Finite Resolutions Research Articles

Impact of atmospheric small‐scale fluctuations on climate sensitivity

Climate change scenarios are based on numerical models with finite spatial and temporal resolutions. The impact of unresolved processes is parameterized without taking the variability induced by subscale processes into account. This drawback could lead to an over‐/underestimation of the climate sensitivity. The aim of this study is to investigate the impact of small‐scale atmospheric fluctuations on the modeled climate sensitivity to increased CO2 concentration. Using a complex coupled atmosphere‐ocean general circulation model (ECHAM5/MPI‐OM) climate response experiments with enhanced small‐scale fluctuations are performed. Our results show that the strength of the global warming due to a CO2 doubling depends on the representation of small‐scale fluctuations. Reducing the horizontal diffusion by a factor of 3 leads to an increase of the equilibrium climate sensitivity by 13%. If white noise is added to the small scales, the climate sensitivity tends to weaken. The largest changes in responses occur in the upper troposphere.

Bias‐free Measurement of Giant Molecular Cloud Properties

ABSTRACTWe review methods for measuring the sizes, line widths, and luminosities of giant molecular clouds (GMCs) in molecular‐line data cubes with low resolution and sensitivity. We find that moment methods are robust and sensitive, making full use of both position and intensity information, and we recommend a standard method to measure the position angle, major and minor axis sizes, line width, and luminosity using moment methods. Without corrections for the effects of beam convolution and sensitivity to GMC properties, the resulting properties may be severely biased. This is particularly true for extragalactic observations, where resolution and sensitivity effects often bias measured values by 40% or more. We correct for finite spatial and spectral resolutions with a simple deconvolution, and we correct for sensitivity biases by extrapolating properties of a GMC to those we would expect to measure with perfect sensitivity (i.e., the 0 K isosurface). The resulting method recovers the properties of a GMC to within 10% over a large range of resolutions and sensitivities, provided the clouds are marginally resolved with a peak signal‐to‐noise ratio greater than 10. We note that interferometers systematically underestimate cloud properties, particularly the flux from a cloud. The degree of bias depends on the sensitivity of the observations and the (u,v) coverage of the observations. In an Appendix to the paper we present a conservative, new decomposition algorithm for identifying GMCs in molecular‐line observations. This algorithm treats the data in physical rather than observational units (i.e., parsecs rather than beams or arcseconds), does not produce spurious clouds in the presence of noise, and is sensitive to a range of morphologies. As a result, the output of this decomposition should be directly comparable among disparate data sets.

Higgs pair production at a linear e+e- collider in models with large extra dimensions

In this paper, we will derive the cross section formula for the Higgs pair production at a linear $e^+e^-$ collider in models with large extra dimensions and study the feasibility of its measurement through realistic Monte Carlo simulations. Since the process has essentially no Standard Model background, once produced, it will provide us with a very clean signature of physics beyond the Standard Model. Moreover, since the final state particles are spinless, the spin 2 of the intermediate virtual KK gravitons has to be conserved by the orbital angular momentum of the Higgs pair. This results in a very characteristic angular distribution of the final states. Taking into account finite detector acceptance and resolutions as well as initial state radiation and beamstrahlung, we demonstrate in this paper that, given a sufficiently high center of mass energy, the angular distribution of the Higgs pair is indeed measurable at the linear collider and will allow us to prove the spin 2 nature of the KK gravitons exchanged in the s-channel.

The free cover of a row contraction

We establish the existence and uniqueness of finite free resolutions - and their attendant Betti numbers - for graded commuting d -tuples of Hilbert space operators. Our approach is based on the notion of free cover of a (perhaps noncommutative) row contraction. Free covers provide a flexible replacement for minimal dilations that is better suited for higher-dimensional operator theory.

Stiffness of finite free resolutions and the canonical element conjecture

Sur un anneau local noethérien, les résolutions libres minimales finies ont une certaine propriété de rigidité lorsque la caractéristique de l'anneau égale la caractéristique de son corps résiduel (théorème 1). Cette propriété évoque le critère de Buchsbaum et Eisenbud où il n'est pas question de caractéristique ; cependant, personne ne sait si rigidité reste valable en caractéristique mixte. Conjecturons seulement ceci : sur tout anneau local de Gorenstein, d'égale caractéristique ou non, toutes les résolutions libres minimales finies sont rigides (au sens de la définition 1). L'objet de cette note est de montrer, au théorème 2, que la conjecture précédente, qui est en fait une conjecture d'algèbre linéaire pour une classe restreinte d'anneaux, équivaut a la conjecture de l'élément canonique de Hochster qui, elle, porte sur toute la classe des anneaux locaux noethériens. Ceci est fait via l'approximation par les modules de Cohen–Macaulay maximaux et les enveloppes de dimension injective finie. Cette théorie, due à Auslander et Buchweitz, s'applique notamment aux modules de type fini sur un anneau de Gorenstein.

Reed-Muller Codes on Complete Intersections

By using results and techniques from commutative algebra such as the vanishing ideal of a set of points, its a-invariant, the Hilbert polynomial and series, as well as finite free resolutions and the canonical module, some results about Reed-Muller codes defined on a zero-dimensional complete intersection in the n-projective dimensional space are given. Several examples of this class of codes are presented in order to illustrate the ideas.

Coexistent observables and effects in quantum mechanics

A pair of coexistent observables which can be expressed as a function of a third observable is characterised in terms of positive operator bimeasures and joint measures. A characterisation of sets of coexistent observables in terms of consistent families of finite (nonorthogonal) resolutions of the identity is worked out

Computing minimal finite free resolutions

In this paper we address the basic problem of computing minimal finite free resolutions of homogeneous submodules of graded free modules over polynomial rings. We develop a strategy, which keeps the resolution minimal at every step. Among the relevant benefits is a marked saving of time, as the first reported experiments in CoCoA show. The algorithm has been optimized using a variety of techniques, such as minimizing the number of critical pairs and employing an “ad hoc” Hilbert-driven strategy. The algorithm can also take advantage of various a priori pieces of information, such as the knowledge of the Castelnuovo regularity.

Applications of Koszul homology to numbers of generators and syzygies

Several spectral sequence techniques are used in order to derive information about the structure of finite free resolutions of graded modules. These results cover estimates of the minimal number of generators of defining ideals of projective varieties. In fact there are generalizations of a classical result of Dubreil. On the other hand there are investigations about the shifts and the dimension of Betti numbers. To this end there is a local analogue of Green's considerations developed in [5].

Information Flow in Chaotic Symbolic Dynamics for Finite and Infinitesimal Resolution

We introduce an information-theory-based concept for the characterization of the information flow in chaotic systems in the framework of symbolic dynamics for finite and infinitesimal measurement resolutions. The information flow characterizes the loss of information about the initial conditions, i.e. the decay of statistical correlations (i.e. nonlinear and non-Gaussian) between the entire past and a point p steps into the future as a function of p. In the case where the partition generating the symbolic dynamics is finite, the information loss is measured by the mutual information that measures the statistical correlations between the entire past and a point p steps into the future. When the partition used is a generator and only one step ahead is observed (p = 1), our definition includes the Kolmogorov–Sinai entropy concept. The profiles in p of the mutual information describe the short- and long-range forecasting possibilities for the given partition resolution. For chaos it is more relevant to study the information loss for the case of infinitesimal partitions which characterizes the intrinsic behavior of the dynamics on an extremely fine scale. Due to the divergence of the mutual information for infinitesimal partitions, the "intrinsic" information flow is characterized by the conditional entropy which generalizes the Kolmogorov–Sinai entropy for the case of observing the uncertainty more than one step into the future. The intrinsic information flow offers an instrument for characterizing deterministic chaos by the transmission of information from the past to the future.

Cartan decompositions and BGG-resolutions

Necessary and sufficient criteria are given for the existence of BGG-resolutions (finite resolutions of modules by finite direct sums of Weyl modules) for simple modules over quasi-hereditary algebras, which have strong exact Borel subalgebras and strong Δ-subalgebras. Our main technical tool is the existence of Cartan decompositions for these algebras. The results apply to simple objects in the BGG-categoryO of a finite-dimensional semisimple complex Lie algebra and to finite dimensional simple rational modules over simply connected semisimple algebraic groups.

Unfolding of true distributions from experimental data distorted by detectors with finite resolutions

A new procedure for unfolding the true distribution from experimental data distorted by a detector is proposed. For the given detector a result can be found by the least squares method, hence without bias and involving minimal statistical errors. Stability of the result is achieved at the expense of its information content and/or using additional information on the shape of the distributions to be measured. The method may be applied for detectors with linear or nonlinear distortions.

Finite free resolutions of length two and koszul generalized complex

Finite free resolutions of length two and koszul generalized complex

Invariant factors of an endomorphism and finite free resolutions

Polynomial invariants are associated to an endomorphism u of a module M that has a finite free resolution. The first invariant χ( u, X) (characteristic polynomial of u) is a monic polynomial of degree the Euler characteristic of M. Its construction is based on the MacRae invariant. When M is a finite free module, χ( u, X) is the classical characteristic polynomial of u. With additional assumptions there is constructed a finite sequence of monic polynomials { d i ( u, X)} i ⩾ 0 such that their product is χ( u, X) and d i ( u, X) divides d i + 1 ( u, X). When R is a field, these polynomials are the invariant factors of u. A generalized Cayley-Hamilton theorem is given. The generic behavior of the polynomials χ( u, X) and { d i ( u, X)} i ⩾ 0 in Spec R is proved. Finally it is shown, under certain assumptions that there exists a free submodule F of M, invariant with respect to u, such that the restriction of u to F is similar to the endomorphism of F defined by the diagonal block matrix where the ith block is the companion matrix of d i ( u, X).

Integrated circuits for the S560 undersea transmission system : M. Campeseet al. Elect. Commun.65, 188 (1992)

The sensitivity of optical orthogonal-frequency division multiplexing (OFDM) transmission to the finite resolutions of the digital-to-analog (DAC) and analog-to-digital converters (ADC) has been analyzed using numerical simulations. We show that for back-to-back configuration the requirements regarding the number of quantization bits for the DACs are similar to the ones of the ADCs. After transmission over metro and long-haul distances a higher resolution of the DAC/ADC is required compared to back-to-back configuration. We show that after transmission it is sufficient to enhance either the ADC resolution or the DAC resolution by 1 bit.

On the structure of free resolutions of length 3

The sources of structure theory of finite free resolutions go back to Hilbert. In his famous paper on invariant theory he proved that every ideal of projective dimension 1 over a polynomial ring is determinantal. This theorem was then extended by Burch to arbitrary commutative rings. There are various motivations for extending this results. One of them is a variety of applications of the Hilbert-Burch theorem in deformation theory and algebraic geometry (cf., for example, [S]). Another reason comes from Serre’s conjecture. One would like to understand in terms of resolutions why the lifting property of Grothendieck does not hold. In 1974 Buchsbaum and Eisenbud in [BEl] introduced so-called structure theorems-various factorisations which occur for acyclic complexes of arbitrary length. Then Hochster in [H] generalized the Hilbert-Burch theorem to arbitrary free complexes of length 2. His results were then improved in [Br, PW]. Still, the structure of complexes of length bigger than 2 is not understood. The first steps in this direction are the results of [B] and the construction of a generic ring for complexes of length 3 of type (1,rz 1, 1) given in [PW]. In this paper I construct a ring which, I believe, is the generic ring for complexes of length 3. The construction involves new structure theorems and a new idea-the notion of defects of structure theorems. The paper is organized as follows. Section 1 is introductory. I give there the background from representation theory and the cohomology groups of a Lie algebra. Section 2 contains the proof of the new structure theorems we need, the construction of the generic ring R,, and of the defect Lie algebra L. In Section 3 I show that genericity of R,, follows from vanising of the homology of a certain family of complexes over U(L)-the enveloping algebra of L. In Section 4 I apply the results of Section 2 to study multi-

Algebra structures on some canonical resolutions

The study of algebra structures on finite free resolutions of cyclic modules begins with Buchsbaum and Eisenbud [B-El]. The classes of cyclic modules whose minimal resolutions are known to admit an algebra structure include the residue field [Gu], complete intersections (Koszul complex), modules of homological dimension atmost 3 [B-El], Gorenstein of codimension four [K-M], and Herzog algebras [K-M23. Avramov [Al] gave examples to show that there are cyclic modules whose minimal resolutions do not admit an algebra structure. Let R be a noetherian local ring with maximal ideal m. In this paper we construct algebra structures on the minimal resolutions of two classes of cyclic modules R/I, namely, when Z is of the form Jk, where J is an ideal generated by a regular sequence, and when Z is the ideal of maximal minors of a generic n x m matrix, provided R contains the rationals. In [Al] Avramov defined certain obstructions to the existance of an algebra structure on the minimal resolution of a module and then produced modules with non-zero obstructions. The general question is whether the vanishing of these obstructions is also sufficient for the existence of a minimal algebra resolution of a cyclic module. When Z is an ideal generated by a regular sequence in R and M= R/Z” for any positive integer k, then these obstructions are all zero. Therefore, Avramov and Schlessinger asked whether the minimal resolutions of R/Z’ admit an algebra structure. Corollary 3.6 of this paper provides an affirmative answer to this question.

A remark on the structure maps for finite free resolutions.

A remark on the structure maps for finite free resolutions.

Nonlinear interactions among internal gravity waves

This paper reviews the nonlinear interaction calculations for the internal gravity wave field in the deep ocean. The nonlinear interactions are a principal part of the dynamics of internal waves and are an important link in the overall energy cascade from large to small scales. Four approaches have been taken for their analysis: the evaluation of the transfer integral describing weakly and resonantly interacting waves, the application of closure hypotheses from turbulence theories to more strongly interacting waves, the integration of the eikonal or ray equations describing the propagation of small‐scale internal waves in a background of large‐scale internal waves, and the direct numerical simulation of the basic hydrodynamic equations of motion. The weak resonant interaction calculations have provided most of the conventional wisdom. Specific interaction processes and their role in shaping the internal wave spectrum have been unveiled and a comprehensive inertial range theory developed. The range of validity of the resonant interaction approximation, however, is not known and must be seriously doubted for high‐wave number, high‐frequency waves. The turbulence closure calculations and the direct numerical modeling are not yet in a state to be directly applicable to the oceanic internal wave field. The closure models are too complex and rest on conjectures that are not demonstrably justified. Numerical modeling can treat strongly interacting waves and buoyant turbulence, but is severely limited by finite computer resolutions. Extensive suites of experiments have only been carried out for two‐dimensional flows. The eikonal calculations provide an efficient and versatile tool to study the interaction of small‐scale internal waves, but it is not clear to what extent the scale‐separated interactions with larger‐scale internal waves compete with and might be overwhelmed by interactions among like scales. The major shortcoming of all four approaches is that they neglect the interaction with the vortical (=potential vorticity carrying) mode of motion that must be expected to exist in addition to internal waves at small scales. This interaction is intrinsically neglected in all Lagrangian‐based studies and in the non‐rotating two‐dimensional simulations. The most promising approach for the future that can handle both arbitrarily strong interactions and the interaction with the vortical mode is numerical modeling once the resolution problem is overcome.

Finite resolutions of modules for reductive algebraic groups

of a suitable G&-module M by modules which are required to be direct sums of tensor products of exterior powers of the natural representation. In particular, in [Z, 31, for partitions 1 and p, resolutions are constructed for the modules L,(F) and L,,,(F) (the Schur functor corresponding to 1 and the skew Schur functor corresponding to (A, p), evaluated at F), where F is a free module over a field or the integers. The purpose of this paper is to characterise the GL,-modules which admit such a resolution as those modules which have a filtration in which each successive quotient has the form L,(F) for some partition p. By contrast with [2,3] we do not produce resolutions explicitly. Our methods are independent of those of Akin and Buchsbaum (we give a new proof of their result on the existence of a resolution for L,(F)) and are algebraic group theoretic in nature. We have therefore cast our main result (the theorem of Section 1) as a statement about resolutions for reductive algebraic groups over an algebraically closed field. One may frequently wish to work over a more general coefficient ring (see [l-3]) and so we treat in some detail Chevalley group schemes and general linear group schemes over a principal ideal domain (2.6 and 2.7), giving the appropriate versions of our main result and relating it to the work of Akin, Buchsbaum and Weyman.