Notes on cluster algebras and some all-loop Feynman integrals

We study cluster algebras for some all-loop Feynman integrals, including box-ladder, penta-box-ladder, and (seven-point) double-penta-ladder integrals. In addition to the well-known box ladder whose symbol alphabet is $D_2\simeq A_1^2$, we show that penta-box ladder has an alphabet of $D_3\simeq A_3$ and provide strong evidence that the alphabet of double-penta ladder can be identified with a $D_4$ cluster algebra. We relate the symbol letters to the ${\bf u}$ variables of cluster configuration space, which provide a gauge-invariant description of the cluster algebra, and we find various sub-algebras associated with limits of the integrals. We comment on constraints similar to extended-Steinmann relations or cluster adjacency conditions on cluster function spaces. Our study of the symbol and alphabet is based on the recently proposed Wilson-loop ${\rm d}\log$ representation, which allows us to predict higher-loop alphabet recursively; by applying such recursions to six-dimensional hexagon integrals, we also find $D_5$ and $D_6$ cluster functions for the two-mass-easy and three-mass-easy case, respectively.Comment: 28 pages, several figures; v2: typos corrected, functions of ladder integrals computed to higher loops; v3: more examples of double-penta-ladder integrals and discussions about their alphabet adde

STUDY OF STABLE MOTION PERCEPTION

The goal of my PhD project was to combine approaches from human psychophysics, Bayesian modeling and animal electrophysiology to study the mechanisms underlying motion perception. The specific goals of the current project are to: i) investigate the relationship between perceptual bias and stabilization with contextual regularity and density in human observers, and develop novel Bayesian models of motion perception that can account for the data; ii) explore motion duration dependence of offset neural activity and its layer specificity in primary visual cortex (V1) of alert monkeys

Modal Analysis of Cylindrical Gears with Arcuate Tooth Trace

In this paper, the forming principle, meshing features and tooth surface equation were introduced. And the modal parameters distribution of cylindrical gears with arcuate tooth trace was researched. The results show: 1. The modulus was the biggest impact factor for modal and natural frequency of cylindrical gears with arcuate tooth trace, then tooth width, and the radius of tooth line have the minimum influence; 2. When the modulus increased, natural frequency of cylindrical gears with arcuate tooth reduced rapidly; 3. When the tooth width increased, natural frequency of cylindrical gears with arcuate tooth has a tendency to rise except for first-order modal; 4. The influence of radius of tooth line can be basic ignored; 5. The second-order modal and third-order modal, fifth-order modal and sixth-order modal was very close. The research on cylindrical gears with arcuate tooth trace in this paper has a certain reference value on gear design and selection

Neural activity dissociation between thought-based and perception-based response conflict

Based on the idea that intentions have different penetrability to perception and thought (Fodor, 1983), four Stroop-like tasks, AA, AW, WA, and WW are used, where the A represents an arrow and the CPPR (closest processing prior to response) is perception, and the W represents a word and the CPPR is thought. Event-related brain potentials were recorded as participants completed these tasks, and sLORETA (standardized low resolution brain electromagnetic tomography) was used to localize the sources at specific time points. These results showed that there is an interference effect in the AA and WA tasks, but not in the AW or WW tasks. The activated brain areas related to the interference effect in the AA task were the PFC and ACC, and PFC activation took place prior to ACC activation; but only PFC in WA task. Combined with previous results, a new neural mechanism of cognitive control is proposed

Bootstrapping octagons in reduced kinematics from A2A_2 cluster algebras

Multi-loop scattering amplitudes/null polygonal Wilson loops in ${\mathcal N}=4$ super-Yang-Mills are known to simplify significantly in reduced kinematics, where external legs/edges lie in an $1+1$ dimensional subspace of Minkowski spacetime (or boundary of the $\rm AdS_3$ subspace). Since the edges of a $2n$-gon with even and odd labels go along two different null directions, the kinematics is reduced to two copies of $G(2,n)/T \sim A_{n{-}3}$. In the simplest octagon case, we conjecture that all loop amplitudes and Feynman integrals are given in terms of two overlapping $A_2$ functions (a special case of two-dimensional harmonic polylogarithms): in addition to the letters $v, 1+v, w, 1+w$ of $A_1 \times A_1$, there are two letters $v-w, 1- v w$ mixing the two sectors but they never appear together in the same term; these are the reduced version of four-mass-box algebraic letters. Evidence supporting our conjecture includes all known octagon amplitudes as well as new computations of multi-loop integrals in reduced kinematics. By leveraging this alphabet and conditions on first and last entries, we initiate a bootstrap program in reduced kinematics: within the remarkably simple space of overlapping $A_2$ functions, we easily obtain octagon amplitudes up to two-loop NMHV and three-loop MHV. We also briefly comment on the generalization to $2n$-gons in terms of $A_2$ functions and beyond.Comment: 26 pages, several figures and tables, an ancilary fil

Feynman Integrals and Scattering Amplitudes from Wilson Loops

We study Feynman integrals and scattering amplitudes in ${\cal N}=4$ super-Yang-Mills by exploiting the duality with null polygonal Wilson loops. Certain Feynman integrals, including one-loop and two-loop chiral pentagons, are given by Feynman diagrams of a supersymmetric Wilson loop, where one can perform loop integrations and be left with simple integrals along edges. As the main application, we compute analytically for the first time, the symbol of the generic ($n\geq 12$) double pentagon, which gives two-loop MHV amplitudes and components of NMHV amplitudes to all multiplicities. We represent the double pentagon as a two-fold $\mathrm{d} \log$ integral of a one-loop hexagon, and the non-trivial part of the integration lies at rationalizing square roots contained in the latter. We obtain a remarkably compact "algebraic words" which contain $6$ algebraic letters for each of the $16$ square roots, and they all nicely cancel in combinations for MHV amplitudes and NMHV components which are free of square roots. In addition to $96$ algebraic letters, the alphabet consists of $152$ dual conformal invariant combinations of rational letters.Comment: 8 pages, 4 figures, 1 ancillary file; v3: important updates, a compact form for the symbol of double pentagon integral added; typos correcte

Nonlinear vibration modeling and bifurcation characteristic study of a planetary gear train processing device

In this paper, a nonlinear torsional vibration model with meshing errors, time varying meshing stiffness, damping coefficients and gear backlashes was established and dimensionless equations of the system are derived in the planetary gear train processing device. The paper analyzed the nonlinear dynamic behavior of the device which was used to machine the Circular-Arc-Tooth-Trace cylindrical gear. By using the method of numerical integration, the bifurcation diagrams are obtained and the results indicate that the processing device has abundant bifurcation characteristics with the change of the dimensionless speed, and the damping ratios, gear backlashes and meshing errors of meshing pairs could influence the vibration greatly. The bifurcation diagrams reveal that increasing the damping ratios can change the bifurcation and the chaos can be avoid when the damping ratios are bigger enough, reducing the gear backlashes can reduce the dimensionless displacement amplitudes, increasing the meshing errors can make the bifurcation diagrams shift left for a distance, and alternating load torque with large amplitude will cause complex chaos phenomenon. The study can help to avoid the fatigue failure and instabilities caused by chaos and it also contribute to improving the performance of the processing device

Scene regularity interacts with individual biases to modulate perceptual stability

Sensory input is inherently ambiguous but our brains achieve remarkable perceptual stability. Prior experience and knowledge of the statistical properties of the world are thought to play a key role in the stabilization process. Individual differences in responses to ambiguous input and biases towards one or the other interpretation could modulate the decision mechanism for perception. However, the role of perceptual bias and its interaction with stimulus spatial properties such as regularity and element density remain to be understood. To this end, we developed novel bi-stable moving visual stimuli in which perception could be parametrically manipulated between two possible mutually exclusive interpretations: transparently or coherently moving. We probed perceptual stability across three composite stimulus element density levels with normal or degraded regularity using a factorial design. We found that increased density led to the amplification of individual biases and consequently to a stabilization of one interpretation over the alternative. This effect was reduced for degraded regularity, demonstrating an interaction between density and regularity. To understand how prior knowledge could be used by the brain in this task, we compared the data with simulations coming from four different hierarchical models of causal inference. These models made different assumptions about the use of prior information by including conditional priors that either facilitated or inhibited motion direction integration. An architecture that included a prior inhibiting motion direction integration consistently outperformed the others. Our results support the hypothesis that direction integration based on sensory likelihoods maybe the default processing mode with conditional priors inhibiting integration employed in order to help motion segmentation and transparency perception