Arbitrary Ground Research Articles

Can a successor comparator be used in an equal pay claim under Section 6(4) of the EEA?

One of the purposes of the Employment Equity Act 55 of 1998 ('EEA') is to achieve equity in the workplace by promoting fair treatment and equal opportunity through eliminating unfair discrimination (section 2(a) of the EEA). Section 5 of the EEA obliges every employer to take steps to promote equal opportunity in the workplace by eliminating unfair discrimination in any employment practice or policy. Section 6(1) of the EEA then prohibits unfair discrimination against employees on a number of grounds which includes an arbitrary ground. Van Niekerk et al state that sections 5 and 6 of the EEA contain a generally expressed obligation to promote equality through the elimination of unfair discrimination as well as the prohibition of unfair discrimination on a list of specified and other grounds (Van Niekerk et al Law@work (2023) 131). Du Toit et al state that the EEA proscribes unfair discrimination as part of its wider purpose of promoting employment equity (Du Toit et al Labour Relations Law: A Comprehensive Guide (2023) 766).- please remove 'et al' and list all the authors.

Representation theorems for simplicial complexes and matroidal-like properties of minimal partitioners

A pairing on an arbitrary ground set Ω is a triple P:=(U,F,Λ), with U,Λ two sets and F:U×Ω⟶Λ a map. Several properties of pairings arise after considering the Moore set system MP and the abstract simplicial complex NP on Ω, defined by taking the maximum and the minimal elements of the equivalence collections with respect to a specific equivalence relation ≈P, respectively called minimal and maximum partitioners.In the present work we first detect various sufficient conditions allowing us to represent specific subfamilies of abstract simplicial complexes as the family of all the minimal partitioners of some pairing on the same ground set. Next, we classify two suitable subcollections of pairings by using generalized matroidal-like properties of NP. More in detail, we first determine a sufficient condition on P ensuring that the family NP is a closable finitary simplicial complex and call the resulting pairings attractive. On an arbitrary ground set Ω, attractiveness, together with a finiteness condition, implies that the minimal members of the equivalence collections of each X∈MP with respect to ≈P all have the same cardinality. Nevertheless, the converse does not hold, neither in the finite case. To this regard, we find some counterexamples inducing us to introduce the class of quasi-attractive pairings. We carried out a detailed analysis of quasi-attractive pairings: for instance we characterize them from a lattice-theoretic point of view and, on a finite ground set Ω, also in term of exchange properties of suitable set systems.Finally, by taking the adjacence matrix of a simple undirected graph G as a model of pairing, we show that the Petersen graph induces an attractive pairing, while the Erdös' friendship graphs induce a quasi-attractive, but not attractive, one.

Presentation of rational Schur algebras

We present rational Schur algebra S(n,r,s) over an arbitrary ground field K as a quotient of the distribution algebra Dist(G) of the general linear group G=GL(n) by an ideal I(n,r,s) and provide an explicit description of the generators of I(n,r,s). Over fields K of characteristic zero, this corrects and completes a presentation of S(n,r,s) in terms of generators and relations originally considered by Dipper and Doty. The explicit presentation over ground fields of positive characteristics appears here for the first time.

Anyon quantum dimensions from an arbitrary ground state wave function

Realizing topological orders and topological quantum computation is a central task of modern physics. An important but notoriously hard question in this endeavor is how to diagnose topological orders that lack conventional order parameters. A breakthrough in this problem is the discovery of topological entanglement entropy, which can be used to detect nontrivial topological order from a ground state wave function, but is far from enough for fully determining the topological order. In this work, we take a key step further in this direction: We propose a simple entanglement-based protocol for extracting the quantum dimensions of all anyons from a single ground state wave function in two dimensions. The choice of the space manifold and the ground state is arbitrary. This protocol is both validated in the continuum and verified on lattices, and we anticipate it to be realizable in various quantum simulation platforms.

Caracal: A Versatile Ring Polymer Molecular Dynamics Simulation Package.

A new open-source program package named Caracal covering simulations of molecular systems with ring polymer molecular dynamics (RPMD) is presented. It combines a powerful RPMD implementation including chemical reaction rate calculations and biased periodic and nonperiodic samplings with a collection of easy to set up potential energy surface (PES) methodologies, thus delivering an all-inclusive approach. Most implemented PESs are based on the QMDFF and EVB-QMDFF methods. Where the quantum mechanically derived force field (QMDFF) can be set up for an arbitrary molecular system in a black-box fashion, the empirical valence bond (EVB)-QMDFF connects two QMDFFs and is able to represent the PES of a chemical reaction. With our previously published flavors of this composite method, PESs for almost arbitrary gas phase thermal ground state reactions can be set up. Given an optimized reaction path, the mechanism of the reaction can be classified and RPMD rate constants can be obtained via umbrella sampling and recrossing calculations on an EVB-QMDFF PES. Further, QMDFFs can be polymerized for the description of liquid systems. In this paper, the internal structure as well as the handling philosophy of Caracal are outlined. Further, examples of the different possible kinds of calculations are given.

Noise Level Free Regularization of General Linear Inverse Problems under Unconstrained White Noise

In this note we solve a general statistical inverse problem under absence of knowledge of both the noise level and the noise distribution via application of the (modified) heuristic discrepancy principle. Hereby the unbounded (non-Gaussian) noise is controlled via introducing an auxiliary discretization dimension and choosing it in an adaptive fashion. We first show convergence for completely arbitrary compact forward operator and ground solution. Then the uncertainty of reaching the optimal convergence rate is quantified in a specific Bayesian-like environment. We conclude with numerical experiments.

Mechanical responses of multi-layered ground due to shallow tunneling with arbitrary ground surface load

Mechanical responses of multi-layered ground due to shallow tunneling with arbitrary ground surface load

Efficient Approximation of Gromov-Wasserstein Distance Using Importance Sparsification

As a valid metric of metric-measure spaces, Gromov-Wasserstein (GW) distance has shown the potential for matching problems of structured data like point clouds and graphs. However, its application in practice is limited due to the high computational complexity. To overcome this challenge, we propose a novel importance sparsification method, called Spar-GW, to approximate GW distance efficiently. In particular, instead of considering a dense coupling matrix, our method leverages a simple but effective sampling strategy to construct a sparse coupling matrix and update it with few computations. The proposed Spar-GW method is applicable to the GW distance with arbitrary ground cost, and it reduces the complexity from O ( n 4 ) to O ( n 2 + δ ) for an arbitrary small δ > 0 . Theoretically, the convergence and consistency of the proposed estimation for GW distance are established under mild regularity conditions. In addition, this method can be extended to approximate the variants of GW distance, including the entropic GW distance, the fused GW distance, and the unbalanced GW distance. Experiments show the superiority of our Spar-GW to state-of-the-art methods in both synthetic and real-world tasks. Supplementary materials for this article are available online.

Stability of exact solutions of the (2 + 1)-dimensional nonlinear Schrödinger equation with arbitrary nonlinearity parameter κ

In this work, we consider the nonlinear Schrödinger equation (NLSE) in 2+1 dimensions with arbitrary nonlinearity exponent κ in the presence of an external confining potential. Exact solutions to the system are constructed, and their stability as we increase the ‘mass’ (i.e., the L 2 norm) and the nonlinearity parameter κ is explored. We observe both theoretically and numerically that the presence of the confining potential leads to wider domains of stability over the parameter space compared to the unconfined case. Our analysis suggests the existence of a stable regime of solutions for all κ as long as their mass is less than a critical value M*(κ). Furthermore, we find that there are two different critical masses, one corresponding to width perturbations and the other one to translational perturbations. The results of Derrick’s theorem are also obtained by studying the small amplitude regime of a four-parameter collective coordinate (4CC) approximation. A numerical stability analysis of the NLSE shows that the instability curve M*(κ) versus κ lies below the two curves found by Derrick’s theorem and the 4CC approximation. In the absence of the external potential, κ = 1 demarcates the separation between the blowup regime and the stable regime. In this 4CC approximation, for κ < 1, when the mass is above the critical mass for the translational instability, quite complicated motions of the collective coordinates are possible. Energy conservation prevents the blowup of the solution as well as confines the center of the solution to a finite spatial domain. We call this regime the ‘frustrated’ blowup regime and give some illustrations. In an appendix, we show how to extend these results to arbitrary initial ground state solution data and arbitrary spatial dimension d.

A height gap theorem for coefficients of Mahler functions

We study the asymptotic growth of coefficients of Mahler power series with algebraic coefficients, as measured by their logarithmic Weil height. We show that there are five different growth behaviors, all of which being reached. Thus, there are gaps in the possible growths. In proving this height gap theorem, we find that a k -Mahler function is k -regular if and only if its coefficients have height in O(\log n) . Moreover, we deduce that, over an arbitrary ground field of characteristic 0, a k -Mahler function is k -automatic if and only if its coefficients belong to a finite set. As a by-product of our results, we also recover a conjecture of Becker which was recently settled by Bell, Chyzak, Coons, and Dumas.

Superfiltered A∞$A_\infty$‐deformations of the exterior algebra, and local mirror symmetry

The exterior algebra $E$ on a finite-rank free module $V$ carries a $\mathbb{Z}/2$-grading and an increasing filtration, and the $\mathbb{Z}/2$-graded filtered deformations of $E$ as an associative algebra are the familiar Clifford algebras, classified by quadratic forms on $V$. We extend this result to $A_\infty$-algebra deformations $\mathcal{A}$, showing that they are classified by formal functions on $V$. The proof translates the problem into the language of matrix factorisations, using the localised mirror functor construction of Cho-Hong-Lau, and works over an arbitrary ground ring. We also compute the Hochschild cohomology algebras of such $\mathcal{A}$. By applying these ideas to a related construction of Cho-Hong-Lau we prove a local form of homological mirror symmetry: the Floer $A_\infty$-algebra of a monotone Lagrangian torus is quasi-isomorphic to the endomorphism algebra of the expected matrix factorisation of its superpotential.

On BGG resolutions of unitary modules for quiver Hecke and Cherednik algebras

We provide a homological construction of unitary simple modules of Cherednik and Hecke algebras of type A via BGG resolutions, solving a conjecture of Berkesch–Griffeth–Sam. We vastly generalize the conjecture and its solution to cyclotomic Cherednik and Hecke algebras over arbitrary ground fields, and calculate the Betti numbers and Castelnuovo–Mumford regularity of certain symmetric linear subspace arrangements.

On the dynamics of protected ramond ground states in the D1-D5 CFT

We examine the behavior of the Ramond ground states in the D1-D5 CFT after a deformation of the free-orbifold sigma model on target space ( {mathbbm{T}}^4 )N/SN by a marginal interaction operator. These states are compositions of Ramond ground states of the twisted and untwisted sectors. They are characterized by a conjugacy class of SN and by the set of their “spins”, including both R-charge and “internal” SU(2) charge. We compute the four-point functions of an arbitrary Ramond ground state with its conjugate and two interaction operators, for genus-zero covering surfaces representing the leading orders in the large-N expansion. We examine short distance limits of these four-point functions, shedding light on the dynamics of the interacting theory. We find the OPEs and a collection of structure constants of the ground states with the interaction operators and a set of resulting non-BPS twisted operators. We also calculate the integrals of the four-point functions over the positions of the interaction operators and show that they vanish. This provides an explicit demonstration that the Ramond ground states remain protected against deformations away of the free orbifold point, as expected from algebraic considerations using the spectral flow of the mathcal{N} = (4, 4) superconformal algebra with central charge c = 6N.

Mental illness, harassment and labour laws: Some thoughts on harassment by employees suffering from mental illness

Section 23 of the Constitution of the Republic of South Africa, 1996 provides that everyone has the right to fair labour practices. Section 9 of the Constitution prohibits unfair discrimination directly or indirectly against anyone on one or more grounds, including among others disability. In terms of section 6(1) of the Employment Equity Act (EEA), no person may unfairly discriminate, directly or indirectly, against an employee, in any employment policy or practice, on one or more grounds, including among others disability or on any other arbitrary ground. Section 6(1) applies to employees, which includes applicants; but it is only limited to conduct occurring within the scope of an “employment policy or practice”. In Marsland v New Way Motor &amp; Diesel Engineering (2009) 30 ILJ 169 (LC), the court concluded that discrimination based on the fact that a person suffers from a mental health problem, has the potential to impair the fundamental dignity of that person as a human being, or to affect them in a comparably serious manner. Consequently, discrimination based on mental illness must be treated as a prohibited ground of discrimination. However, as it was pointed out in Hoffmann v South African Airways 2001 (1) SA 1 (CC), it may in some instances be justified to discriminate on the ground of mental illness, if it is proved that the discrimination is based on an inherent requirement of a job. Section 15 of the EEA requires that, when the employer implements affirmative action measures, he/she must make reasonable accommodation for people from designated groups, in order to ensure that they enjoy equal opportunities and are equitably represented in the workforce of a designated employer. Section 1 defines “reasonable accommodation” as “any modification or adjustment to a job or to the working environment that will enable a person from a designated group to have access to or participate or advance in employment”. Section 6(3) of the EEA provides that harassment is a form of discrimination and is prohibited among others on the ground of disability or any other arbitrary ground. Harassment is also a form of misconduct. The employer is required to take reasonable steps to prevent harassment and failure to do so, the employer is liable for such harassment. Where an employee who has a mental illness, commits an act of harassment against another employee, the employer should take into account its duty to reasonably accommodate the offending employee, the duty to take steps to prevent harassment and the fact that it may be automatically unfair to dismiss an employee for misconduct which was committed because of mental illness.

A New Reconstruction Method for a Source Above an Arbitrarily Shaped Ground Plane

Source reconstruction using equivalent dipole moments to replace the original noise source is widely used in radio frequency interference. The conventional reconstruction methods only work well in the cases of free space or with a large ground plane. In this article a new method is proposed for source reconstruction over an arbitrarily shaped ground plane. It uses the field below the ground plane to eliminate the arbitrary ground effect. Comparing to the method of solving surface integral equation, it is simple and does not need the ground geometric information. The new method is validated in the full-wave simulations.

A strictly commutative model for the cochain algebra of a space

AbstractThe commutative differential graded algebra$A_{\mathrm {PL}}(X)$of polynomial forms on a simplicial set$X$is a crucial tool in rational homotopy theory. In this note, we construct an integral version$A^{\mathcal {I}}(X)$of$A_{\mathrm {PL}}(X)$. Our approach uses diagrams of chain complexes indexed by the category of finite sets and injections$\mathcal {I}$to model$E_{\infty }$differential graded algebras (dga) by strictly commutative objects, called commutative$\mathcal {I}$-dgas. We define a functor$A^{\mathcal {I}}$from simplicial sets to commutative$\mathcal {I}$-dgas and show that it is a commutative lift of the usual cochain algebra functor. In particular, it gives rise to a new construction of the$E_{\infty }$dga of cochains. The functor$A^{\mathcal {I}}$shares many properties of$A_{\mathrm {PL}}$, and can be viewed as a generalization of$A_{\mathrm {PL}}$that works over arbitrary commutative ground rings. Working over the integers, a theorem by Mandell implies that$A^{\mathcal {I}}(X)$determines the homotopy type of$X$when$X$is a nilpotent space of finite type.

Development of a new method for online parameter identification in seismically excited smart building structures using virtual synchronization and adaptive control design

In this paper a new method for online parameter identification and damage detection in smart building structures that are subjected to arbitrary seismic excitation is proposed. It uses real-time measurements of a structure's motion to identify its unknown constant or piecewise constant parameters such as stiffness, damping and mass over the time. The method is based on elements of system synchronization and adaptive control theories. First, a computational system, called the virtual system, is defined. Next, by using properly designed controller and estimations for the unknown parameters, the state of the virtual system is forced to follow the measured motion of the real structure. The mentioned estimations are computed from a proposed update law which depends on the measured motion of the real structure and the virtual system’s state. A major theoretical novelty of this paper is a proposed convergence condition which is applicable in case of arbitrary external forces or ground acceleration. It is shown that upon the satisfaction of that condition, as the synchronization completes, the computed estimation function converges to the true value of the vector of unknown parameters. In addition, an important practical contribution presented in this study is the introduction of a technique called scale factors. It helps to use available initial guesses of the unknown parameters to improve the speed of online identification. Numerical examples show that the proposed method is promising and has a good performance in both online identification and online damage detection problems.

Coherent orders

A preference relation is a binary endorelation B over an arbitrary ground set X of alternatives. The requisite that B is a strict partial order relation is equivalent to positing that the implication y∈B(x)⇒B(y)⊂B(x) – with B(t) denoting {s∈X:s,t∈B} for all t∈X and with ⊂ denoting strict inclusion – holds true for every pair y,x in X×X. The paper introduces the notion of a coherent order relation, defined as one that satisfies the previous one-way implication as a double implication. Examined are some properties of coherent order relations and how they connect with strict weak order relations and with transitive relations. In particular, the paper shows various conditions for a transitive relation to be a coherent order relation and for a coherent order relation to be a strict weak order relation.

Sobolev inequalities for the symmetric gradient in arbitrary domains

A form of Sobolev inequalities for the symmetric gradient of vector-valued functions is proposed, which allows for arbitrary ground domains in Rn. In the relevant inequalities, boundary regularity of domains is replaced with information on boundary traces of trial functions. The inequalities so obtained exhibit the same exponents as in classical inequalities for the full gradient of Sobolev functions, in regular domains. Furthermore, they involve constants independent of the geometry of the domain, and hence yield novel results yet for smooth domains. Our approach relies upon a pointwise estimate for the functions in question via a Riesz potential of their symmetric gradient and an unconventional potential depending on their boundary trace.

Integral u-deformed involution modules

Let (W,S) be a Coxeter system and ⁎ an automorphism of W with order ≤2 and S⁎=S. Lusztig and Vogan ([20], [23]) have introduced a u-deformed version Mu of Kottwitz's involution module over the Iwahori–Hecke algebra Hu(W) with Hecke parameter u2, where u is an indeterminate. Lusztig has proved that Mu is isomorphic to the left Hu(W)-submodule of Hˆu generated by X∅:=∑w⁎=w∈Wu−ℓ(w)Tw, where Hˆu is the vector space consisting of all formal (possibly infinite) sums ∑x∈WcxTx (cx∈Q(u) for each x). He speculated that one can extend this by replacing u with any λ∈C∖{0,1,−1}. In this paper, we give a positive answer to his speculation for any λ∈K∖{0,1,−1} and any W, where K is an arbitrary ground field.