ISOTROPIC MOTIVES

In this article we introduce the local versions of the Voevodsky category of motives with $\mathbb{F} _p$ -coefficients over a field k, parametrized by finitely generated extensions of k. We introduce the so-called flexible fields, passage to which is conservative on motives. We demonstrate that, over flexible fields, the constructed local motivic categories are much simpler than the global one and more reminiscent of a topological counterpart. This provides handy ‘local’ invariants from which one can read motivic information. We compute the local motivic cohomology of a point for $p=2$ and study the local Chow motivic category. We introduce local Chow groups and conjecture that over flexible fields these should coincide with Chow groups modulo numerical equivalence with $\mathbb{F} _p$ -coefficients, which implies that local Chow motives coincide with numerical Chow motives. We prove this conjecture in various cases.

simplePHENOTYPES: SIMulation of Pleiotropic, Linked and Epistatic PHENOTYPES

MotivationAdvances in genotyping and phenotyping techniques have enabled the acquisition of a great amount of data. Consequently, there is an interest in multivariate statistical analyses that identify genomic regions likely to contain causal mutations affecting multiple traits (i.e., pleiotropy). As the demand for multivariate analyses increases, it is imperative that optimal tools are available to compare different implementations of these analyses. To facilitate the testing and validation of these multivariate approaches, we developed simplePHENOTYPES, an R package that simulates pleiotropy, partial pleiotropy, and spurious pleiotropy in a wide range of genetic architectures, including additive, dominance and epistatic models.ResultsWe illustrate simplePHENOTYPES’ ability to simulate thousands of phenotypes in less than one minute. We then provide a vignette illustrating how to simulate a set of correlated traits in simplePHENOTYPES. Finally, we demonstrate the use of results from simplePHENOTYPES in a standard GWAS software, as well as the numerical equivalence of simulated phenotypes from simplePHENOTYPES and other packages with similar capabilities.ConclusionssimplePHENOTYPES is a CRAN package that makes it possible to simulate multiple traits controlled by loci with varying degrees of pleiotropy. Its ability to interface with both commonly-used marker data formats and downstream quantitative genetics software and packages should facilitate a rigorous assessment of both existing and emerging statistical GWAS and GS approaches. simplePHENOTYPES is also available at https://github.coin/sainuelbfernandes/siinplePHENOTYPES.

Fourier–Mukai and autoduality for compactified Jacobians. I

To every singular reduced projective curve X one can associate, following Esteves, many fine compactified Jacobians, depending on the choice of a polarization on X, each of which yields a modular compactification of a disjoint union of the generalized Jacobian of X. We prove that, for a reduced curve with locally planar singularities, the integral (or Fourier–Mukai) transform with kernel the Poincaré sheaf from the derived category of the generalized Jacobian of X to the derived category of any fine compactified Jacobian of X is fully faithful, generalizing a previous result of Arinkin in the case of integral curves. As a consequence, we prove that there is a canonical isomorphism (called autoduality) between the generalized Jacobian of X and the connected component of the identity of the Picard scheme of any fine compactified Jacobian of X and that algebraic equivalence and numerical equivalence of line bundles coincide on any fine compactified Jacobian, generalizing previous results of Arinkin, Esteves, Gagné, Kleiman, Rocha, and Sawon.The paper contains an Appendix in which we explain how our work can be interpreted in view of the Langlands duality for the Higgs bundles as proposed by Donagi–Pantev.

Five Red Apples are Larger than Five Green Apples: The Effect of Object Features on Numerosity Recognition

representation of number is a cornerstone of maturity, by which humans perceive numerical equivalence between various sets of objects. However, it is still unclear how humans perceive and retain the abstract nature of number from concrete numerical stimuli. Two experiments were conducted using a novel memory paradigm to clarify this issue. In Experiment 1, participants were asked to study sets of concrete objects and identify either familiar numerosity or familiar object shape, which were independently manipulated to create congruent and incongruent pairs. Results showed that the numerical cues interfered with object shape recognition and the object shape cues interfered with numerosity recognition. However, the magnitude of interference on numerosity recognition was larger than that on object recognition. These results suggest that individuals tend to integrate all of the visual properties present in the stimulus and use them to process numerical quantity even when the integrated input is not required for a given task. In Experiment 2, the instructions were the same as in Experiment 1 except that the participants were offered an attention-biased training session. The results suggest that independent processing of numerosity tends to be utilized only with cued attention later in development. Therefore, we concluded that unlike symbolic number processing, non-symbolic numbers are processed in a non-abstract manner, which may change later depending on observers’ expertise.

On Heckits, LATE, and Numerical Equivalence

Structural econometric methods are often criticized for being sensitive to functional form assumptions. We study parametric estimators of the local average treatment effect (LATE) derived from a widely used class of latent threshold crossing models and show they yield LATE estimates algebraically equivalent to the instrumental variables (IV) estimator. Our leading example is Heckman's (1979) two‐step (“Heckit”) control function estimator which, with two‐sided non‐compliance, can be used to compute estimates of a variety of causal parameters. Equivalence with IV is established for a semiparametric family of control function estimators and shown to hold at interior solutions for a class of maximum likelihood estimators. Our results suggest differences between structural and IV estimates often stem from disagreements about the target parameter rather than from functional form assumptions per se. In cases where equivalence fails, reporting structural estimates of LATE alongside IV provides a simple means of assessing the credibility of structural extrapolation exercises.

SOME NEW EXAMPLES OF SMASH-NILPOTENT ALGEBRAIC CYCLES

Voevodsky has conjectured that numerical equivalence and smash-equivalence coincide for algebraic cycles on any smooth projective variety. Building on work of Vial and Kahn–Sebastian, we give some new examples of varieties where Voevodsky's conjecture is verified.