Rational Equivalence Research Articles

Tautological rings on Jacobian varieties of curves with automorphisms

Let $J$ be the Jacobian of a smooth projective complex curve $C$ which admits non-trivial automorphisms, and let $A(J)$ be the ring of algebraic cycles on $J$ with rational coefficients modulo algebraic equivalence. We present new tautological rings in $A(J)$ which extend in a natural way the tautological ring studied by Beauville (Compos Math 140(3):683-688, 2004). We then show there exist tautological rings induced on special complementary abelian subvarieties of $J$.

Diagnostic causal reasoning with verbal information

In diagnostic causal reasoning, the goal is to infer the probability of causes from one or multiple observed effects. Typically, studies investigating such tasks provide subjects with precise quantitative information regarding the strength of the relations between causes and effects or sample data from which the relevant quantities can be learned. By contrast, we sought to examine people’s inferences when causal information is communicated through qualitative, rather vague verbal expressions (e.g., “X occasionally causes A”). We conducted three experiments using a sequential diagnostic inference task, where multiple pieces of evidence were obtained one after the other. Quantitative predictions of different probabilistic models were derived using the numerical equivalents of the verbal terms, taken from an unrelated study with different subjects. We present a novel Bayesian model that allows for incorporating the temporal weighting of information in sequential diagnostic reasoning, which can be used to model both primacy and recency effects. On the basis of 19,848 judgments from 292 subjects, we found a remarkably close correspondence between the diagnostic inferences made by subjects who received only verbal information and those of a matched control group to whom information was presented numerically. Whether information was conveyed through verbal terms or numerical estimates, diagnostic judgments closely resembled the posterior probabilities entailed by the causes’ prior probabilities and the effects’ likelihoods. We observed interindividual differences regarding the temporal weighting of evidence in sequential diagnostic reasoning. Our work provides pathways for investigating judgment and decision making with verbal information within a computational modeling framework.

Fourier–Mukai and autoduality for compactified Jacobians. I

Abstract 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.

SOME NEW EXAMPLES OF SMASH-NILPOTENT ALGEBRAIC CYCLES

AbstractVoevodsky 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.

A comparative study on two different approaches of bulk viscosity in the Polyakov–Nambu–Jona-Lasinio model

We have gone through a comparative study on two different kinds of bulk viscosity expressions by using a common dynamical model. The Polyakov–Nambu–Jona-Lasinio (PNJL) model in the realm of mean-field approximation, including up to eight quark interactions for 2+1 flavor quark matter, is treated for this common dynamics. We have probed the numerical equivalence as well as discrepancy of two different expressions for bulk viscosity at vanishing quark chemical potential. Our estimation of bulk viscosity to entropy density ratio follows a decreasing trend with temperature, which is observed in most of the earlier investigations. We have also extended our estimation for finite values of quark chemical potential.

Multislice Analysis of Blood Flow Values in CT Perfusion Studies of Lung Cancer.

Objectives. Tumour heterogeneity represents a key issue in CT perfusion (CTp), where all studies are usually based on global mean or median values of perfusion maps, often computed on whole tumour. We sought to determine whether, and to what extent, such global values can be representative of tumour heterogeneity, with respect to single slices, and could be used for therapy assessment. Materials and Methods. Twelve patients with one primary non-small cell lung cancer lesion were enrolled in this study, for a total amount of 26 CTp examinations and 118 slices. Mean and median blood flow (BF) values, calculated voxel-based, were computed on each slice and the whole tumour. To measure functional heterogeneity, entropy was calculated on BF values as well. Results. Most of the slices were not represented by the global BF values computed on the whole tumour. In addition, there are a number of lesions having equivalent global BF values, but they are composed of slices having very different heterogeneity distributions, that is, entropy values. Conclusions. Global mean/median BF values of the single slices separately should be considered for clinical assessment, only if interpreted through entropy computed on BF values. The numerical equivalence between global BF values of different lesions may correspond to different clinical status, thus inducing possible errors in choice of therapy when considering global values only.

The Research on network Losses Allocation of Power Market based on Improved REI Network Numerical Equivalence

The Research on network Losses Allocation of Power Market based on Improved REI Network Numerical Equivalence

Characterizing volume via cone duality

For divisors over smooth projective varieties, we show that the volume can be characterized by the duality between the pseudo-effective cone of divisors and the movable cone of curves. Inspired by this result, we define and study a natural intersection-theoretic volume functional for 1-cycles over compact Kahler manifolds. In particular, for numerical equivalence classes of curves over projective varieties, it is closely related to the mobility functional studied by Lehmann.

Volume-type functions for numerical cycle classes

A numerical equivalence class of k-cycles is said to be big if it lies in the interior of the closed cone generated by effective classes. We construct analogues for arbitrary cycle classes of the volume function for divisors which distinguishes big classes from boundary classes.

Adjoint groups over ℚ_{𝕡}(𝕏) and R-equivalence - revisited

We obtain a class of examples of non-rational adjoint classical groups of type 2 A n ^2A_n and a group of type 2 D 3 ^2D_3 over the function field F F of a smooth geometrically integral curve over a p p -adic field with p ≠ 2 p \neq 2 . We also show that for any group of type C n C_n over F F , the group of rational equivalence classes of G G over F F is trivial, i.e., G ( F ) / R = ( 1 ) G(F)/R=(1) .

Birational motives, I: pure birational motives

This is a considerably expanded version of the "pure" part of our 2002 preprint. We define a category of pure birational motives over a field, depending on the choice of an adequate equivalence relation on algebraic cycles. It is obtained by "killing" the Lefschetz motive in the corresponding category of effective motives. For rational equivalence, it encompasses Bloch's decomposition of the diagonal. We study the induced Chow-K\"unneth decompositions in this category, and establish relationships with Rost's cycle modules and the Albanese functor for smooth projective varieties.

Where next for the reproducibility agenda in computational biology?

BackgroundThe concept of reproducibility is a foundation of the scientific method. With the arrival of fast and powerful computers over the last few decades, there has been an explosion of results based on complex computational analyses and simulations. The reproducibility of these results has been addressed mainly in terms of exact replicability or numerical equivalence, ignoring the wider issue of the reproducibility of conclusions through equivalent, extended or alternative methods.ResultsWe use case studies from our own research experience to illustrate how concepts of reproducibility might be applied in computational biology. Several fields have developed ‘minimum information’ checklists to support the full reporting of computational simulations, analyses and results, and standardised data formats and model description languages can facilitate the use of multiple systems to address the same research question. We note the importance of defining the key features of a result to be reproduced, and the expected agreement between original and subsequent results. Dynamic, updatable tools for publishing methods and results are becoming increasingly common, but sometimes come at the cost of clear communication. In general, the reproducibility of computational research is improving but would benefit from additional resources and incentives.ConclusionsWe conclude with a series of linked recommendations for improving reproducibility in computational biology through communication, policy, education and research practice. More reproducible research will lead to higher quality conclusions, deeper understanding and more valuable knowledge.

Local positivity in terms of Newton–Okounkov bodies

In recent years, the study of Newton–Okounkov bodies on normal varieties has become a central subject in the asymptotic theory of linear series, after its introduction by Lazarsfeld–Mustaţă and Kaveh–Khovanskii. One reason for this is that they encode all numerical equivalence information of divisor classes (by work of Jow). At the same time, they can be seen as local positivity invariants, and Küronya–Lozovanu have studied them in depth from this point of view.We determine what information is encoded by the set of all Newton–Okounkov bodies of a big divisor with respect to flags centered at a fixed point of a surface, by showing that it determines and is determined by the numerical equivalence class of the divisor up to negative components in the Zariski decomposition that do not go through the fixed point.

On Rational Equivalence in Tropical Geometry

AbstractThis article discusses the concept of rational equivalence in tropical geometry (and replaces an older, imperfect version). We give the basic definitions in the context of tropical varieties without boundary points and prove some basic properties. We then compute the “bounded” Chow groups of Rn by showing that they are isomorphic to the group of fan cycles. The main step in the proof is of independent interest. We show that every tropical cycle in Rn is a sum of (translated) fan cycles. This also proves that the intersection ring of tropical cycles is generated in codimension 1 (by hypersurfaces).

Assimilation of transformed retrievals obtained from clear‐sky IASI measurements

A recent study, which provided theoretical justifications to show that the assimilation of radiances and retrievals can be equivalent when some key requirements are met – namely when remote‐sounding data are related to model variable perturbations in an approximately linear way and when the same prior constraints are used to solve the inverse problem – has brought new insight into the problem of how to assimilate satellite measurements in numerical weather prediction. In this article, we explore this new theoretical framework in practical terms and investigate the viability of assimilating transformed retrievals from hyperspectral infrared sounders such as the Infrared Atmospheric Sounding Interferometer (IASI) in an optimal way. First, the dependence of a measure of information content of temperature‐ and humidity‐sensitive observations from IASI in clear‐sky conditions on the forecast (or ‘background’) uncertainty used to constrain the retrievals is assessed. Subsequently, numerical comparisons between profiles obtained from direct radiance assimilation and those obtained from assimilating appropriately transformed retrievals are performed, for different atmospheric conditions. By making use of the new concept of degrees‐of‐freedom (DFS) weighting function, it is shown that the background error covariance matrix plays a crucial role in determining the conditions for equivalence between assimilation of radiances and transformed retrievals. The numerical comparisons confirm the theoretical results stating that the temperature and humidity profiles obtained from both the direct assimilation of radiances and the assimilation of transformed retrievals are equal within the analysis error as long as the same prior constraints are used. Finally, when the prior differs, it is shown how the characteristics of background error matrices are crucial to guarantee the numerical equivalence between the two methods; the best is when the background matrix utilised to calculate the transformed retrievals is characterised by larger variances than the one utilised in the assimilation.

Demonstration of numerical equivalence of ensemble and spectral averaging in electromagnetic scattering by random particulate media.

The numerically exact superposition T-matrix method is used to model far-field electromagnetic scattering by two types of particulate object. Object 1 is a fixed configuration that consists of N identical spherical particles (with N=200 or 400) quasi-randomly populating a spherical volume V having a median size parameter of 50. Object 2 is a true discrete random medium (DRM) comprising the same number N of particles randomly moving throughout V. The median particle size parameter is fixed at 4. We show that if Object 1 is illuminated by a quasi-monochromatic parallel beam then it generates a typical speckle pattern having no resemblance to the scattering pattern generated by Object 2. However, if Object 1 is illuminated by a parallel polychromatic beam with a 10% bandwidth then it generates a scattering pattern that is largely devoid of speckles and closely reproduces the quasi-monochromatic pattern generated by Object 2. This result serves to illustrate the capacity of the concept of electromagnetic scattering by a DRM to encompass fixed quasi-random particulate samples provided that they are illuminated by polychromatic light.

Corporate Crocodile Tears? On the Reactive Attitudes of Corporate Agents

Recently, a number of people have argued that certain entities embodied by groups of agents themselves qualify as agents, with their own (analogs of) beliefs, desires, and intentions; even, some cl ...

$$\mathbb {A}^{1}$$ A 1 -connectivity on Chow monoids versus rational equivalence of algebraic cycles

Let k be a field of characteristic zero, and let X be a projective variety embedded into a projective space over k. For two natural numbers r and d let $$C_{r,d}(X)$$ be the Chow scheme parametrizing effective cycles of dimension r and degree d on the variety X. Choosing an r-cycle of minimal degree gives rise to a chain of embeddings of Chow schemes, whose colimit is the connective Chow monoid $${C}_r^{\infty }(X)$$ of r-cycles on X. Let be the classifying motivic space of this monoid. In the paper we establish an isomorphism between the Chow group of degree 0 dimension r algebraic cycles modulo rational equivalence on X, and the group of sections of the Nisnevich sheaf of $${\mathbb {A}^{1}}$$ -path connected components of the loop space of at $${{\mathrm{Spec}}(k)}$$ . Equivalently, is isomorphic to the group of sections of the stabilized motivic fundamental group at $${{\mathrm{Spec}}(k)}$$ .

General equilibrium models with Morishima elasticities of substitution in production

Analytical general equilibrium (AGE) models are important tools that economists use to answer questions about theory and policy. When a production function has three or more inputs, the traditional modeling technique employs Allen elasticities of substitution to represent general functional forms. This paper builds an analytical general equilibrium model using the Morishima elasticity of substitution (MES). Specifically, an existing model using Allen elasticities is reformulated to employ the MES and the new, closed-form solutions are interpreted with additional insights from the reformulation. Importantly, the special case of constant elasticity of substitution (CES) production follows directly when using Morishima elasticities, but not Allen elasticities. This paper also provides a general technique for switching from Allen to Morishima elasticities in any existing AGE model and demonstrates a one-to-one numerical equivalence regardless of the elasticity measure employed. Replicating prior results, plausible parameter values are applied to the reformulated model to analyze the source-side, distributional effects of a pollution tax and highlight how the Allen and Morishima elasticities differ.

Numerical Equivalences among Krylov Subspace Algorithms for Skew-Symmetric Matrices

We briefly review Krylov subspace methods based on the Galerkin and minimum residual conditions for solving $Ax=b$ with real $A$ and $b$, followed by two implementations: the conjugate gradient (CG) based methods CGNE and CGNR. We then show the numerical equivalence of Lanczos tridiagonalization and Golub--Kahan bidiagonalization for any real skew-symmetric matrix $A$. We give short derivations of two algorithms for solving $Ax=b$ with skew-symmetric $A$ and use the above equivalence to show that these are numerically equivalent to the Golub--Kahan bidiagonalization variants of CGNE and CGNR. These last two numerical equivalences add to the theoretical equivalences in the work by Eisenstat [Equivalence of Krylov Subspace Methods for Skew-Symmetric Linear Systems, Department of Computer Science, Yale University, preprint, arXiv:1512.00311, 2015] that unified and extended earlier work. We next present a method based on the Lanczos tridiagonalization process for minimizing $\|A^T (b-Ax_k)\|_2$ when $A^T\! =-A$ and show that for skew-symmetric systems it is numerically equivalent to LSMR developed by Fong and Saunders [SIAM J. Sci. Comput., 33 (2011), pp. 2950--2971]. Finally, we illustrate the typical convergence behaviors of these algorithms with a numerical example and use these and an analysis to give new insights into algorithm choices for general large sparse matrix solution of equations problems.