Permutation Of Variables Research Articles

Control of continuous digester kappa number using generalized model predictive control

Kappa number variability at the digester impacts pulp yield, physical strength properties, and lignin content for downstream delignification processing. Regulation of the digester kappa number is therefore of great importance to the pulp and paper industry. In this work, an industrial application of model-based predictive control (MPC), based on generalized prediction control, was developed for kappa number feedback control and applied to a dual vessel continuous digester located in Western Canada. The problem was complicated by the need to apply heat at multiple locations in the cook. In this study, the problem was reduced from a multiple to a single input system by identifying three potential single variable permutations for temperature adjustment. In the end, a coordinated approach to the heaters was adopted. The process was perturbed and modeled as a simple first order plus dead time model and implemented in generalized predictive control (GPC). The GPC was then configured to be equivalent to Dahlin’s controller, which reduced tuning parameterization to a single closed loop time constant. The controller was then tuned based on robustness towards a worst-case dead time mismatch of 50%. The control held the mean value of the kappa number close to the setpoint, and a 40% reduction in the kappa number’s standard deviation was achieved. Different kappa number trials were run, and the average fiberline yield for each period was evaluated. Trial results suggested yield gains of 0.3%–0.5% were possible for each 1 kappa number target increase.

Products and powers of principal symmetric ideals

Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case.

Polynomial decompositions with invariance and positivity inspired by tensors

We present a framework to decompose real multivariate polynomials while preserving invariance and positivity. This framework has been recently introduced for tensor decompositions, in particular for quantum many-body systems. Here we transfer results about decomposition structures, invariance under permutations of variables, positivity, rank inequalities and separations, approximations, and undecidability to real polynomials. Specifically, we define invariant decompositions of polynomials, and characterize which polynomials admit such decompositions. We then include positivity: We define invariant separable and sum-of-squares decompositions, and characterize the polynomials similarly. We provide inequalities and separations between the ranks of the decompositions, and show that the separations are not robust with respect to approximations. For cyclically invariant decompositions, we show that it is undecidable whether the polynomial is nonnegative or sum-of-squares for all system sizes. Our framework is different from existing approaches for polynomial decompositions, since it covers symmetry and positivity combined, in a clean and uniform way. Also, our work sheds new light on polynomials by putting them on an equal footing with tensors, and opens the door to extending this framework to other tensor product structures.

A STATISTICAL ANALYSIS OF GEOLOGICAL AND ENGINEERING PREDICTORS OF OILFIELD PERFORMANCE RESPONSE: A CASE STUDY OF OILFIELDS ON THE UK CONTINENTAL SHELF

Oilfield production is controlled by a wide range of geological and engineering parameters, many of which are at least partially interrelated. This paper uses multivariate statistical methods (principal component analysis, regression analysis and analysis of variance) to determine how these parameters are related, and which of them are most significant in controlling and predicting oilfield performance. The analysis is based on a database of publicly available oilfield data from the United Kingdom Continental Shelf (UKCS), from which a series of geological, engineering and fluid‐related control variables from 136 fields were pre‐processed and analyzed. This dataset is a subset of a much wider project database for UKCS oil, gas and condensate fields. For this study, the project database was divided into two datasets: a first dataset with 10 parameters from 136 fields, and a second, more detailed dataset with 27 parameters from 38 fields. Both datasets were analysed using principal component analysis in order to investigate possible correlations between numerically/statistically interrogable predictor variables such as porosity, permeability, number of production wells, gas‐oil ratio and reservoir temperature. A regression analysis was then carried out on the predictor variables in order to obtain a ranking of predictability (i.e. how indicative a predictor is of a particular outcome) and sensitivity (how sensitive an outcome is to slight changes in a predictor) in relation to recovery factor based on R‐squared and regression coefficient values. The results showed that key variables from the principal component analysis included field size, number of production wells, PVT, gross depositional environment and reservoir quality. High‐ranking parameters of predictability and sensitivity from the regression analysis were found to include API, net‐to‐gross, porosity and reservoir depth. These results are consistent with previous studies and suggest that it should be possible to forecast oilfield recovery based on only a few selected input variables. As a preliminary test of forecasting ability of the variable permutations put forward, a best‐subsets multiple regression was carried out using a statistical software package and yielded results which corroborated the main findings.

Submaximal clones over a three-element set up to minor-equivalence

We study clones modulo minor homomorphisms, which are mappings from one clone to another preserving arities of operations and respecting permutation and identification of variables. Minor-equivalent clones satisfy the same sets of identities of the form f(x1,⋯,xn)≈g(y1,⋯,ym)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(x_1,\\dots ,x_n)\\approx g(y_1,\\dots ,y_m)$$\\end{document}, also known as minor identities, and therefore share many algebraic properties. Moreover, it was proved that the complexity of the CSP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {CSP}}$$\\end{document} of a finite structure A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {A}$$\\end{document} only depends on the set of minor identities satisfied by the polymorphism clone of A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {A}$$\\end{document}. In this article we consider the poset that arises by considering all clones over a three-element set with the following order: we write C⪯mD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}\\ {\\preceq _{\ extrm{m}}}\\ \\mathcal {D}$$\\end{document} if there exists a minor homomorphism from C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}$$\\end{document} to D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {D}$$\\end{document}. We show that the aforementioned poset has only three submaximal elements.

Machine Learning Deciphered Molecular Mechanistics with Accurate Kinetic and Thermodynamic Prediction.

Time-lagged independent component analysis (tICA) and the Markov state model (MSM) have been extensively employed for extracting conformational dynamics and kinetic community networks from unbiased trajectory ensembles. However, these techniques may not be the optimal choice for elucidating transition mechanisms within low-dimensional representations, especially for intricate biosystems. Unraveling the association mechanism in such complex systems always necessitates permutations of several essential independent components or collective variables, a process that is inherently obscure and may require empirical knowledge for selection. To address these challenges, we have implemented an integrated unsupervised dimension reduction model: uniform manifold approximation and projection (UMAP) with hierarchy density-based spatial clustering of applications with noise (HDBSCAN). This approach effectively generates low-dimensional configurational embeddings. The hierarchical application of this architecture, in conjunction with MSM, reveals global kinetic connectivity while identifying local conformational states. Consequently, our methodology establishes a multiscale mechanistic elucidation framework. Leveraging the benefits of the uniform sample distribution and a denoising approach, our model demonstrates robustness in preserving global and local data structures compared to traditional dimension reduction methods in the field of MD analysis area. The interpretability of hyperparameter selection and compatibility with downstream tasks are cross-validated across various simulation data sets, utilizing both computational evaluation metrics and experimental kinetic observables. Furthermore, the predicted Mcl1-BH3 association kinetics (0.76 s-1) is in close agreement with surface plasmon resonance experiments (0.12 s-1), affirming the plausibility of the identified pathway composed of representative conformations. We anticipate that the devised workflow will serve as a foundational framework for studying recognition patterns in complex biological systems. Its contributions extend to the exploration of protein functional dynamics and rational drug design, offering a potent avenue for advancing research in these domains.

Data analysis on the three defect wavelengths of a MoS2-based defective photonic crystal using machine learning

To reduce the dimension of optoelectronic devices, recently, Molybdenum disulfide (MoS2) monolayers with direct bandgap in the visible range are widely used in designing a variety of photonic devices. In these applications, adjustability of the working wavelength and bandwidth with optimum absorption value plays an important role. This work proposes a symmetric defective photonic crystal with three defects containing MoS2 monolayer to achieve triple narrowband defect modes with wavelength adjustability throughout the Photonic Band Gap (PBG) region, 560 to 680 nm. Within one of our designs remarkable FWHM approximately equal to 5 nm with absorption values higher than 90% for the first and third defect modes are achieved. The impacts of varying structural parameters on absorption value and wavelength of defect modes are investigated. Due to the multiplicity of structural parameters which results in data plurality, the optical properties of the structure are also predicted by machine learning techniques to assort the achieved data. Multiple Linear Regression (MLR) modeling is used to predict the absorption and wavelength of defect modes for four datasets based on various permutations of structural variables. The machine learning modeling results are highly accurate due to the obtained R2-score and cross-validation score values higher than 90%.

An efficient method of finding new symplectic schemes for Hamiltonian mechanics problems with the aid of parametric Gröbner bases

The explicit symplectic difference schemes are considered for the numerical solution of molecular dynamics problems described by systems with separable Hamiltonians. A general method for finding symplectic schemes of high order of accuracy using parametric Gröbner bases, resultants, permutations of variables, and optimization techniques is proposed. The implementation of the method is described by the example of four-stage partitioned Runge–Kutta (PRK) schemes of the Forest–Ruth and Yoshida families. 97 new PRK schemes of Forest–Ruth family have been obtained. The error functionals for five new schemes FR47, FR50, FR51, FR52, and FR59 are smaller by more than two orders of magnitude than in the case of the fourth-order schemes, which were obtained previously in the explicit form in this family. In the family of Yoshida schemes, 115 new schemes have been found. Verification of the schemes was carried out on the Kepler's problem, which has the exact solution. Calculations of this task performed at large time steps showed that the FR50 scheme significantly exceeds in terms of accuracy the four-stage Forest–Ruth schemes known in the literature, the Verlet scheme, schemes from the Yoshida family Y7, Y95, Y110, and a six-stage fourth-order scheme.

The type-oriented algorithm for baseline correction of acceleration time histories

This paper presents the development of an exhaustive approach to baseline correction of acceleration signals that amalgamates several existing methods to provide a more powerful and flexible toolkit for scientific software implementation. The proposed type-oriented baseline correction algorithm employs the traditional methodology of subtracting a least squares polynomial from the raw time series, but it is seeded by a specified combination of correction types that can vary in both polynomial order and kinematic variable considered (i.e., acceleration, velocity, or displacement). The result is a dramatic expansion in the number of corrections possible with current commercial software, which tend to consider only one kinematic variable while also restricting the polynomial order used. The proposed algorithm incorporates seven unique correction types based on permutations of kinematic variables adjusted in sequence. Different numerical methods are evaluated to solve the system of normal equations of the least squares polynomial approximation of the one-dimensional real function. The result is an algorithm that achieves practically unlimited polynomial order. The external behavior of the algorithm is evaluated by presenting a set of parametric tests that use different correction types. It is shown that the optimal correction of some vibration signals, particularly those with low frequency content occurring over many cycles, cannot be obtained without adjusting the time histories by high order best-fits of more than one kinematic variable. The importance of baseline correction in the context of finite element analysis is discussed, and several implementation examples are provided in different programming languages.

The Inhibition and Equilibrium of Policy Effectiveness of the Low-Carbon Economy: Evidence from Liaoning Province of China.

Excessive emissions of carbon dioxide and other greenhouse gases have seriously affected the ecological environment, public health, and the normal operation of the social economy, and the development of the low-carbon economy has become an international consensus. The policy norms are one of the important factors affecting the development of the low-carbon economy; however, the implementation of low-carbon economic policy in many countries has been inhibited. This study chose Liaoning Province of China for the case study, and the policy system, the policy tools, the administrative system, the low-carbon technology, and the low-carbon concept are found to be factors that led to the inhibition of the policy effectiveness of the low-carbon economy in Liaoning Province. We applied the modified Schweller Neoclassical Realist Theory to establish a multi-factor linkage model to demonstrate the overall relationship among various variables. The results show that the equilibrium of policy effectiveness of the low-carbon economy in Liaoning Province depends on different permutations of variables. We also discussed the problems of the policy system, the policy tools, the administrative system, the low-carbon technology, and the low-carbon concept that lead to policy effectiveness inhibition, and used the economic method to set a special mathematical model for maximizing the equilibrium of policy effectiveness of the low-carbon economy in Liaoning Province. In response to the problems of the above factors, strategies to promote the development of the low-carbon economy in Liaoning Province are proposed. This study enriches the research on the policy effectiveness of the low-carbon economy in China and provides some inspiration for the goal of carbon neutrality and other developing countries with high carbon emissions.

Parametric Vine Copula Framework in the Trivariate Probability Analysis of Compound Flooding Events

The interaction between oceanographic, meteorological, and hydrological factors can result in an extreme flooding scenario in the low-lying coastal area, called compound flooding (CF) events. For instance, rainfall and storm surge (or high river discharge) can be driven by the same meteorological forcing mechanisms, tropical or extra-tropical cyclones, resulting in a CF phenomenon. The trivariate distributional framework can significantly explain compound events’ statistical behaviour reducing the associated high-impact flood risk. Resolving heterogenous dependency of the multidimensional CF events by incorporating traditional 3D symmetric or fully nested Archimedean copula is quite complex. The main challenge is to preserve all lower-level dependencies. An approach based on decomposing the full multivariate density into simple local building blocks via conditional independence called vine or pair-copulas is a much more comprehensive way of approximating the trivariate flood dependence structure. In this study, a parametric vine copula of a drawable (D-vine) structure is introduced in the trivariate modelling of flooding events with 46 years of observations of the west coast of Canada. This trivariate framework searches dependency by combining the joint impact of annual maximum 24-h rainfall and the highest storm surge and river discharge observed within the time ±1 day of the highest rainfall event. The D-vine structures are constructed in three alternative ways by permutation of the conditioning variables. The most appropriate D-vine structure is selected using the fitness test statistics and estimating trivariate joint and conditional joint return periods. The investigation confirms that the D-vine copula can effectively define the compound phenomenon’s dependency. The failure probability (FP) method is also adopted in assessing the trivariate hydrologic risk. It is observed that hydrologic events defined in the trivariate case produce higher FP than in the bivariate (or univariate) case. It is also concluded that hydrologic risk increases (i) with an increase in the service design life of the hydraulic facilities and (ii) with a decrease in return periods.

Mirror symmetry on levels of non-abelian Landau–Ginzburg orbifolds

We consider the Berglund–Hübsch–Henningson–Takahashi duality of Landau–Ginzburg orbifolds with a symmetry group generated by some diagonal symmetries and some permutations of variables. We study the orbifold Euler characteristics, the orbifold monodromy zeta functions and the orbifold E-functions of such dual pairs. We conjecture that we get a mirror symmetry between these invariants even on each level, where we call level the conjugacy class of a permutation. We support this conjecture by giving partial results for each of these invariants.

Minimization of binary decision diagrams for systems of completely defined Boolean functions using Shannon expansions and algebraic representations of cofactors

In the systems of digital VLSI design (Very Large Integrated Circuits), the BDD (Binary Decision Diagram) is used for VLSI verification, as well as for technologically independent optimization as the first stage in the synthesis of logic circuits in various technological bases. The BDD is an acyclic graph defining a Boolean function or a system of Boolean functions. Each vertex of this graph corresponds to the complete or reduced Shannon expansion formula. When BDD representation for systems of Boolean functions is constructed, it is possible to perform additional logical optimization based on the proposed method of searching for algebraic representations of cofactors (subfunctions) of the same BDD level in the form of a disjunction, conjunction either exclusive-or of cofactors of the same level or lower levels of BDD. A directed BDD graph for a system of functions is constructed on the basis of Shannon expansion of all component functions of the system by the same permutation of variables. The method allows to reduce the number of literals by replacing the Shannon expansion formulas with simpler formulas that are disjunctions or conjunctions of cofactors, and to reduce the number of literals in specifying a system of Boolean functions. The number of literals in algebraic multilevel representations of systems of fully defined Boolean functions is the main optimization criterion in the synthesis of combinational circuits from librarian logic elements.

Symmetric ideals, Specht polynomials and solutions to symmetric systems of equations

An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the leading monomials of polynomials in the ideal and the Specht polynomials contained in the ideal. This provides applications in several contexts. Most notably, this connection gives information about the solutions of the corresponding set of equations. From another perspective, it restricts the isotypic decomposition of the ideal viewed as a representation of the symmetric group.

On the Number of Equivalence Classes of Boolean and Invertible Boolean Functions

The number U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> of equivalence classes of Boolean functions of n variables and the number V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> of equivalence classes of vectorial Boolean functions of n variables under the action of four groups of transformations are considered. The four groups are the group S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ' of permutations of variables, the group G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> of permutations and complementations, the linear group GL(n, 2) and the affine group AGL(n, 2). Harrison obtained cycle indexes for these groups and the expressions for U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> and V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> in terms of the corresponding cycle index. He also tabulated the numbers U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> and the cycle indexes for n≤6 for S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ' and Gn, and for n≤5 for GL(n, 2) and AGL(n, 2). This bound was only recently slightly exceeded. Fripertinger implemented computation of cycle indexes for GL(n, q) and AGL(n, q); if q = 2 this implementation works for about n 5≤21. By introducing appropriate precomputed tables, we reduced the cycle index computation to evaluation of a sum over partitions of n for all the four groups. Using this more efficient procedure, we obtained values of U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> and the explicit cycle index expressions for larger values of n.

Dual Invertible Polynomials with Permutation Symmetries and the Orbifold Euler Characteristic

P. Berglund, T. H\"ubsch, and M. Henningson proposed a method to construct mirror symmetric Calabi-Yau manifolds. They considered a pair consisting of an invertible polynomial and of a finite (abelian) group of its diagonal symmetries together with a dual pair. A. Takahashi suggested a method to generalize this construction to symmetry groups generated by some diagonal symmetries and some permutations of variables. In a previous paper, we explained that this construction should work only under a special condition on the permutation group called parity condition (PC). Here we prove that, if the permutation group is cyclic and satisfies PC, then the reduced orbifold Euler characteristics of the Milnor fibres of dual pairs coincide up to sign.

Unistructurality of cluster algebras

We prove that any skew-symmetrizable cluster algebra is unistructural, which is a conjecture by Assem, Schiffler and Shramchenko. As a corollary, we obtain that a cluster automorphism of a cluster algebra ${\mathcal{A}}({\mathcal{S}})$ is just an automorphism of the ambient field ${\mathcal{F}}$ which restricts to a permutation of the cluster variables of ${\mathcal{A}}({\mathcal{S}})$.

Data-driven intelligent optimisation of discontinuous composites

Fibre composites, and especially aligned discontinuous composites (ADCs), offer enormous versatility in composition, microstructure, and performance, but are difficult to optimise, due to their inherent variability and myriad permutations of microstructural design variables. This work combines an accurate yet efficient virtual testing framework (VTF) with a data-driven intelligent Bayesian optimisation routine, to maximise the mechanical performance of ADCs for a number of single- and multi-objective design cases. The use of a surrogate model helps to minimise the number of optimisation iterations, and provides a more accurate insight into the expected performance of materials which feature significant variability. Results from the single-objective optimisation study show that a wide range of structural properties can be achieved using ADCs, with a maximum stiffness of 505 GPa, maximum ultimate strain of 3.94%, or a maximum ultimate strength of 1.92 GPa all possible. A moderate trade-off in performance can be achieved when considering multi-objective optimisation design cases, such as an optimal ultimate strength & ultimate strain combination of 982 MPa and 3.27%, or an optimal combination of 720 MPa yield strength & 1.91% pseudo-ductile strain.

Dimension formula for the space of relative symmetric polynomials of $$\varvec{D_n}$$ with respect to any irreducible representation

For positive integers d and n, the vector space $$H_d(x_1,x_2,\ldots ,x_n)$$ of homogeneous polynomials of degree d is a representation of the symmetric group $$S_n$$ acting by permutation of variables. Regarding this as a representation for the dihedral subgroup $$D_n$$ , we prove a formula for the dimension of all the isotypical subrepresentations. Our formula is simpler than the existing one found by Zamani and Babaei (Bull. Iranian Math. Soc. 40(4) (2014) 863–874). By varying the degrees d we compute the generating functions for these dimensions. Further, our formula leads us naturally to a specific supercharacter theory of $$D_n$$ . It turns out to be a $$*$$ -product of a specific supercharacter theory studied in depth by Fowler et al. (The Ramanujan Journal (2014)), with the unique supercharacter theory of a group of order 2.

On the orbifold Euler characteristics of dual invertible polynomials with non-abelian symmetry groups

In the framework of constructing mirror symmetric pairs of Calabi-Yau manifolds, P. Berglund, T. Hubsch and M. Henningson considered a pair $(f,G)$ consisting of an invertible polynomial $f$ and a finite abelian group $G$ of its diagonal symmetries and associated to this pair a dual pair $(\widetilde{f}, \widetilde{G})$. A. Takahashi suggested a generalization of this construction to pairs $(f, G)$ where $G$ is a non-abelian group generated by some diagonal symmetries and some permutations of variables. In a previous paper, the authors showed that some mirror symmetry phenomena appear only under a special condition on the action of the group $G$: a parity condition. Here we consider the orbifold Euler characteristic of the Milnor fibre of a pair $(f,G)$. We show that, for an abelian group $G$, the mirror symmetry of the orbifold Euler characteristics can be derived from the corresponding result about the equivariant Euler characteristics. For non-abelian symmetry groups we show that the orbifold Euler characteristics of certain extremal orbit spaces of the group $G$ and the dual group $\widetilde{G}$ coincide. From this we derive that the orbifold Euler characteristics of the Milnor fibres of dual periodic loop polynomials coincide up to sign.