The amplituhedron from momentum twistor diagrams

We propose a new diagrammatic formulation of the all-loop scattering amplitudes/Wilson loops in planar N=4 SYM, dubbed the "momentum-twistor diagrams". These are on-shell-diagrams obtained by gluing trivalent black and white vertices defined in momentum twistor space, which, in the reduced diagram case, are known to be related to diagrams in the original twistor space. The new diagrams are manifestly Yangian invariant, and they naturally represent factorization and forward-limit contributions in the all-loop BCFW recursion relations in momentum twistor space, in a fashion that is completely different from those in momentum space. We show how to construct and evaluate momentum-twistor diagrams, and how to use them to obtain tree-level amplitudes and loop-level integrands; in particular for the latter we identify an isolated bubble-structure for each loop variable, arising from a forward limit, or entangled removal of particles. From a given diagram one can directly read off the C, D matrices via a generalized "boundary measurement"; this in turn determines a cell in the amplituhedron associated with the amplitude, and our diagrammatic representations of the amplitude can provide triangulations of the amplituhedron with generally very intricate geometries. To demonstrate the computational power of the formalism, we give explicit results for general two-loop integrands, and the cells of the complete amplituhedron for two-loop MHV amplitudes.Comment: 39 pages, 34 figure

Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet

The search for a theory of the S-Matrix has revealed surprising geometric structures underlying amplitudes ranging from the worldsheet to the amplituhedron, but these are all geometries in auxiliary spaces as opposed to kinematic space where amplitudes live. In this paper, we propose a novel geometric understanding of amplitudes for a large class of theories. The key is to think of amplitudes as differential forms directly on kinematic space. We explore this picture for a wide range of massless theories in general spacetime dimensions. For the bi-adjoint cubic scalar, we establish a direct connection between its "scattering form" and a classic polytope--the associahedron--known to mathematicians since the 1960's. We find an associahedron living naturally in kinematic space, and the tree amplitude is simply the "canonical form" associated with this "positive geometry". Basic physical properties such as locality, unitarity and novel "soft" limits are fully determined by the geometry. Furthermore, the moduli space for the open string worldsheet has also long been recognized as an associahedron. We show that the scattering equations act as a diffeomorphism between this old "worldsheet associahedron" and the new "kinematic associahedron", providing a geometric interpretation and novel derivation of the bi-adjoint CHY formula. We also find "scattering forms" on kinematic space for Yang-Mills and the Non-linear Sigma Model, which are dual to the color-dressed amplitudes despite having no explicit color factors. This is possible due to a remarkable fact--"Color is Kinematics"--whereby kinematic wedge products in the scattering forms satisfy the same Jacobi relations as color factors. Finally, our scattering forms are well-defined on the projectivized kinematic space, a property that provides a geometric origin for color-kinematics duality.Comment: 77 pages, 25 figures; v2, corrected discussion of worldsheet associahedron canonical for

Quality Risk Evaluation of the Food Supply Chain Using a Fuzzy Comprehensive Evaluation Model and Failure Mode, Effects, and Criticality Analysis

Evaluating the quality risk level in the food supply chain can reduce quality information asymmetry and food quality incidents and promote nationally integrated regulations for food quality. In order to evaluate it, a quality risk evaluation indicator system for the food supply chain is constructed based on an extensive literature review in this paper. Furthermore, a mathematical model based on the fuzzy comprehensive evaluation model (FCEM) and failure mode, effects, and criticality analysis (FMECA) for evaluating the quality risk level in the food supply chain is developed. A computational experiment aimed at verifying the effectiveness and feasibility of this proposed model is conducted on the basis of a questionnaire survey. The results suggest that this model can be used as a general guideline to assess the quality risk level in the food supply chain and achieve the most important objective of providing a reference for the public and private sectors when making decisions on food quality management

Hands-off, artificial construction, or penalty? How to deal with the increasingly polluted coastal wetland ecosystem in China

As the destruction of coastal wetlands in China becomes more and more serious, the government needs to strengthen the management of the wetland ecosystem. Considering that pollution is an important factor in the destruction of coastal wetlands in China, the government can deal with the problem of wetland pollution through three modes: hands-off, artificial construction, and penalty. In this article, the differential game model is constructed under these three modes. The balanced social benefits of the government and polluting enterprises under the three modes are obtained, and the applicable conditions of various wetland treatment paths are compared. The results show that when the revenue generated by taxation and the indirect income generated by artificial construction are small, the government will choose the laissez-faire mode. However, with the gradual increase of indirect income generated by artificial construction, the government will be inclined to choose the artificial construction mode. When the income from government fines is small, the social benefits of polluters in the hands-off mode are greater than those in the penalty mode. With the increase of fines, the social benefits of polluting enterprises will first decrease and then increase, which will eventually be greater than the social benefits under the hands-off mode

Psychological screening, providing social security or strengthening training? How can government and labor unions protect the mental health of young employees

As the mental health problems of young employees become more and more prominent, the government and labor unions need to take measures to protect the mental health of young employees. Considering that the main mental health safeguard measures are divided into psychological screening, providing social security and strengthening training, this article constructs a differential game model under these three modes. The balanced efforts and social benefits of the government and labor unions under the three modes are obtained, and the applicable conditions of various mental health protection modes are compared. The results show that if the government pays a lower cost, the government gets the highest economic benefit under the training mode, followed by the security mode, and the government gets the lowest economic benefit under the psychological screening mode. If the reputation of the labor unions improved by its efforts is low, the equilibrium benefits of the labor unions under psychological screening and providing security are the same, and greater than the equilibrium benefits under the intensive training mode. If the labor unions have a higher reputation enhanced by its efforts, the balance return of the labor unions under the guarantee mode is the highest, followed by the balance return under the intensive training mode, and the balance return of the labor unions under the psychological screening mode is the lowest

Pesticide control, physical control, or biological control? How to manage forest pests and diseases more effectively

The frequent occurrence of forest diseases and insect pests has a significant impact on the forest ecosystem. The government needs to take measures to protect the forest ecosystem. The common management modes for forest pests and diseases include pesticide control, physical control, and biological control. In the process of governance, governments need to consider not only cost-effectiveness but also the impact on the ecosystem. In this article, the differential game model under these three modes is constructed, and the equilibrium results are compared and analyzed. Finally, the research conclusion is drawn that under the biological control mode, the income generated by the unit control quantity is inversely proportional to the balanced control quantity. However, under pesticide control and physical control modes, the revenue generated by the unit control quantity is proportional to the balanced control quantity. At the same time, under the biological control mode, the unit governance cost is proportional to the balanced control quantity. Under the pesticide control and physical control modes, the unit control cost is inversely proportional to the balanced control quantity. Social forces tend to adopt pesticide control. The government prefers physical control

How to reduce the impact of contaminated seafood on public health with the discharge of Fukushima nuclear wastewater

As wastewater from the Fukushima nuclear disaster continues to drain into the sea, the supply system for healthy seafood is being seriously challenged. To protect public health, it is necessary to restructure the seafood supply mode. The seafood supply mode is divided into the original mode, land farming mode, and strengthen monitoring mode. To derive the applicable scope of the various modes of the seafood supply chain and to provide recommendations for the safety and sustainability of seafood supply chains for governments and enterprises, three differential game models are constructed in this study. Then, the equilibrium results obtained by the models are compared and analyzed. Based on the findings, the health impact of seafood pollution is relatively small, and the government tends to choose the original supply mode. As the health impact of seafood grows, governments tend to prefer land-based farming. The social benefit to the government is directly proportional to the monitoring cost of seafood. To protect public health, enterprises tend to choose the mode of strengthen monitoring if the proportion of unqualified seafood is low. In addition, if sea products show a high degree of adaptation to the land environment, they tend to choose land farming

Phase-sensitive optomechanical amplifier for quantum noise reduction in laser interferometers

The sensitivity of future gravitational wave interferometers is expected to be limited throughout the detection band by quantum vacuum fluctuations, which can be reduced by applying quantum optics techniques such as squeezed vacuum injection. However, decoherence caused by optical losses in the readout chain will severely limit the effectiveness of such schemes. It was proposed that effect of losses in the final stage of detection can be mitigated by a phase-sensitive amplifier placed in between the output port of the interferometer and the photodetector. In this paper we propose to implement such amplification using an optomechanical device, study some of its practical limitations, and discuss its applicability to next-generation gravitational-wave detectors