Descent Equations Research Articles

Algebraic study of chiral anomalies†

The algebraic structure of chiral anomalies ismade globally valid on non-trivial bundles by the introduction of a fixed background connection. Some of the techniques used in the study of the anomaly are improved or generalized, including a systematic way of generating towers of ‘descent equations’.

MODIFIED PARTITION FUNCTIONS, CONSISTENT ANOMALIES AND CONSISTENT SCHWINGER TERMS

A gauge invariant partition function is defined for gauge theories which leads to the standard quantization. It is shown that the descent equations and consequently the consistent anomalies and Schwinger terms can be extracted from this modified partition function naturally.

Formula omitted]-invariant decomposition of ten-dimensional Yang–Mills supersymmetry

The N=1, d=10 super Yang–Mills action is constructed in a twisted form, using SU(5) invariant decomposition of spinors in 10 dimensions. The action and its off-shell closed twisted scalar supersymmetry operator Q is derived from a Chern–Simons term. The action can be decomposed as the sum of a term in the cohomology of Q and of a term that is Q-exact. The first term is a fermionic Chern–Simons term for a twisted component of the Majorana–Weyl gluino and it is related to the second one by a twisted vector supersymmetry with 5 parameters. The cohomology of Q and some topological observables are defined from descent equations. In this SU(5)⊂SO(10) invariant decomposition, the N=1, d=10 theory is determined by only 6 supersymmetry generators, as in the twisted N=4, d=4 theory. There is a superspace with 6 twisted fermionic directions, with solvable constraints.

A comparison principle for a Sobolev gradient semi-flow

We consider gradient descent equations for energy functionals of the type $S(u) = \frac{1}{2} _{L^2} + \int_{\Omega} V(x,u) dx$, where $A$ is a uniformly elliptic operator of order 2, with smooth coefficients. The gradient descent equation for such a functional depends on the metric under consideration. We consider the steepest descent equation for $S$ where the gradient is an element of the Sobolev space $H^{\beta}$, $\beta \in (0,1)$, with a metric that depends on $A$ and a positive number $\gamma >$sup$|V_{2 2}|$. We prove a weak comparison principle for such a gradient flow. We extend our methods to the case where $A$ is a fractional power of an elliptic operator, and provide an application to the Aubry-Mather theory for partial differential equations and pseudo-differential equations by finding plane-like minimizers of the energy functional.

Effective Level Set Image Segmentation With a Kernel Induced Data Term

This study investigates level set multiphase image segmentation by kernel mapping and piecewise constant modeling of the image data thereof. A kernel function maps implicitly the original data into data of a higher dimension so that the piecewise constant model becomes applicable. This leads to a flexible and effective alternative to complex modeling of the image data. The method uses an active curve objective functional with two terms: an original term which evaluates the deviation of the mapped image data within each segmentation region from the piecewise constant model and a classic length regularization term for smooth region boundaries. Functional minimization is carried out by iterations of two consecutive steps: 1) minimization with respect to the segmentation by curve evolution via Euler-Lagrange descent equations and 2) minimization with respect to the regions parameters via fixed point iterations. Using a common kernel function, this step amounts to a mean shift parameter update. We verified the effectiveness of the method by a quantitative and comparative performance evaluation over a large number of experiments on synthetic images, as well as experiments with a variety of real images such as medical, satellite, and natural images, as well as motion maps.

From massive gravity to modified general relativity

Massive gravity which has been constructed from a cohomological formulation of gauge invariance by means of the descent equations is here investigated in the classical limit. Gauge invariance requires a vector-graviton field v coupled to the massive tensor field h μν . In the limit of vanishing graviton mass the v-field does not decouple. On the classical level this leads to a modification of general relativity. The contribution of the v-field to the energy-momentum tensor can be interpreted as dark matter density and pressure. We solve the modified field equations in the simplest spherically symmetric geometry.

Massive gravity from descent equations

Both massless and massive gravity are derived from descent equations (Wess–Zumino consistency conditions). The massive theory is a continuous deformation of the massless one.

Supersymmetric Adler–Bardeen anomaly in [formula omitted] super-Yang–Mills theories

We provide a study of the supersymmetric Adler–Bardeen anomaly in the N = 1 , d = 4 , 6 , 10 super-Yang–Mills theories. We work in the component formalism that includes shadow fields, for which Slavnov–Taylor identities can be independently set for both gauge invariance and supersymmetry. We find a method with improved descent equations for getting the solutions of the consistency conditions of both Slavnov–Taylor identities and finding the local field polynomials for the standard Adler–Bardeen anomaly and its supersymmetric counterpart. We give the explicit solution for the ten-dimensional case.

A new approach to dynamic self-tuning of database buffers

Current businesses rely heavily on efficient access to their databases. Manual tuning of these database systems by performance experts is increasingly infeasible: For small companies, hiring an expert may be too expensive; for large enterprises, even an expert may not fully understand the interaction between a large system and its multiple changing workloads. This trend has led major vendors to offer tools that automatically and dynamically tune a database system. Many database tuning knobs concern the buffer pool for caching data and disk pages. Specifically, these knobs control the buffer allocation and thus the cache miss probability, which has direct impact on performance. Previous methods for automatic buffer tuning are based on simulation, black-box control, gradient descent, and empirical equations. This article presents a new approach, using calculations with an analytically-derived equation that relates miss probability to buffer allocation; this equation fits four buffer replacement policies, as well as twelve datasets from mainframes running commercial databases in large corporations. The equation identifies a buffer-size limit that is useful for buffer tuning and powering down idle buffers. It can also replace simulation in predicting I/O costs. Experiments with PostgreSQL illustrate how the equation can help optimize online buffer partitioning, ensure fairness in buffer reclamation, and dynamically retune the allocation when workloads change. It is also used, in conjunction with DB2's interface for retrieving miss data, for tuning DB2 buffer allocation to achieve targets for differentiated service.

On torsional observables in topological 4D gravity

We construct, in topological 4D gravity with torsion, a set of observables involving the topological invariant given by integrating the Nieh–Yan (NY) 4-form. The method relies as usual on the determination of a hierarchy of generalized descent equations based on the BRST algebra. The latter is derived by using a superconnection for a SO(5) gauge group. This and its associated supercurvature lead to the fields occurring in the quantized theory by exploiting an appropriate embedding of SO(4) into SO(5).

Unsupervised Variational Image Segmentation/Classification Using a Weibull Observation Model

Studies have shown that the Weibull distribution can model accurately a wide variety of images. Its parameters index a family of distributions which includes the exponential and approximations of the Gaussian and the Raleigh models widely used in image segmentation. This study investigates the Weibull distribution in unsupervised image segmentation and classification by a variational method. The data term of the segmentation functional measures the conformity of the image intensity in each region to a Weibull distribution whose parameters are determined jointly with the segmentation. Minimization of the functional is implemented by active curves via level sets and consists of iterations of two consecutive steps: curve evolution via Euler-Lagrange descent equations and evaluation of the Weibull distribution parameters. Experiments with synthetic and real images are described which verify the validity of method and its implementation.

Affine and projective active contour models

We investigate the evolution of active contours in terms of progressive modification of an initial contour following the chosen Lie group of object-to-image transformations. Because of non-fronto-parallel viewing of an object or due to relative motion between the camera and the object, the resultant image may undergo affine or projective object-to-image transformations. In a recent paper we have shown that in the case of object tracking, frame-to-frame deformations of an initial curve obtained through Euler–Lagrange descent equations of a curve functional can be used to enact a desired Lie group of plane transformations [A.-R. Mansouri, D.P. Mukherjee, S.T. Acton, Constraining active contour evolution via Lie groups of transformation, IEEE Trans. Image Process. 13 (2004) 853–863]. In this work, we propose an energy functional that encodes the Lie group transformation parameters, which in turn guide shape distortion due to oblique viewing. Additional constraints, such as transformation smoothness, are imposed on the active contour by modifying the energy functional. The functional is minimized using numerical schemes similar to the conjugate gradient technique, and the convergence properties are discussed. The success of the technique for affine and projective scenes is demonstrated with both synthetic and real image examples and compared with the related approaches.

Visual reconstruction of ground plane obstacles in a sparse view robot environment

The purpose of this study is to investigate a geometric/level set method to locate ground plane objects in a robot environment and reconstruct their structure from a collection of sparse views. In a first step, a model of the ground plane surface, on which the robot is operating, is obtained through the matching of the available views. This wide-baseline matching of the ground plane views allows also to compute camera pose information associated with each of these views. Based on the information obtained, reconstruction of the obstacles proceeds by minimizing an energy functional containing three terms: a term of shape-from-silhouettes consistency to characterize the ground plane objects structure and to account for possibly non-textured object surfaces; a term of visual information consistency to measure the conformity of the objects surface visual information to the acquired images; and finally, a term of regularization to bias the solution toward smooth object surfaces. The functional is minimized following the associated Euler–Lagrange surface evolution descent equations, implemented via level set PDEs to allow changes in topology while ensuring numerical stability. We provide examples of verification of the scheme on real data.

Multi-frame simultaneous motion estimation and segmentation

This study proposes a new approach to jointly perform motion estimation and segmentation in the spatiotemporal domain. An energy functional that embeds the constraints of both the motion estimation and segmentation problems is minimized through the Euler-Lagrange equations yielding a set of descent equations executed via spatiotemporal level-sets representation. The resulting motion field is smooth yet boundary preserving and the motion segmentation map reflects the motion field enhancement.

Spatiotemporal Motion Boundary Detection and Motion Boundary Velocity Estimation for Tracking Moving Objects With a Moving Camera: A Level Sets PDEs Approach With Concurrent Camera Motion Compensation

The purpose of this study is to investigate a method of tracking moving objects with a moving camera. This method estimates simultaneously the motion induced by camera movement. The problem is formulated as a Bayesian motion-based partitioning problem in the spatiotemporal domain of the image quence. An energy functional is derived from the Bayesian formulation. The Euler-Lagrange descent equations determine imultaneously an estimate of the image motion field induced by camera motion and an estimate of the spatiotemporal motion undary surface. The Euler-Lagrange equation corresponding to the surface is expressed as a level-set partial differential equation for topology independence and numerically stable implementation. The method can be initialized simply and can track multiple objects with nonsimultaneous motions. Velocities on motion boundaries can be estimated from geometrical properties of the motion boundary. Several examples of experimental verification are given using synthetic and real-image sequences.

Constraining Active Contour Evolution via Lie Groups of Transformation

We present a novel approach to constraining the evolution of active contours used in image analysis. The proposed approach constrains the final curve obtained at convergence of curve evolution to be related to the initial curve from which evolution begins through an element of a desired Lie group of plane transformations. Constraining curve evolution in such a way is important in numerous tracking applications where the contour being tracked in a certain frame is known to be related to the contour in the previous frame through a geometric transformation such as translation, rotation, or affine transformation, for example. It is also of importance in segmentation applications where the region to be segmented is known up to a geometric transformation. Our approach is based on suitably modifying the Euler-Lagrange descent equations by using the correspondence between Lie groups of plane actions and their Lie algebras of infinitesimal generators, and thereby ensures that curve evolution takes place on an orbit of the chosen transformation group while remaining a descent equation of the original functional. The main advantage of our approach is that it does not necessitate any knowledge of nor any modification to the original curve functional and is extremely straightforward to implement. Our approach therefore stands in sharp contrast to other approaches where the curve functional is modified by the addition of geometric penalty terms. We illustrate our algorithm on numerous real and synthetic examples.

N=2 SUPER-YANG–MILLS ACTION AND BRST COHOMOLOGY

The extended BRST cohomology of N=2 super-Yang–Mills theory is discussed in the framework of Algebraic Renormalization. In particular, N=2 supersymmetric descent equations are derived from the cohomological analysis of linearized Slavnov–Taylor operator ℬ. It is then shown that both off- and on-shell N=2 super-Yang–Mills actions are related to a lower-dimensional gauge-invariant field polynomial Tr ϕ2 by solving these descent equations. Moreover, it is found that these off- and on-shell solutions differ only by a ℬ-exact term, which can be interpreted as a consequence of the fact that the cohomology of both cases are the same.

Holographic renormalization and Ward identities with the Hamilton–Jacobi method

A systematic procedure for performing holographic renormalization, which makes use of the Hamilton–Jacobi method, is proposed and applied to a bulk theory of gravity interacting with a scalar field and a U(1) gauge field in the Stückelberg formalism. We describe how the power divergences are obtained as solutions of a set of “descent equations” stemming from the radial Hamiltonian constraint of the theory. In addition, we isolate the logarithmic divergences, which are closely related to anomalies. The method allows to determine also the exact one-point functions of the dual field theory. Using the other Hamiltonian constraints of the bulk theory, we derive the Ward identities for diffeomorphisms and gauge invariance. In particular, we demonstrate the breaking of U(1) R current conservation, recovering the holographic chiral anomaly recently discussed in hep-th/0112119 and hep-th/0202056.

N = 1 SYM ACTION AND BRST COHOMOLOGY

The relation between BRST cohomology and the N = 1 supersymmetric Yang–Mills action in four dimensions is discussed. In particular, it is shown that both off- and on-shell N = 1 SYM actions are related to a lower-dimensional field polynomial by solving the descent equations, which is obtained from the cohomological analysis of linearized Slavnov–Taylor operator ℬ, in the framework of algebraic renormalization. Furthermore we show that off- and on-shell solutions differ only by a ℬ-exact term, which is a consequence of the fact that the cohomology of both cases are the same.

Remarks on the BRST cohomology of supersymmetric gauge theories

The supersymmetric version of the descent equations following from the Wess-Zumino consistency condition is discussed. A systematic framework in order to solve them is proposed.