Invariance Properties Research Articles

Beyond the Blackwell order in dichotomies

I establish a translation invariance property of the Blackwell order for dichotomies and show that the norm of the distance from the identity matrix may be interpreted as a measure of informativeness. The better experiment is closer to the fully revealing experiment; this measure extends the Blackwell order, is complete, and prior-independent.

Melody transposition tolerance in the human cortex: An fMRI adaptation and MVPA investigation

Abstract Melody perception involves constant relational representations, enabling most people to recognize the melodies after being transposed. This invariant property of melody transposition has been supported in many previous behavioral studies, and we hypothesize that there are brain regions showing tolerance toward melody transposition when processing melodies. To test the hypothesis, we adopted an event-related adaptation approach and a multivariate pattern cross-classification (MVCC) analysis approach. Consistent with our prediction, we discovered clusters in the right superior temporal gyrus (STG) and the left middle temporal gyrus (MTG) that exhibited adaptation when participants listened to both the same and transposed-same melodies after the original ones. An ROI and searchlight-based cross-classification analysis also revealed that BOLD pattern in the bilateral precentral gyrus (PreCG), the left inferior frontal gyrus (IFG), the left inferior parietal lobule (IPL), the right angular gyrus (AG), and the right superior temporal gyrus (STG) showed tolerance to melody transposition. These findings suggest that tolerance to melody transposition exists throughout the music processing pathway from auditory to motor cortices.

Invariant Checking for SMT-Based Systems with Quantifiers

This article addresses the problem of checking invariant properties for a large class of symbolic transition systems defined by a combination of SMT theories and quantifiers. State variables can be functions from an uninterpreted sort (finite but unbounded) to an interpreted sort, such as the integers under the theory of linear arithmetic. This formalism is very expressive and can be used for modeling parameterized systems, array-manipulating programs, and more. We propose two algorithms for finding universal inductive invariants for such systems. The first algorithm combines an IC3-style loop with a form of implicit predicate abstraction to construct an invariant in an incremental manner. The second algorithm constructs an under-approximation of the original problem and searches for a formula which is an inductive invariant for this case; then, the invariant is generalized to the original case and checked with a portfolio of techniques. We have implemented the two algorithms and conducted an extensive experimental evaluation, considering various benchmarks and different tools from the literature. As far as we know, our method is the first capable of handling in a large class of systems in a uniform way. The experiment shows that both algorithms are competitive with the state of the art.

Validation of the Warwick-Edinburgh Mental Well-Being Scale for the Mental Health Surveillance (MHS) of German adults

BackgroundMental health encompasses more than just the absence of mental disorders. Thus, a Mental Health Surveillance (MHS) and reporting system for Germany should monitor mental well-being in addition to psychopathology to capture a more complete picture of population mental health. The Warwick-Edinburgh Mental Well-Being Scale (WEMWBS) is an internationally established inventory for the integrated assessment of different aspects of mental well-being (i.e., hedonic and eudaimonic) in population samples that has not yet been validated for Germany.MethodsUsing data from a cross-sectional online survey of a convenience sample of N = 1.048 adults aged 18–79 years (51% female) living in Germany, the factorial structure, measurement invariance (age, sex) and psychometric properties of the WEMWBS in its long (14 items) and short (7 items) versions were analyzed. Additionally, correlations to relevant factors (e.g., health-related quality of life, psychological distress) were investigated as indicators of criterion validity.ResultsMeans of model fit indices did not confirm a unidimensional factor structure for either version. The three-factor-correlative models showed moderate to good fit while the bifactor model with one general mental well-being factor and three grouping factors fitted the data best. The full range of possible responses was used for all items, and the distribution of both scales was approximately normal. Moreover, the results revealed measurement invariance across sex and age groups. Initial evidence of criterion validity was obtained. Internal consistencies were α = 0.95 and α = 0.89, respectively. Average mental well-being was comparable to that of other European countries at 3.74 for the long version and 3.84 for the short version. While there were no differences by sex, comparisons between age groups revealed higher mental well-being among the older age groups.ConclusionsBoth versions of the WEMWBS showed sound psychometric characteristics in the present German sample. The findings indicate that the instrument is suitable for measuring mental well-being at the population level due to its distributional properties. These results are promising, suggesting that the scale is suitable for use in a national MHS that aims to capture positive mental health in the population as a foundation for prevention and promotion efforts within public mental health.

An Algebraic Approach to Light–Matter Interactions

AbstractA theoretical and computational framework for the study and engineering of light–matter interactions is reviewed in here. The framework rests on the invariance properties of electromagnetism, and is formalized in a Hilbert space whose conformally invariant scalar product provides connections to physical quantities, such as the energy or momentum of a given field, or the outcome of measurements. The light–matter interaction is modeled by the polychromatic scattering operator, which establishes a natural connection to a popular computational formalism, the transition matrix, or T‐matrix. This review contains a succinct yet comprehensive description of the main theoretical ideas, and illustrates some of the practical benefits of the approach.

Dynamic Sliding Mode Control of Spherical Bubble for Cavitation Suppression

Cavitation is a disadvantageous phenomenon that occurs when fluid pressure drops below its vapor pressure. Under these conditions, bubbles form in the fluid. When these bubbles flow into a high-pressure area or tube, they erupt, causing harm to mechanical parts such as centrifugal pumps. The difference in pressure in a fluid is the result of varying temperatures. One way to eliminate cavitation is to reduce the radius of the bubbles to zero before they reach high-pressure areas, using a robust approach. In this paper, sliding mode control is used for this purpose due to its invariance property. To force the radius of the bubbles toward zero and prevent chattering, a new dynamic sliding mode control approach is used. In dynamic sliding mode control, chattering is removed by passing the input control through a low-pass filter, such as an integrator. A general model of the spherical bubble is used, transferred to the state space, and then a state proportional-integral feedback is applied to obtain a linear system with a new input control signal. A comparison is also made with traditional sliding mode control using state feedback, providing a trusted comparison.

Rugged magneto-hydrodynamic invariants in weakly collisional plasma turbulence: Two-dimensional hybrid simulation results

Aims. We investigated plasma turbulence in the context of solar wind. We concentrated on properties of ideal second-order magneto-hydrodynamic (MHD) and Hall MHD invariants. Methods. We studied the results of a two-dimensional hybrid simulation of decaying plasma turbulence with an initial large cross helicity and a negligible magnetic helicity. We investigated the evolution of the combined energy and the cross, kinetic, mixed, and magnetic helicities. For the combined (kinetic plus magnetic) energy and the cross, kinetic, and mixed helicities, we analysed the corresponding Kármán-Howarth-Monin (KHM) equation in the hybrid (kinetic proton and fluid electron) approximation. Results. The KHM analysis shows that the combined energy decays at large scales. At intermediate scales, this energy cascades (from large to small scales) via the MHD non-linearity and this cascade partly continues via Hall coupling to sub-ion scales. The cascading combined energy is transferred (dissipated) to the internal energy at small scales via the resistive dissipation and the pressure-strain effect. The Hall term couples the cross helicity with the kinetic one, suggesting that the coupled invariant, referred to here as the mixed helicity, is a relevant turbulence quantity. However, when analysed using the KHM equations, the kinetic and mixed helicities exhibit very dissimilar behaviours to that of the combined energy. On the other hand, the cross helicity, in analogy to the energy, decays at large scales, cascades from large to small scales via the MHD+Hall non-linearity, and is dissipated at small scales via the resistive dissipation and the cross-helicity equivalent of the pressure-strain effect. In contrast to the combined energy, the Hall term is important for the cross helicity over a wide range of scales (even well above ion scales). In contrast, the magnetic helicity is scantily generated through the resistive term and does not exhibit any cascade.

Representing Noisy Image Without Denoising.

A long-standing topic in artificial intelligence is the effective recognition of patterns from noisy images. In this regard, the recent data-driven paradigm considers 1) improving the representation robustness by adding noisy samples in training phase (i.e., data augmentation) or 2) pre-processing the noisy image by learning to solve the inverse problem (i.e., image denoising). However, such methods generally exhibit inefficient process and unstable result, limiting their practical applications. In this paper, we explore a non-learning paradigm that aims to derive robust representation directly from noisy images, without the denoising as pre-processing. Here, the noise-robust representation is designed as Fractional-order Moments in Radon space (FMR), with also beneficial properties of orthogonality and rotation invariance. Unlike earlier integer-order methods, our work is a more generic design taking such classical methods as special cases, and the introduced fractional-order parameter offers time-frequency analysis capability that is not available in classical methods. Formally, both implicit and explicit paths for constructing the FMR are discussed in detail. Extensive simulation experiments and robust visual applications are provided to demonstrate the uniqueness and usefulness of our FMR, especially for noise robustness, rotation invariance, and time-frequency discriminability.

Fractional-Order Correlation between Special Functions Inspired by Bone Fractal Operators

In recent years, our research on biomechanical and biophysical problems has involved a series of symmetry issues. We found that the fundamental laws of the aforementioned problems can all be characterized by fractal operators, and each type of operator possesses rich invariant properties. Based on the invariant properties of fractal operators, we discovered that the symmetry evolution laws of functional fractal trees in the physical fractal space can reveal the intrinsic correlations between special functions. This article explores the fractional-order correlation between special functions inspired by bone fractal operators. Specifically, the following contents are included: (1) showing the intrinsic expression in the convolutional kernel function of bone fractal operators and its correlation with special functions; (2) proving the following proposition: the convolutional kernel function of bone fractal operators is still related to the special functions under different input signals (external load, external stimulus); (3) using the bone fractal operators as the background and error function as the core, deriving the fractional-order correlation between different special functions.

Structure of Multi-State Correlation in Electronic Systems.

Beyond the Hohenberg-Kohn density functional theory for the ground state, it has been established that the Hamiltonian matrix for a finite number (N) of lowest eigenstates is a matrix density functional. Its fundamental variable─the matrix density D(r)─can be represented by, or mapped to, a set of auxiliary, multiconfigurational wave functions expressed as a linear combination of no more than N2 determinant configurations. The latter defines a minimal active space (MAS), which naturally leads to the introduction of the correlation matrix functional, responsible for the electronic correlation effects outside the MAS. In this study, we report a set of rigorous conditions in the Hamiltonian matrix functional, derived by enforcing the symmetry of a Hilbert subspace, namely the subspace invariance property. We further establish a fundamental theorem on the correlation matrix functional. That is, given the correlation functional for a single state in the N-dimensional subspace, all elements of the correlation matrix functional for the entire subspace are uniquely determined. These findings reveal the intricate structure of electronic correlation within the Hilbert subspace of lowest eigenstates and suggest a promising direction for efficient simulation of excited states.

Differential Subordination and Coefficient Functionals of Univalent Functions Related to $\cos z$

Differential subordination in the complex plane is the generalization of a differential inequality on the real line. In this paper, we consider two subclasses of univalent functions associated with the trigonometric function $\cos z$. Using some properties of the hypergeometric functions, we determine the sharp estimate on the parameter $\beta$ such that the analytic function $p(z)$ satisfying $p(0)=1$, is subordinate to $\cos z$ when the differential expression $p(z)+\beta z (dp(z)/dz)$ is subordinate to the Janowski function. We compute sharp bounds on coefficient functional Hermitian--Toeplitz determinants of the third and the fourth order with an invariance property for such functions. In addition, we estimate bound on Hankel determinants of the second and the third order.

Characterization and Exploitation of the Rotational Memory Effect in Multimode Fibers

In an ideal perfectly straight multimode fiber with a circular core, the symmetry ensures that rotating the input wave front leads to a corresponding rotation of the output wave front. This invariant property, known as the rotational memory effect (RME), remains independent of the typically unknown output profile. The RME thus offers significant potential for imaging and telecommunication applications. However, in real-life fibers, this effect is degraded by intrinsic imperfections and external perturbations, and is challenging to observe because of its acute sensitivity to misalignments and aberrations in the optical setup. Building on a previously established method for precisely estimating fiber transmission properties, we demonstrate an accurate extraction of RME properties. Additionally, we introduce a comprehensive theoretical framework for both qualitative and quantitative analysis, which specifically links the angular-dependent correlation of the RME to the core deformation’s geometrical properties and the fiber’s mode characteristics. Our theoretical predictions align well with experimental data and simulations for various amounts of fiber distorsion. Finally, we demonstrate the ability to engineer wave fronts with significantly enhanced correlation across all rotation angles. This work enables accurate characterization of distributed disorder from the fabrication process and facilitates calibration-free imaging in multimode fibers. Published by the American Physical Society 2024

Hydrogel Composition Effects on Performance as Single-Walled Carbon Nanotube Purification Media.

Hydrogel microsphere media allows for postsynthetic purification of single-walled carbon nanotubes (SWNTs), affording characterization and application of their unique (n,m) chirality-dependent properties. This work reports the characterization of five hydrogel resins, Sephacryl S-100, S-200, S-300, S-400, and S-500, and the implementation of each as a SWNT purification medium. The physiochemical properties of each resin were explored spectroscopically through elemental analyses and with both light and electron microscopy. Both surface porosity and hydrogel swelling ratio were found to increase as the concentration of component allyl dextran (aDEX) decreased, each with an increasing Sephacryl S-number. Conversely, invariant properties included a hydrogel microsphere size distribution and concentrations of components methylenebisacrylamide and ammonium persulfate. When employed within gel-based SWNT purification schemes in overloading conditions, Sephacryl formulations of larger S-number adsorbed fewer SWNTs, but the chirality dependence of SWNT adsorption and elution was approximately consistent across all resins. In underloading conditions, approximately one-third of introduced SWNTs passed through each resin unabsorbed, while the resins showed varying chirality-dependent adsorption efficiencies. These observations collectively identify aDEX-rich gel regions as being responsible for SWNT purification, along with a SWNT-exclusive parameter other than chirality (speculated as length) that convolutes the effectiveness of gel-based single-chirality purification.

Cooperative coevolution for non-separable large-scale black-box optimization: Convergence analyses and distributed accelerations

Given the ubiquity of non-separable optimization problems in real worlds, in this paper we analyze and extend the large-scale version of the well-known cooperative coevolution (CC), a divide-and-conquer black-box optimization framework, on non-separable functions. First, we reveal empirical reasons of when decomposition-based methods are preferred or not in practice on some non-separable large-scale problems, which have not been clearly pointed out in many previous CC papers. Then, we formalize CC to a continuous-game model via simplification, but without losing its essential property. Different from previous evolutionary game theory for CC, our new model provides a much simpler but useful viewpoint to analyze its convergence, since only the pure Nash equilibrium concept is needed and more general fitness landscapes can be explicitly considered. Based on convergence analyses, we propose a hierarchical decomposition strategy for better generalization, as for any decomposition, there is a risk of getting trapped into a suboptimal Nash equilibrium. Finally, we use powerful distributed computing to accelerate it under the recent multi-level learning framework, which combines the fine-tuning ability from decomposition with the invariance property of CMA-ES. Experiments on a set of high-dimensional test functions validate both its search performance and scalability (w.r.t. CPU cores) on a clustering computing platform with 400 CPU cores.

Pivotal-Aware Principal Component Analysis.

A conventional principal component analysis (PCA) frequently suffers from the disturbance of outliers, and thus, spectra of extensions and variations of PCA have been developed. However, all the existing extensions of PCA derive from the same motivation, which aims to alleviate the negative effect of the occlusion. In this article, we design a novel collaborative-enhanced learning framework that aims to highlight the pivotal data points in contrast. As for the proposed framework, only a part of well-fitting samples are adaptively highlighted, which indicates more significance during training. Meanwhile, the framework can collaboratively reduce the disturbance of the polluted samples as well. In other words, two contrary mechanisms could work cooperatively under the proposed framework. Based on the proposed framework, we further develop a pivotal-aware PCA (PAPCA), which utilizes the framework to simultaneously augment positive samples and constrain negative ones by retaining the rotational invariance property. Accordingly, extensive experiments demonstrate that our model has superior performance compared with the existing methods that only focus on the negative samples.

Invariant correlation under marginal transforms

A useful property of independent samples is that their correlation remains the same after applying marginal transforms. This invariance property plays a fundamental role in statistical inference, but does not hold in general for dependent samples. In this paper, we study this invariance property on the Pearson correlation coefficient and its applications. A multivariate random vector is said to have an invariant correlation if its pairwise correlation coefficients remain unchanged under any common marginal transforms. For a bivariate case, we characterize all models of such a random vector via a certain combination of comonotonicity—the strongest form of positive dependence—and independence. In particular, we show that the class of exchangeable copulas with invariant correlation is precisely described by what we call positive Fréchet copulas. In the general multivariate case, we characterize the set of all invariant correlation matrices via the clique partition polytope. We also propose a positive regression dependent model that admits any prescribed invariant correlation matrix. Finally, we show that all our characterization results of invariant correlation, except one special case, remain the same if the common marginal transforms are confined to the set of increasing ones.

(α+2)-Dimensional fractional evolution equation: group classification, symmetries, reduction and conservation laws

ABSTRACT A preliminary group classification based on symmetry operators is applied to study invariance properties of the time-fractional ( α + 2 ) -dimensional fractional evolution equation. The concepts of Riemann–Liouville and Caputo fractional derivatives are used in this study. The similarity variables obtained from symmetries and one-dimensional optimal systems of constructed Lie algebras are used in order to find the group-invariant solutions of the equation. Finally conservation laws of the equation are derived via a modified version of Noether’s theorem.

Invariant neural architecture for learning term synthesis in instantiation proving

The development of strong CDCL-based propositional (SAT) solvers has greatly advanced several areas of automated reasoning (AR). One of the directions in AR is therefore to make use of SAT solvers in expressive formalisms such as first-order logic, for which large corpora of general mathematical problems exist today. This is possible due to Herbrand's theorem, which allows reduction of first-order problems to propositional problems by instantiation. The core challenge is synthesizing the appropriate instances from the typically infinite Herbrand universe.In this work, we develop a machine learning system targeting this task, addressing its combinatorial and invariance properties. In particular, we develop a GNN2RNN architecture based on a graph neural network (GNN) that learns from problems and their solutions independently of many symmetries and symbol names (addressing the abundance of Skolems), combined with a recurrent neural network (RNN) that proposes for each clause its instantiations. The architecture is then combined with an efficient ground solver and, starting with zero knowledge, iteratively trained on a large corpus of mathematical problems. We show that the system is capable of solving many problems by such educated guessing, finding proofs for 32.12% of the training set. The final trained system solves 19.74% of the unseen test data on its own. We also observe that the trained system finds solutions that the iProver and CVC5 systems did not find.

Multi-Target Pairing Method Based on PM-ESPRIT-like DOA Estimation for T/R-R HFSWR

The transmit/receive-receive (T/R-R) synergetic High Frequency Surface Wave Radar (HFSWR) has increasingly attracted attention due to its high localization accuracy, but multi-target pairing needs to be performed before localization in multi-target scenarios. However, existing multi-target parameter matching methods have primarily focused on track association, which falls under the category of information-level fusion techniques, with few methods based on detected points. In this paper, we propose a multi-target pairing method with high computational efficiency based on angle information for T/R-R synergetic HFSWR. To be more specific, a dual-receiving array signal model under long baseline condition is firstly constructed. Then, the amplitude and phase differences of the same target reaching two subarrays are calculated to establish the cross-correlation matrix. Subsequently, in order to extract the rotation factor matrices containing pairing information and improve angle estimation performance, we utilize the conjugate symmetry properties of the uniform linear array (ULA) manifold matrix for generalized virtual aperture extension. Ultimately, azimuths estimation and multi-target pairing are accomplished by combining the propagator method (PM) and the ESPRIT algorithm. The proposed method relies solely on angle information for multi-target pairing and leverages the rotational invariance property of Vandermonde matrices to avoid peak searching or iterations, making it computationally efficient. Furthermore, the proposed method maintains superb performance regardless of whether the spatial angles are widely separated or very close. Simulation results validate the effectiveness of the proposed method.

A new family of copulas based on probability generating functions

Abstract We propose a method to obtain a new class of copulas using a probability generating function (PGF) of positive-integer valued random variable. Some existing copulas in the literature are sub-families of the proposed copulas. Various dependence measures and invariant property of the tail dependence coefficient under PGF transformation are also discussed. We propose an algorithm for generating random numbers from the PGF copula. The bivariate concavity properties, such as Schur concavity and quasi-concavity, associated with the PGF copula are studied. Two new generalized FGM copulas are introduced using PGFs of geometric and discrete Mittag-Leffler distributions. The proposed two copulas improved the Spearman’s rho of FGM copula by (−0.3333, 0.4751) and (−0.3333, 0.9573). Finally, we analyse a real dataset to illustrate the practical application of the proposed copulas.