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We have computed the simplest five point function in $\mathcal{N} = 4$ SYM at two loops using the hexagonalization approach to correlation functions. Along the way we have determined all two-particle mirror contributions at two loops and we have computed all the integrals involved in the final result. As a test of our results we computed a few four-point functions and they agree with the perturbative results computed previously. We have also obtained $l$ loop results for some parts of the two-particle contributions with $l$ arbitrary. We also derive differential equations for a class of integrals that should appear at higher loops in the five point function., Comment: 24 pages + appendices + 12 nice figures

We consider a self-avoiding walk model (SAW) on the faces of the square lattice $\mathbb{Z}^2$. This walk can traverse the same face twice, but crosses any edge at most once. The weight of a walk is a product of local weights: each square visited by the walk yields a weight that depends on the way the walk passes through it. The local weights are parametrised by angles $\theta\in[\frac{\pi}{3},\frac{2\pi}{3}]$ and satisfy the Yang-Baxter equation. The self-avoiding walk is embedded in the plane by replacing the square faces of the grid with rhombi with corresponding angles. By means of the Yang-Baxter transformation, we show that the 2-point function of the walk in the half-plane does not depend on the rhombic tiling (i.e. on the angles chosen). In particular, this statistic coincides with that of the self-avoiding walk on the hexagonal lattice. Indeed, the latter can be obtained by choosing all angles $\theta$ equal to $\frac{\pi}{3}$. For the hexagonal lattice, the critical fugacity of SAW was recently proved to be equal to $1+\sqrt{2}$. We show that the same is true for any choice of angles. In doing so, we also give a new short proof to the fact that the partition function of self-avoiding bridges in a strip of the hexagonal lattice tends to 0 as the width of the strip tends to infinity. This proof also yields a quantitative bound on the convergence., Comment: 25 pages, 10 figures

Quantum cryptography has developed some fundamental primitives such as quantum one-time pad and quantum IND (indistinguishability)-security. Compared with other terms in quantum cryptography, quantum obfuscation attracts less attention and is still in its infancy due to its difficulty in implementation and application. In this chapter, we provide a positive result of quantum obfuscation. To analyze the obfuscatability of quantum point functions, we introduce the strict definition of a quantum point function and discuss its variants of multiple points and multiple qubits. Furthermore, we discuss the application of quantum obfuscation in quantum zero-knowledge and quantum symmetric encryption.

We present a new derivation of the distance-dependent two-point function for planar Eulerian triangulations and give expressions for more refined generating functions where we also control hull perimeters. These results are obtained in the framework of a new recursion relation for slice generating functions and extend similar results obtained recently for triangulations and quadrangulations. A number of explicit formulas are given for the statistics of hull perimeters in infinitely large random planar Eulerian triangulations., Comment: 30 pages, 17 figures

This study examines the accuracy of a wayside train wheel profile measurement system. This was accomplished by an evaluation of the contact-point function for the wheel–rail interface. The wheel profile measurement system in question generates data about the wheel profiles of passing trains. These data are used for improving the wheel maintenance procedures for the rolling stock operator. Recent work shows that there are differences between the data from the two different units in the system, but how this influences further use of the data, e.g. in wheel–rail contact analysis, has not been investigated so far. Accordingly, this article shows how two key wheel measures (the wheel flange thickness and the wheel profile) impact on the contact-point function and which of these measures has the largest impact on the contact-point function. The data used in this study were generated by two different measurement units for the same wheel and with the same wheel status. The results show that the different units produce different results and that these differences are more prominent when a difference in the flange thickness is detected, with a resulting shift of the front side of the flange and of the tread. With no difference in the flange thickness, i.e. no shift of the front side of the flange and of the tread, a difference was still detected in the contact conditions. Furthermore, this investigation shows that the shape of the tread has a greater impact on the contact-point conditions compared to a change in the flange thickness of up to 2.5 mm. This difference in the tread shape could have originated in measurement noise or different wheel measurement positions. The results of the study also show the importance of managing the measurement quality before using the data, for example for maintenance decisions.

We compute all three and four point couplings of BPS $D_{p}$-branes for all different nonzero $p$-values on the entire world volume and transverse directions. We start finding out all four point function supersymmetric Wess-Zumino (WZ) actions of one closed string Ramond-Ramond (RR) field with two fermions, either with the same (IIB) or different chirality (IIA) as well as their all order $\alpha'$ corrections. The closed form of S-matrices of two closed string RR in both IIB, IIA, including their all order $\alpha'$ corrections have also been addressed. Our results confirm that, not only the structures of $\alpha'$ corrections but also their coefficients of IIB are quite different from their IIA ones. The S-matrix of an RR and two gauge (scalar) fields and their all order corrections in antisymmetric picture of RR have been carried out as well. Various remarks on the restricted Bianchi identities as well as all order $\alpha'$ corrections to all different supersymmetric WZ couplings in both type IIA and IIB superstring theory are also released. Lastly, different singularity structures as well as all order contact terms for all non-vanishing traces in type II have also been constructed., Comment: 19 pages,V3: Clarifications on 4-point functions are added, Published in PLB

In this paper, using the formula for the integrals of the $\psi$-classes over the double ramification cycles found by S. Shadrin, L. Spitz, D. Zvonkine and the author, we derive a new explicit formula for the $n$-point function of the intersection numbers on the moduli space of curves., Comment: Revised version, 10 pages, to be published in the Moscow Mathematical Journal

We explore data analysis techniques for signatures from heavy particle production during inflation. Heavy particules can be produced by time dependent masses and couplings, which are ubiquitous in string theory. These localized excitations induce curvature perturbations with non-zero correlation functions at all orders. In particular, Flauger et. al. 2016 has shown that the signal-to-noise as a function of the order $N$ of the correlation function can peak for $N$ of order $\mathcal{O}(1)$ to $\mathcal{O}(100)$ for an interesting space of models. As previous non-Gaussianity analyses have focused on $N=\{3,4\}$, in principle this provides an unexplored data analysis window with new discovery potential. We derive estimators for arbitrary $N$-point functions in this model and discuss their properties and covariances. To lowest order, the heavy particle production phenomenology reduces to a classical Poisson process, which can be implemented as a search for spherically symmetric profiles in the curvature perturbations. We explicitly show how to recover this result from the $N$-point functions and their estimators. Our focus in this paper is on method development, but we provide an initial data analysis using WMAP data, which illustrates the particularities of higher $N$-point function searches., 19 pages, 9 figures

This chapter presents point function-based logic locking techniques, namely SARLock, Anti-SAT, and AND-tree detection that thwart the SAT attack by controlling the distinguishing ability of the DIPs. All these techniques integrate with the original netlist a point function that sets a limit on the number of incorrect key values that a DIP can eliminate. While these techniques cost-effectively thwart the SAT attack, their main limitation is the susceptibility to removal attacks. Moreover, these techniques fail to achieve a high output error rate.

We identify the formulas of Buryak and Okounkov for the n-point functions of the intersection numbers of psi-classes on the moduli spaces of curves. This allows us to combine the earlier known results and this one into a principally new proof of the famous Witten conjecture / Kontsevich theorem, where the link between the intersection theory of the moduli spaces and integrable systems is established via the geometry of double ramification cycles., 11 pages, some changes in the introduction

Differing-inputs obfuscation (diO), first introduced by Barak et al. (CRYPTO 2001) and then revisited by Ananth et al. (ePrint 2013) and Boyle et al. (TCC 2014), is a natural extension of indistinguishability obfuscation (iO), which captures a security notion that the obfuscations of two efficiently generated programs \(C_0\) and \(C_1\) are indistinguishable if it is hard for an adversary to find an input x such that \(C_0(x)\ne C_1(x)\), even in the presence of auxiliary information aux that is generated together with \(C_0\) and \(C_1\). A variant notion of diO, called public-coin diO, introduced by Ishai, Pandey and Sahai (TCC 2015) relaxes the original definition of diO by requiring that only the actual random coins that were used to sample programs \(C_0\) and \(C_1\) can be used as the auxiliary input. Public-coin diO is indeed of great interest, since it not only allows to evade the implausible results of diO, but also yields several useful applications. However, as far as we know, there was no approach known to build a public-coin differing-input obfuscator neither for general-purpose programs/circuits such as NC\(^1\) circuits nor for special-purpose function such as some variant of point function.

In five-axis machining processes, tool center point (TCP) control is often used to adjust the orientation of the ball end milling while maintaining constant tool center point coordinates in the workpiece frame. Therefore, TCP contouring error decreases the quality of the machined surface, but very few efforts have been made so far to minimize the contouring error of five-axis TCP by feedback control. To this end, a model predictive control (MPC) method is developed here for rolling optimization. More precisely, the contouring error model of five-axis TCP function is developed, where a Jacobian matrix is used to approximate the relationship between the contouring error and five drivers’ tracking errors. The MPC algorithm is afterwards derived, where constraints of control signals are considered as well to avoid violation of the power limitation of the drivers. Finally, the proposed MPC method is examined by extensive experiments using testing tool path of ISO 10791-6: 2014 to verify its effectiveness and superiority.

We prove that the correction to exponential decay of the truncated two points function in the homogeneous positive field Ising model is $c\|x\|^{-(d-1)/2}$. The proof is based on the development in the random current representation of a "modern" Ornstein-Zernike theory, as developed by Campanino, Ioffe and Velenik., 26 pages, 7 figures, details added

The postprocessor is essential for machining with five-axis machine tools. This paper develops one universal postprocessor for table-tilting type of five-axis machine tools without rotational tool center point (RTCP) function. Firstly, positions of two rotary axes and the workpiece in the machine coordinate system (MCS) are introduced into the kinematic chain of the five-axis machine tools. The uniform product of exponential (POE) formula of the tool relative to the workpiece is established to obtain the universal forward kinematics. On this basis, the postprocessor of table-tilting type of five-axis machine tools is developed. The calculation of rotation angles of rotation axes is proposed in details, including the calculation of double solutions, the determination of rotation angles of C-axis and the selection principle of the shortest path of rotation angles. Movements of linear axes are calculated with rotation angles of rotary axes. The generated movements of all axes are actual positions of all axes relative to their zero positions, which can be used for machining directly. The postprocessor does not rely on RTCP function with positions of rotary axes and the workpiece in MCS. Finally, cutting test in VERICUT and real cutting experiments on SmartCNC500_DRTD five-axis machine tool are carried out to verify the effectiveness of the proposed postprocessor.

Leading (large) logarithms in non-renormalizable theories have been investigated in the recent past. Besides some general considerations, explicit results for the expansion coefficients (in terms of leading logarithms) of partial wave amplitudes and of scalar and vector form factors have been given. Analyticity and unitarity constraints haven been used to obtain the expansion coefficients of partial waves in massless theories, yielding form factors and the scalar two-point function to five-loop order in the O(4)/O(3) model. Later, the all order solutions for the partial waves in any O(N+1)/O(N) model were found. Also, results up to four-loop order exist for massive theories. Here we extend the implications of analyticity and unitarity constraints on the leading logarithms to arbitrary loop order in massless theories. We explicitly obtain the scalar and vector form factors as well as to the scalar two-point function in any O(N) and SU(N) type models. We present relations between the expansion coefficients of these quantities and those of of the relevant partial waves. Our work offers a consistency check on the published results in the O(N) models for form factors, and new results for the scalar two-point function. For the SU(N) type models, we use the known expansion coefficients for partial waves to obtain those for scalar and vector form factors as well as for the scalar two-point function. Our results for the form factor offer a check for the known and future results for massive O(N) and SU(N) type models when the massless limit is taken. Mathematica notebooks which can be used to calculate the expansion coefficients are provided as ancillary files., 10 pages; 2 ancillary Mathematica files

We present a description of electrons propagating in an elliptically polarised plane wave background which includes circular and linear polarisations as special cases. We calculate to all orders in the background field the two point function and relate it to various expressions found in the literature. The background field induced mass shift of the electron is shown to be polarisation independent in the full elliptic class. The matrix nature of this mass shift in the fermionic theory is discussed. The extent to which a momentum space description is possible for this system is clarified., 17 pages Title changed in journal. Some references added

We study the long-distance behaviour of the massless vector propagator in (n)-dimensional de Sitter spacetime, where $n \geq 4$. Specifically, we consider the massless limit of the vector propagator in the Stueckelberg theory, which is an extension of Proca theory, with an additional gauge-fixing term. We work to leading order in the de Sitter-invariant distance $Z$ to show that, in the large $Z$ limit, this propagator tends to a gauge dependent constant, where the gauge worked in is described by the Stueckelberg parameter $\xi$. In the Landau gauge, where $\xi=0$, this constant is found to be 0. This result is in agreement with the 4 dimensional case discussed by Youssef., Comment: 14 pages, no figures, to appear in a Special Issue of the IJMPD on Selected Papers of the III Amazonian Symposium on Physics

We study the flow equation for the $$ \mathcal{N} $$ = 1 supersymmetric O(N ) nonlinear sigma model in two dimensions, which cannot be given by the gradient of the action, as evident from dimensional analysis. Imposing the condition on the flow equation that it respects both the supersymmetry and the O(N ) symmetry, we show that the flow equation has a specific form, which however contains an undetermined function of the supersymmetric derivatives D and $$ \overline{D} $$ . Taking the most simple choice, we propose a flow equation for this model. As an application of the flow equation, we give the solution of the equation at the leading order in the large N expansion. The result shows that the flow of the superfield in the model is dominated by the scalar term, since the supersymmetry is unbroken in the original model. It is also shown that the two point function of the superfield is finite at the leading order of the large N expansion.

We introduce an $\imath \epsilon$ prescription for the SYK model both at finite and at zero temperature. This prescription regularizes all the naive ultraviolet divergences of the model. As expected the prescription breaks the conformal invariance, but the latter is restored in the $\epsilon \to 0$ limit. We prove rigorously that the Schwinger Dyson equation of the resummed two point function at large $N$ and low momentum is recovered in this limit. Based on this $\imath \epsilon$ prescription we introduce an effective field theory Lagrangian for the infrared SYK model., Comment: Second version: the effective field theory part of the paper (subsections 2.1 and 3.1 and discussion) added

Minimal realizations of an interesting third order impedance function are discussed. The solution, based on an elegant algebraic identity, illustrates several basic concepts of driving point function synthesis.

In this chapter we shift our focus to three point functions. The first and second section are a review of two computational methods: the Tailoring procedure, which provides the weak-coupling limit of the three point function using a spin-chain construction, and the Hexagon proposal, which provides us the all-loop three point function. The last section is devoted to the rewriting of the hexagon proposal in a language more resembling the algebraic Bethe ansatz by means of Zamolodchikov-Faddeev operators.

Synthesis of an LC driving point function is one of the initial topics in the study of network synthesis. This article gives a practical example of application of such synthesis in the design of a ...

In this paper, we study algebraic and analytic properties of Fourier coefficients, expressed as q-series, of the so-called Bloch-Okounkov n-point function. We prove several results about these series and explain how they relate to Rogers' false theta function. Then we obtain their full asymptotics, as ? ? 0 , and use this result to derive asymptotic properties of the coefficients in the q-expansion. At the end, we also introduce and discuss higher rank generalization of Bloch-Okounkov's functions.

It is shown how the geometrical splitting of N-point Feynman diagrams can be used to simplify the parametric integrals and reduce the number of variables in the occurring functions. As an example, a calculation of the dimensionally-regulated one-loop four-point function in general kinematics is presented., 8 pages, 9 figures, contribution for proceedings of ACAT 2017 (Seattle, USA, August 21-25, 2017). arXiv admin note: substantial text overlap with arXiv:1605.04828

We study large $N$ tensor models on the lattice without disorder. We introduce techniques which can be applied to a wide class of models, and illustrate it by studying some specific rank-3 tensor models. In particular, we study Klebanov-Tarnopolsky model on lattice, Gurau-Witten model (by treating it as a tensor model on four sites) and also a new model which interpolates between these two models. In each model, we evaluate various four point functions at large $N$ and strong coupling, and discuss their spectrum and long time behaviors. We find similarities as well as differences from SYK model. We also generalize our analysis to rank-$D$ tensor models where we obtain analogous results as $D=3$ case for the four point functions which we computed. For $D>5$, we are able to compute the next-to-subleading ${1 \over N}$ corrections for a specific four point function., 46 pages, 29 figures; v2:typos corrected, reference added; v3:minor revisions, to be published in JHEP

We compute the weak magnetic field correction to the four-point quark-gluon vertex at finite temperature and chemical potential within the hard thermal loop approximation. The resulting four-point quark-gluon vertex satisfies the Ward identity with the three-point quark-gluon vertex., 11 pages, 1 figure, v2, published in PRD

The two point gluon and ghost correlation functions and the three gluon vertex are investigated, in the Landau gauge, using lattice simulations. For the two point functions, we discuss the approach to the continuum limit looking at the dependence on the lattice spacing and volume. The analytical structure of the propagators is also investigated by computing the corresponding spectral functions using an implementation of the Tikhonov regularisation to solve the integral equation. For the three point function we report results when the momentum of one of the gluon lines is set to zero and discuss its implications., Comment: Proceedings of Light Cone 2016, held in Lisbon, September 2016. Minor changes in text. To appear in Few B Sys