scholarly journals A note on sign of a self-dual representation

2021 ◽  
pp. 1-6
Author(s):  
Manish Mishra
Keyword(s):  
Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 66
Author(s):  
Aviv Gibali ◽  
Oleg Kelis

In this paper we present an appropriate singular, zero-sum, linear-quadratic differential game. One of the main features of this game is that the weight matrix of the minimizer’s control cost in the cost functional is singular. Due to this singularity, the game cannot be solved either by applying the Isaacs MinMax principle, or the Bellman–Isaacs equation approach. As an application, we introduced an interception differential game with an appropriate regularized cost functional and developed an appropriate dual representation. By developing the variational derivatives of this regularized cost functional, we apply Popov’s approximation method and show how the numerical results coincide with the dual representation.


2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Renato Maria Prisco ◽  
Francesco Tramontano

Abstract We propose a novel local subtraction scheme for the computation of Next-to-Leading Order contributions to theoretical predictions for scattering processes in perturbative Quantum Field Theory. With respect to well known schemes proposed since many years that build upon the analysis of the real radiation matrix elements, our construction starts from the loop diagrams and exploits their dual representation. Our scheme implements exact phase space factorization, handles final state as well as initial state singularities and is suitable for both massless and massive particles.


2021 ◽  
Author(s):  
Sarah Shiraz ◽  
Krishna Regmi ◽  
Shruti Vyas ◽  
Yogesh S. Rawat ◽  
Mubarak Shah

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Guanda Lin ◽  
Gang Yang

Abstract Form factors, as quantities involving both local operators and asymptotic particle states, contain information of both the spectrum of operators and the on-shell amplitudes. So far the studies of form factors have been mostly focused on the large Nc planar limit, with a few exceptions of Sudakov form factors. In this paper, we discuss the systematical construction of full color dependent form factors with generic local operators. We study the color decomposition for form factors and discuss the general strategy of using on-shell unitarity cut method. As concrete applications, we compute the full two-loop non-planar minimal form factors for both half-BPS operators and non-BPS operators in the SU(2) sector in $$ \mathcal{N} $$ N = 4 SYM. Another important aspect is to investigate the color-kinematics (CK) duality for form factors of high-length operators. Explicit CK dual representation is found for the two-loop half-BPS minimal form factors with arbitrary number of external legs. The full-color two-loop form factor result provides an independent check of the infrared dipole formula for two-loop n-point amplitudes. By extracting the UV divergences, we also reproduce the known non-planar SU(2) dilatation operator at two loops. As for the finite remainder function, interestingly, the non-planar part is found to contain a new maximally transcendental part beyond the known planar result.


Development ◽  
1987 ◽  
Vol 99 (3) ◽  
pp. 393-410
Author(s):  
J.S. Taylor

This study concerns the retinotopic organization of the ganglion cell fibres in the visual system of the frog Xenopus laevis. HRP was used to trace the pathways taken by fibres from discrete retinal positions as they pass from the retina, along the optic nerve and into the chiasma. The ganglion cell fibres in the retina are arranged in fascicles which correspond with their circumferential positions of origin. Within the fascicles the fibres show little age-related layering and do not have a strict radial organization. As the fascicles of fibres pass into the optic nerve head there is some exchange of position resulting in some loss of the retinal circumferential organization. The poor radial organization of the fibres in the retinal fascicles persists as the fibres pass through the intraocular part of the nerve. At a position just behind the eye there is a major fibre reorganization in which fibres arising from cells of increasingly peripheral retinal locations are found to have passed into increasingly peripheral positions in the nerve. Thus, fibres from peripheral-most retina are located at the nerve perimeter, whilst fibres from central retina are located in the nerve core. It is at this point that the radial, chronotopic, ordering of the ganglion cell axons, found throughout the rest of the optic pathway, is established. This annular organization persists along the length of the nerve until a position just before the nerve enters the brain. Here, fibres from each annulus move to form layers as they pass into the optic chiasma. This change in the radial organization appears to be related to the pathway followed by all newly growing fibres, in the most superficial part of the optic tract, adjacent to the pia. Just behind the eye, where fibres become radially ordered, the circumferential organization of the projection is largely lost. Fibres from every circumferential retinal position, which are of similar radial position, are distributed within the same annulus of the nerve. At the nerve-chiasma junction where each annulus forms a single layer as it enters the optic tract, there is a further mixing of fibres from all circumferential positions. However, as the fibres pass through the chiasma some active pathway selection occurs, generating the circumferential organization of the fibres in the optic tract. Additional observations of the organization of fibres in the optic nerve of Rana pipiens confirm previous reports of a dual representation of fibres within the nerve. The difference in the organization of fibres in the optic nerve of Xenopus and Rana pipiens is discussed.


2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Christos E. Kountzakis

We prove a general dual representation form for restricted coherent risk measures, and we apply it to a minimization problem of the required solvency capital for an insurance company.


2018 ◽  
Author(s):  
Naohide Yamamoto ◽  
Dagmara E. Mach ◽  
John W. Philbeck ◽  
Jennifer Van Pelt

Generally, imagining an action and physically executing it are thought to be controlled by common motor representations. However, imagined walking to a previewed target tends to be terminated more quickly than real walking to the same target, raising a question as to what representations underlie the two modes of walking. To address this question, the present study put forward a hypothesis that both explicit and implicit representations of gait are involved in imagined walking, and further proposed that the underproduction of imagined walking duration largely stems from the explicit representation due to its susceptibility to a general undershooting tendency in time production (i.e., the error of anticipation). Properties of the explicit and implicit representations were examined by manipulating their relative dominance during imagined walking through concurrent bodily motions, and also by using non-spatial tasks that extracted the temporal structure of imagined walking. Results showed that the duration of imagined walking subserved by the implicit representation was equal to that of real walking, and a time production task exhibited an equivalent underproduction bias as in imagined walking tasks that were based on the explicit representation. These findings are interpreted as evidence for the dual-representation view of imagined walking.


2017 ◽  
Vol 3 (4) ◽  
Author(s):  
Pieter W. Claeys ◽  
Dimitri Van Neck ◽  
Stijn De Baerdemacker

We present the inner products of eigenstates in integrable Richardson-Gaudin models from two different perspectives and derive two classes of Gaudin-like determinant expressions for such inner products. The requirement that one of the states is on-shell arises naturally by demanding that a state has a dual representation. By implicitly combining these different representations, inner products can be recast as domain wall boundary partition functions. The structure of all involved matrices in terms of Cauchy matrices is made explicit and used to show how one of the classes returns the Slavnov determinant formula.Furthermore, this framework provides a further connection between two different approaches for integrable models, one in which everything is expressed in terms of rapidities satisfying Bethe equations, and one in which everything is expressed in terms of the eigenvalues of conserved charges, satisfying quadratic equations.


1998 ◽  
Vol 08 (05n06) ◽  
pp. 537-575 ◽  
Author(s):  
Jai Menon ◽  
Baining Guo

This paper presents a unified approach for incorporating free-form solids in bilateral Brep and CSG representation schemes, by resorting to low-degree (quadratic, cubic) algebraic surface patches. We develop a general CSG solution that represents a free-form solid as a boolean combination of a direct term and a complicated delta term. This solution gives rise to the trunctet-subshell conditions, under which the delta term computation can be obviated. We use polyhedral smoothing to construct a Brep consisting of quadratic algebraic patches that meet with tangent-plane continuity, such that the trunctet-subshell conditions are guaranteed automatically. This guarantee is not currently available for cubic patches. The general CSG solution thus applies whenever trunctet-subshell conditions are violated, e.g. sometimes for cubic patches or sometimes for patches of any degree that are subject to shape control operations. Manifold solids of arbitrary topology can be represented in our dual representation system. Ensuing CSG constructs are parallel processed on the RayCasting Engine to support a wide range of solid modeling applications, including general sweeping, Minkowski operations, NC machining, and touch-sense probing.


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