study of root-theoretic Young diagrams (RYDs) from [Searles-Yong '13]. We provide an RYD formula for the $GL_n$ Belkale-Kumar product, after [Knutson-Purbhoo '11], and we give a translation of the indexing set of [Buch-Kresch-Tamvakis '09] for Schubert varieties of non-maximal isotropic Grassmannians into RYDs. We then use this translation to prove that the RYD formulas of [Searles-Yong '13] for Schubert calculus of the classical (co)adjoint varieties agree with the Pieri rules of [Buch-Kresch-Tamvakis '09], which were needed in the proofs of the (co)adjoint formulas.
Noncoherent initial ideals in exterior algebras
Dominic Searles,Arkadii Slinko
Mathematics , 2014,
Abstract: We construct a noncoherent initial ideal of an ideal in the exterior algebra of order 6, answering a question of D. Maclagan (2000). We also give a method for constructing noncoherent initial ideals in exterior algebras using certain noncoherent term orders.
Root-theoretic Young diagrams and Schubert calculus: planarity and the adjoint varieties
Dominic Searles,Alexander Yong
Mathematics , 2013,
Abstract: We study root-theoretic Young diagrams to investigate the existence of a Lie-type uniform and nonnegative combinatorial rule for Schubert calculus. We provide formulas for (co)adjoint varieties of classical Lie type. This is a simplest case after the (co)minuscule family (where a rule has been proved by H.Thomas and the second author using work of R.Proctor). Our results build on earlier Pieri-type rules of P.Pragacz-J.Ratajski and of A.Buch-A.Kresch-H.Tamvakis. Specifically, our formula for OG(2,2n) is the first complete rule for a case where diagrams are non-planar. Yet the formulas possess both uniform and non-uniform features. Using these classical type rules, as well as results of P.-E.Chaput-N.Perrin in the exceptional types, we suggest a connection between polytopality of the set of nonzero Schubert structure constants and planarity of the diagrams. This is an addition to work of A.Klyachko and A.Knutson-T.Tao on the Grassmannian and of K.Purbhoo-F.Sottile on cominuscule varieties, where the diagrams are always planar.
Deformed Cohomology of Flag Varieties
Oliver Pechenik,Dominic Searles
Mathematics , 2014,
Abstract: This paper introduces a two-parameter deformation of the cohomology of generalized flag varieties. One special case is the Belkale-Kumar deformation (used to study eigencones of Lie groups). Another picks out intersections of Schubert varieties that behave nicely under projections. Our construction yields a new proof that the Belkale-Kumar product is well-defined. This proof is shorter and more elementary than earlier proofs.
Isoconfigurational thermostat
Igor Rychkov,Debra J. Searles
Physics , 2007,
Abstract: A holonomic constraint is used to enforce a constant instantaneous configurational temperature on an equilibrium system. Three sets of equations of motion are obtained, differing according to the way in which the holonomic constraint is introduced and the phase space distribution function that is preserved by the dynamics. Firstly, Gauss' principle of least constraint is used, and it is shown that it does not preserve the canonical distribution. Secondly, a modified Hamiltonian is used to find a dynamics that provides a restricted microcanonical distribution. Lastly, we provide equations that are specifically designed to both satisfy the temperature constraint and produce a restricted canonical ensemble.
A local fluctuation theorem for large systems
Guillaume Michel,Debra J. Searles
Physics , 2012, DOI: 10.1103/PhysRevLett.110.260602
Abstract: The fluctuation theorem characterizes the distribution of the dissipation in nonequilibrium systems and proves that the average dissipation will be positive. For a large system with no external source of fluctuation, fluctuations in properties will become unobservable and details of the fluctuation theorem are unable to be explored. In this letter, we consider such a situation and show how a fluctuation theorem can be obtained for a small open subsystem within the large system. We find that a correction term has to be added to the large system fluctuation theorem due to correlation of the subsystem with the surroundings. Its analytic expression can be derived provided some general assumptions are fulfilled, and its relevance it checked using numerical simulations.
A Method to Compute Multiplicity Corrected Confidence Intervals for Odds Ratios and Other Relative Effect Estimates
Jimmy Thomas Efird,Susan Searles Nielsen
International Journal of Environmental Research and Public Health , 2008, DOI: 10.3390/ijerph5050394
Abstract: Epidemiological studies commonly test multiple null hypotheses. In some situations it may be appropriate to account for multiplicity using statistical methodology rather than simply interpreting results with greater caution as the number of comparisons increases. Given the one-to-one relationship that exists between confidence intervals and hypothesis tests, we derive a method based upon the Hochberg step-up procedure to obtain multiplicity corrected confidence intervals (CI) for odds ratios (OR) and by analogy for other relative effect estimates. In contrast to previously published methods that explicitly assume knowledge of P values, this method only requires that relative effect estimates and corresponding CI be known for each comparison to obtain multiplicity corrected CI.
A Method to Model Season of Birth as a Surrogate Environmental Risk Factor for Disease
Jimmy Thomas Efird,Susan Searles Nielsen
International Journal of Environmental Research and Public Health , 2008, DOI: 10.3390/ijerph5010049
Abstract: Environmental exposures, including some that vary seasonally, may play a role in the development of many types of childhood diseases such as cancer. Those observed in children are unique in that the relevant period of exposure is inherently limited or perhaps even specific to a very short window during prenatal development or early infancy. As such, researchers have investigated whether specific childhood cancers are associated with season of birth. Typically a basic method for analysis has been used, for example categorization of births into one of four seasons, followed by simple comparisons between categories such as via logistic regression, to obtain odds ratios (ORs), confidence intervals (CIs) and p-values. In this paper we present an alternative method, based upon an iterative trigonometric logistic regression model used to analyze the cyclic nature of birth dates related to disease occurrence. Disease birth-date results are presented using a sinusoidal graph with a peak date of relative risk and a single p-value that tests whether an overall seasonal association is present. An OR and CI comparing children born in the 3-month period around the peak to the symmetrically opposite 3-month period also can be obtained. Advantages of this derivative-free method include ease of use, increased statistical power to detect associations, and the ability to avoid potentially arbitrary, subjective demarcation of seasons.
A Method to Model Season of Birth as a Surrogate Environmental Risk Factor for Disease
Jimmy Thomas Efird,Susan Searles Nielsen
International Journal of Environmental Research and Public Health , 2008,
Abstract: Environmental exposures, including some that vary seasonally, may play a role in the development of many types of childhood diseases such as cancer. Those observed in children are unique in that the relevant period of exposure is inherently limited or perhaps even specific to a very short window during prenatal development or early infancy. As such, researchers have investigated whether specific childhood cancers are associated with season of birth. Typically a basic method for analysis has been used, for example categorization of births into one of four seasons, followed by simple comparisons between categories such as via logistic regression, to obtain odds ratios (ORs), confidence intervals (CIs) and p-values. In this paper we present an alternative method, based upon an iterative trigonometric logistic regression model used to analyze the cyclic nature of birth dates related to disease occurrence. Disease birth-date results are presented using a sinusoidal graph with a peak date of relative risk and a single p-value that tests whether an overall seasonal association is present. An OR and CI comparing children born in the 3-month period around the peak to the symmetrically opposite 3-month period also can be obtained. Advantages of this derivative-free method include ease of use, increased statistical power to detect associations, and the ability to avoid potentially arbitrary, subjective demarcation of seasons.
The Fluctuation Theorem for Stochastic Systems
Debra J. Searles,Denis J. Evans
Physics , 1999, DOI: 10.1103/PhysRevE.60.159
Abstract: The Fluctuation Theorem describes the probability ratio of observing trajectories that satisfy or violate the second law of thermodynamics. It has been proved in a number of different ways for thermostatted deterministic nonequilibrium systems. In the present paper we show that the Fluctuation Theorem is also valid for a class of stochastic nonequilibrium systems. The Theorem is therefore not reliant on the reversibility or the determinism of the underlying dynamics. Numerical tests verify the theoretical result.
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