Exact Representation Research Articles

The Kennicutt–Schmidt law and the main sequence of galaxies in Newtonian and milgromian dynamics

ABSTRACT The Kennicutt–Schmidt law is an empirical relation between the star formation rate surface density (ΣSFR) and the gas surface density (Σgas) in disc galaxies. The relation has a power-law form $\Sigma _{\text{SFR}} \propto \Sigma _{\text{gas}}^{n}$. Assuming that star formation results from gravitational collapse of the interstellar medium, ΣSFR can be determined by dividing Σgas by the local free-fall time tff. The formulation of tff yields the relation between ΣSFR and Σgas, assuming that a constant fraction (εSFE) of gas is converted into stars every tff. This is done here for the first time using Milgromian dynamics (MOND). Using linear stability analysis of a uniformly rotating thin disc, it is possible to determine the size of a collapsing perturbation within it. This lets us evaluate the sizes and masses of clouds (and their tff) as a function of Σgas and the rotation curve. We analytically derive the relation $\Sigma _{\text{SFR}} \propto \Sigma _{\text{gas}}^{n}$ both in Newtonian and Milgromian dynamics, finding that n = 1.4. The difference between the two cases is a change only to the constant pre-factor, resulting in increased ΣSFR of up to 25 per cent using MOND in the central regions of dwarf galaxies. Due to the enhanced role of disc self-gravity, star formation extends out to larger galactocentric radii than in Newtonian gravity, with the clouds being larger. In MOND, a nearly exact representation of the present-day main sequence of galaxies is obtained if $\epsilon _{\text{SFE}} = \text{constant} \approx 1.1{{\ \rm per\ cent}}$. We also show that empirically found correction terms to the Kennicutt–Schmidt law are included in the here presented relations. Furthermore, we determine that if star formation is possible, then the temperature only affects ΣSFR by at most a factor of $\sqrt{2}$.

Analysis of positive fractional-order neutral time-delay systems

In this paper, we consider an initial value problem for linear matrix coefficient systems of the fractional-order neutral differential equations with two incommensurate constant delays in Caputo’s sense. Firstly, we introduce the exact analytical representation of solutions to linear homogeneous and non-homogeneous neutral fractional-order differential-difference equations system by means of newly defined delayed Mittag–Leffler type matrix functions. Secondly, a criterion on the positivity of a class of fractional-order linear homogeneous time-delay systems has been proposed. Furthermore, we prove the global existence and uniqueness of solutions to non-linear fractional neutral delay differential equations system using the contraction mapping principle in a weighted space of continuous functions with regard to classical Mittag–Leffler functions. In addition, Ulam–Hyers stability results of solutions are attained based on fixed-point approach. Finally, we present an example to demonstrate the applicability of our theoretical results.

Neural-Network Quantum States for Spin-1 Systems: Spin-Basis and Parameterization Effects on Compactness of Representations.

Neural network quantum states (NQS) have been widely applied to spin-1/2 systems, where they have proven to be highly effective. The application to systems with larger on-site dimension, such as spin-1 or bosonic systems, has been explored less and predominantly using spin-1/2 Restricted Boltzmann Machines (RBMs) with a one-hot/unary encoding. Here, we propose a more direct generalization of RBMs for spin-1 that retains the key properties of the standard spin-1/2 RBM, specifically trivial product states representations, labeling freedom for the visible variables and gauge equivalence to the tensor network formulation. To test this new approach, we present variational Monte Carlo (VMC) calculations for the spin-1 anti-ferromagnetic Heisenberg (AFH) model and benchmark it against the one-hot/unary encoded RBM demonstrating that it achieves the same accuracy with substantially fewer variational parameters. Furthermore, we investigate how the hidden unit complexity of NQS depend on the local single-spin basis used. Exploiting the tensor network version of our RBM we construct an analytic NQS representation of the Affleck-Kennedy-Lieb-Tasaki (AKLT) state in the spin-1 basis using only hidden units, compared to required in the basis. Additional VMC calculations provide strong evidence that the AKLT state in fact possesses an exact compact NQS representation in the basis with only hidden units. These insights help to further unravel how to most effectively adapt the NQS framework for more complex quantum systems.

Blended isogeometric Kirchhoff–Love and continuum shells

The computational modeling of thin-walled structures based on isogeometric analysis (IGA), non-uniform rational B-splines (NURBS), and Kirchhoff–Love (KL) shell formulations has attracted significant research attention in recent years. While these methods offer numerous benefits over the traditional finite element approach, including exact representation of the geometry, naturally satisfied high-order continuity within each NURBS patch, and computationally efficient rotation-free formulations, they also present a number of challenges in modeling real-world engineering structures of considerable complexity. Specifically, these NURBS-based engineering models are usually comprised of numerous patches, with discontinuous derivatives, non-conforming discretizations, and non-watertight connections at their interfaces. Moreover, the analysis of such structures often requires the full stress and strain tensors (i.e., including the transverse normal and shear components) for subsequent failure analysis and remaining life prediction. Despite the efficiency provided by the KL shell, the formulation cannot accurately predict the response in the transverse directions due to its kinematic assumptions. In this work, a penalty-based formulation for the blended coupling of KL and continuum shells is presented. The proposed approach embraces both the computational efficiency of KL shells and the availability of the full-scale stress/strain tensors of continuum shells where needed by modeling critical structural components using continuum shells and other components using KL shells. The proposed method enforces the displacement and rotational continuities in a variational manner and is applicable to non-conforming and non-smooth interfaces. The efficacy of the developed method is demonstrated through a number of benchmark studies with a variety of analysis configurations, including linear and nonlinear analyses, matching and non-matching discretizations, and isotropic and composite materials. Finally, an aircraft horizontal stabilizer is considered to demonstrate the applicability of the proposed blended shells to real-world engineering structures of significant complexity.

Flowing with the temporal renormalization group

We discuss the far-from-equilibrium evolution of $\phi^3$-theory in $1+1$ dimensions with the temporal functional renormalisation group \cite{Gasenzer:2007za, Gasenzer:2010rq}. In particular, we show that this manifestly causal approach leads to novel one-loop exact equations for fully dressed correlation functions. Within this setup, we numerically compute the dynamical propagator. Its behaviour suggests self-similarity far from equilibrium in a restricted momentum regime. We discuss the scaling exponents for our solution, as well as the numerical satisfaction of energy and particle number conservation. We also derive a simple exact representation of the expectation value of the energy-momentum tensor solely in terms of the propagator.

Waves propagation through a localized axisymmetric vortical flow

We propose a particular model for the axisymmetric vortex in two-dimensional space that can be generated under the rotation of the circular cylinder in a viscous fluid. The velocity field induced by such vortex is similar to the velocity field of the point vortex in the limited region and exponentially decays at infinity. We consider the plane acoustic wave propagation through such flow. The numerical solution for the scattered field is obtained using the exact integral representation. The scattered wave in the far field is an outgoing cylindrical wave peaking at small scattering angles.

Quadratic optimization with switching variables: the convex hull for $$n=2$$

We consider quadratic optimization in variables (x, y) where \(0\le x\le y\), and \(y\in \{ 0,1 \}^n\). Such binary variables are commonly referred to as indicator or switching variables and occur commonly in applications. One approach to such problems is based on representing or approximating the convex hull of the set \(\{ (x,xx^T, yy^T)\,:\,0\le x\le y\in \{ 0,1 \}^n \}\). A representation for the case \(n=1\) is known and has been widely used. We give an exact representation for the case \(n=2\) by starting with a disjunctive representation for the convex hull and then eliminating auxiliary variables and constraints that do not change the projection onto the original variables. An alternative derivation for this representation leads to an appealing conjecture for a simplified representation of the convex hull for \(n=2\) when the product term \(y_1y_2\) is ignored.

IGA Using Offset-based Overlapping Domain Parameterizations

Isogeometric analysis (IGA) is a numerical method, proposed in Hughes et al. (2005), that connects computer-aided design (CAD) with finite element analysis (FEA). In CAD the computational domain is usually represented by B-spline or NURBS patches. Given a B-spline or NURBS parameterization of the domain, an isogeometric discretization is defined on the domain using the same B-spline or NURBS basis as for the domain parameterization. Ideally, such an isogeometric discretization allows an exact representation of the underlying CAD model.CAD models usually represent only the boundary of the object. For planar domains, the CAD model is given as a collection of curves representing the boundary. Finding a suitable parameterization of the interior is one of the major issues for IGA, similar to the mesh generation process in the FEA setting. The objective of this isogeometric parameterization problem is to obtain a set of patches, which exactly represent the boundary of the domain and which are parameterized regularly and without self-intersections. This can be achieved by segmenting the domain into patches which are matching along interfaces, or by covering the domain with overlapping patches. In this paper we follow the second approach.To construct a planar parameterization suitable for IGA from a given boundary curve, we propose an offset-based domain parameterization algorithm. Given a boundary curve, we obtain an inner curve by generalized offsetting. The inner curve, together with the boundary curve, naturally defines a ring-shaped patch with an associated parameterization. By definition, the ring-shaped patch has a hole, which can be covered by a multi-cell domain. Consequently, the domain is represented as a union of two overlapping subdomains which are regularly parameterized. On such a configuration, one can employ the overlapping multi-patch (OMP) method, as introduced in Kargaran et al. (2019), to solve PDEs on the given domain. The performance of the proposed method is reported in several numerical examples, considering different shape properties of the given boundary curve.

Pseudo-Peakon, Periodic Peakons and Compactons on a Shallow Water Model with Coriolis Effect

For a shallow water model with Coriolis effect, by applying the methodologies of dynamical systems and singular traveling wave theory developed by Li and Chen [2007] to its traveling wave system, under different parameter conditions, all possible bounded solutions (solitary wave solution, pseudo-peakon and periodic peakons as well as compactons) are obtained. Some exact explicit parametric representations are presented.

On-shell representations of two-body transition amplitudes: Single external current

This work explores scattering amplitudes that couple two-particle systems via a single external current insertion, $2+\mathcal{J}\ensuremath{\rightarrow}2$. Such amplitudes can provide structural information about the excited QCD spectrum. We derive an exact analytic representation for these reactions. From these amplitudes, we show how to rigorously define resonance and bound-state form factors. Furthermore, we explore the consequences of the narrow-width limit of the amplitudes as well as the role of the Ward-Takahashi identity for conserved vector currents. These results hold for any number of two-body channels with no intrinsic spin, and a current with arbitrary Lorentz structure and quantum numbers. This work and the existing finite-volume formalism provide a complete framework for determining this class of amplitudes from lattice QCD.

GloBiMapsAI: An AI-Enhanced Probabilistic Data Structure for Global Raster Datasets

In the last decade, more and more spatial data has been acquired on a global scale due to satellite missions, social media, and coordinated governmental activities. This observational data suffers from huge storage footprints and makes global analysis challenging. Therefore, many information products have been designed in which observations are turned into global maps showing features such as land cover or land use, often with only a few discrete values and sparse spatial coverage like only within cities. Traditional coding of such data as a raster image becomes challenging due to the sizes of the datasets and spatially non-local access patterns, for example, when labeling social media streams. This article proposes GloBiMap, a randomized data structure, based on Bloom filters, for modeling low-cardinality sparse raster images of excessive sizes in a configurable amount of memory with pure random access operations avoiding costly intermediate decompression. In addition, the data structure is designed to correct the inevitable errors of the randomized layer in order to have a fully exact representation. We show the feasibility of the approach on several real-world datasets including the Global Urban Footprint in which each pixel denotes whether a particular location contains a building at a resolution of roughly 10m globally as well as on a global Twitter sample of more than 220 million precisely geolocated tweets. In addition, we propose the integration of a denoiser engine based on artificial intelligence in order to reduce the amount of error correction information for extremely compressive GloBiMaps.

Boundary-integral representation sans waterline integral for flows around ships steadily advancing in calm water

The boundary-integral flow-representation associated with the boundary-value problem, commonly called Neumann–Kelvin (NK) problem, that corresponds to linear potential flow around a ship steadily advancing in calm water involves an integral around the mean waterline of the ship. This ‘waterline integral’ is a notorious source of numerical difficulties and has been extensively studied. The waterline integral in the NK theory is largely – but not fully – eliminated in the modification, called Neumann–Michell (NM) theory, of the NK theory. Specifically, the NM theory includes a residual waterline-distribution of weak Rankine singularities, ignored in practical applications. A crucial element of the NM theory is a mathematical transformation that is based on a vector Green function, which is associated with the common scalar Green function used in the NK theory. This transformation is revisited in the present study. A rigorous analysis yields an answer to a fifty-year old puzzle: an exact boundary-integral flow representation that does not include a waterline integral. A remarkable feature of this new flow representation, which is a modification of the NM flow representation given previously, is that it explicitly determines the flow potential and the flow velocity at a ship hull surface in terms of the flow velocity at the hull surface, rather than in terms of the hull-surface potential as in usual boundary-integral flow representations obtained via Green’s classical identity.

On evaluation of local fields by fast multipole method employing smooth equivalent/check surfaces

The work aims to increase the efficiency of finding local fields in problems, like those of fracture mechanics, for which extreme, rather than average, values of fields are of prime significance. We focus on using kernel independent fast multipole methods (KI-FMM) when a problem is solved by a boundary element method. To accurately calculate local fields, we employ smooth equivalent surfaces (SES), used in translations of a KI-FMM, instead of non-smooth surfaces having singular lines (12 edges of a cube) and/or points (8 vertices of a cube in 3D; 4 vertices of a square in 2D). We develop and employ exact representations of circular (in 2D) and spherical (in 3D) equivalent surfaces and higher-order approximations of equivalent densities on them. It is established that the accuracy and stability of a KI-FMM with the SES suggested are notably higher than those of a similar KI-FMM employing non-smooth (square in 2D, cubic in 3D) equivalent surfaces. Theoretical considerations presented and numerical tests performed show that, when using the SES, the complexity of a KI-FMM does not exceed that of an analytical FMM of the same accuracy. Numerical examples illustrate the exposition. The subroutines developed are of immediate use in any code of the KI-FMM to increase the accuracy of translations without increasing their time expense.

The topological line of ABJ(M) theory

We construct the one-dimensional topological sector of mathcal{N} = 6 ABJ(M) theory and study its relation with the mass-deformed partition function on S3. Supersymmetric localization provides an exact representation of this partition function as a matrix integral, which interpolates between weak and strong coupling regimes. It has been proposed that correlation functions of dimension-one topological operators should be computed through suitable derivatives with respect to the masses, but a precise proof is still lacking. We present non-trivial evidence for this relation by computing the two-point function at two-loop, successfully matching the matrix model expansion at weak coupling and finite ranks. As a by-product we obtain the two-loop explicit expression for the central charge cT of ABJ(M) theory. Three- and four-point functions up to one-loop confirm the relation as well. Our result points towards the possibility to localize the one-dimensional topological sector of ABJ(M) and may also be useful in the bootstrap program for 3d SCFTs.

Computing Ex Ante Coordinated Team-Maxmin Equilibria in Zero-Sum Multiplayer Extensive-Form Games

Computational game theory has many applications in the modern world in both adversarial situations and the optimization of social good. While there exist many algorithms for computing solutions in two-player interactions, finding optimal strategies in multiplayer interactions efficiently remains an open challenge. This paper focuses on computing the multiplayer Team-Maxmin Equilibrium with Coordination device (TMECor) in zero-sum extensive-form games. TMECor models scenarios when a team of players coordinates ex ante against an adversary. Such situations can be found in card games (e.g., in Bridge and Poker), when a team works together to beat a target player but communication is prohibited; and also in real world, e.g., in forest-protection operations, when coordinated groups have limited contact during interdicting illegal loggers. The existing algorithms struggle to find a TMECor efficiently because of their high computational costs. To compute a TMECor in larger games, we make the following key contributions: (1) we propose a hybrid-form strategy representation for the team, which preserves the set of equilibria; (2) we introduce a column-generation algorithm with a guaranteed finite-time convergence in the infinite strategy space based on a novel best-response oracle; (3) we develop an associated-representation technique for the exact representation of the multilinear terms in the best-response oracle; and (4) we experimentally show that our algorithm is several orders of magnitude faster than prior state-of-the-art algorithms in large games.

RI-IGABEM based on generalized-[formula omitted] method in 2D and 3D elastodynamic problems

The isogeometric analysis boundary element method (IGABEM) has a broad application prospect due to its exact geometric representation and good approximation properties. In this paper, a novel radial integration IGABEM (RI-IGABEM) based on the generalized-α method is proposed to solve 2D and 3D elastodynamic problems of homogeneous and inhomogeneous materials. First of all, the elastostatics Kelvin fundamental solution is used as the fundamental solution of the problem. In order to preserve the advantage of IGABEM, i.e. only boundary is discretized, the radial integration method (RIM) is applied to transform the domain integral caused by the material heterogeneity and the inertia term into an equivalent boundary integral by means of applied points. In addition, using a simple transformation method, the rigid-body technique is applied to solve the strongly singular integrals, and the Telles scheme and the power series expansion method are used to solve the weakly singular integrals in RI-IGABEM respectively. Furthermore, the generalized-α method is adopted to solve the time domain problem, which can improve the stability of numerical results by effectively filtering out the false response of high frequency and minimizing the attenuation of low frequency response. A number of 2D and 3D examples, such as those with homogeneous materials, functionally gradient materials, and material defects and inclusions, are used to demonstrate the ability of the scheme to simulate the elastodynamic problems.

An Exact Axisymmetric Solution in Anisotropic Plasticity

A hollow cylinder of incompressible material obeying Hill’s orthotropic quadratic yield criterion and its associated flow rule is contracted on a rigid cylinder inserted in its hole. Friction occurs at the contact surface between the hollow and solid cylinders. An axisymmetric boundary value problem for the flow of the material is formulated and solved, and the solution is in closed form. A numerical technique is only necessary for evaluating ordinary integrals. The solution may exhibit singular behavior in the vicinity of the friction surface. The exact asymptotic representation of the solution shows that some strain rate components and the plastic work rate approach infinity in the friction surface’s vicinity. The effect of plastic anisotropy on the solution’s behavior is discussed.

Conformal Regge theory at finite boost

The Operator Product Expansion is a useful tool to represent correlation functions. In this note we extend Conformal Regge theory to provide an exact OPE representation of Lorenzian four-point correlators in conformal field theory, valid even away from Regge limit. The representation extends convergence of the OPE by rewriting it as a double integral over continuous spins and dimensions, and features a novel “Regge block”. We test the formula in the conformal fishnet theory, where exact results involving nontrivial Regge trajectories are available.

Bifurcations and Exact Traveling Wave Solutions in Two Nonlinear Wave Systems

This paper studies two nonlinear wave models. From the dynamical systems approach and using the singular traveling wave theory developed by [Li & Chen, 2007], all possible bounded solutions (solitary wave solutions, periodic wave solutions, as well as kink and anti-kink wave solutions) are obtained under different parameter conditions. More than 27 explicit exact parametric representations are derived. The new results complete the recent studies of [Rogers et al., 2020; Zayeda et al., 2020].

Multilayer shallow-water model with stratification and shear

The purpose of this paper is to present a shallow-water-type model with multiple inhomogeneous layers featuring variable linear velocity vertical shear and startificaion in horizontal space and time. This is achieved by writing the layer velocity and buoyancy fields as linear functions of depth, with coefficients that depend arbitrarily on horizontal position and time. The model is a generalization of Ripa's (1995) single-layer model to an arbitrary number of layers. Unlike models with homogeneous layers the present model is able to represent thermodynamics processes driven by heat and freshwater fluxes through the surface or mixing processes resulting from fluid exchanges across contiguous layers. By contrast with inhomogeneous-layer models with depth-independent velocity and buoyancy, the model derived here can sustain explicitly at low frequency a current in thermal wind balance (between the vertical vertical shear and the horizontal density gradient) within each layer. In the absence of external forcing and dissipation, energy, volume, mass, and buoyancy variance constrain the dynamics; conservation of total zonal momentum requires in addition the usual zonal symmetry of the topography and horizontal domain. The inviscid, unforced model admits a formulation suggestive of a generalized Hamiltonian structure, which enables the classical connection between symmetries and conservation laws via Noether's theorem. A steady solution to a system involving one Ripa-like layer and otherwise homogeneous layers can be proved formally (or Arnold) stable using the above invariants. A model configuration with only one layer has been shown previously to provide: a very good representation of the exact vertical normal modes up to the first internal mode; an exact representation of long-perturbation (free boundary) baroclinic instability; and a very reasonable representation of short-perturbation (classical Eady) baroclinic instability. Here it is shown that substantially more accurate overall results with respect to single-layer calculations can be achieved by considering a stack of only a few layers. A similar behavior is found in ageostrophic (classical Stone) baroclinic instability by describing accurately the dependence of the solutions on the Richardson number with only two layers.