Invariant Space Research Articles

A Fock space structure for the diffeomorphism invariant Hilbert space of loop quantum gravity and its applications

Loop quantum gravity (LQG) is a quantization program for gravity based on the principles of QFT and general covariance of general relativity. Quantum states of LQG describe gravitational excitations based on graphs embedded in a spatial slice of spacetime. We show that, under certain assumptions on the class of diffeomorphisms, the space of diffeomorphism invariant states carries a Fock space structure. The role of one-particle excitations for this structure is played by the diffeomorphism invariant states based on graphs with a single (linked) component. This means, however, that a lot of the structure of the diffeomorphism invariant Hilbert space remains unresolved by this structure. We show how the Fock structure allows to write at least some condensate states of group field theory as diffeomorphism invariant coherent states of LQG in a precise sense. We also show how to construct other interesting states using this Fock structure. We finally explore the quantum geometry of single- and multi-particle states and tentatively observe some resemblance to geometries with a single or multiple components, respectively.

Generalized Stević-Sharma operators from the minimal Möbius invariant space into Bloch-type spaces

Abstract The aim of this study is to investigate the boundedness, essential norm, and compactness of generalized Stević-Sharma operator from the minimal Möbius invariant space into Bloch-type space.

Implementation of a triangular probabilistic distribution for optimal parametrization of the SEIR model recovery rates with delay.

Dynamical analysis of epidemiological models received significant attention after the global disaster of the Covid-19 pandemic. Although much of the attention has focused on choosing the appropriate model to describe and interpret the epidemiological data and on predicting the spread of the disease, working on uncovering the hidden dynamics by analyzing the predictive data and studying the algebraic properties of the models such as the invariant spaces, the center manifolds, and the Lyapunov functions in the general parameter space still have the potential to make a significant contribution to our understanding of the dynamics of spread. In this paper, in order to determine the recovery rate based on characteristic model parameters, the triangular probabilistic distribution is implemented in an epidemic delay differential equation with delays. By defining generic coefficients derived from field surveys and population characteristics, we propose a way to reduce the number of model parameters that need to be obtained during model fitting in order to find an effective way to handle the cultural and physiological diversity of societies and isolate the effect of counter measures for the epidemics.

Riesz transform associated with the fractional Fourier transform and applications in image edge detection

The fractional Hilbert transform was introduced by Zayed [30, Zayed, 1998] and has been widely used in signal processing. In view of its connection with the fractional Fourier transform, Chen, the first, second and fourth authors of this paper in [6, Chen et al., 2021] studied the fractional Hilbert transform and other fractional multiplier operators on the real line. The present paper is concerned with a natural extension of the fractional Hilbert transform to higher dimensions: this extension is the fractional Riesz transform and is given by multiplication which a suitable chirp function on the fractional Fourier transform side. In addition to a thorough study of the fractional Riesz transform, in this work we also investigate the boundedness of singular integral operators with chirp functions on rotation invariant spaces, chirp Hardy spaces and their relation to chirp BMO spaces, as well as applications of the theory of fractional multipliers in partial differential equations. Through numerical simulation, we provide physical and geometric interpretations of high-dimensional fractional multipliers. Finally, we present an application of the fractional Riesz transforms in edge detection which verifies a hypothesis insinuated in [26, Xu et al., 2016]. In fact our numerical implementation confirms that amplitude, phase, and direction information can be simultaneously extracted by controlling the order of the fractional Riesz transform.

Optimal Gagliardo–Nirenberg interpolation inequality for rearrangement invariant spaces

We prove the optimality of the Gagliardo–Nirenberg inequality: ‖∇u‖X≲‖∇2u‖Y1/2‖u‖Z1/2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Vert \ abla u\\Vert _{X}\\lesssim \\Vert \ abla ^2 u\\Vert _Y^{1/2}\\Vert u\\Vert _Z^{1/2}, \\end{aligned}$$\\end{document}where Y and Z are rearrangement invariant Banach function spaces, and X = Y^{1/2}Z^{1/2} is the Calderón–Lozanovskii space. By optimality, we mean that for a certain pair of spaces on the right-hand side, it is not possible to reduce the space on the left-hand side while remaining in the class of rearrangement invariant spaces. Our result establishes the optimality for Lorentz and Orlicz spaces, surpassing previous findings. Additionally, we discuss the significance of pointwise inequalities and present a counterexample that prohibits further improvements.

On the values taken by slice torus invariants

AbstractWe study the space of slice torus invariants. In particular we characterise the set of values that slice torus invariants may take on a given knot in terms of the stable smooth slice genus. Our study reveals that the resolution of the local Thom conjecture implies the existence of slice torus invariants without having to appeal to any explicit construction from a knot homology theory.

Switching quantum reference frames in the N-body problem and the absence of global relational perspectives

Given the importance of quantum reference frames (QRFs) to both quantum and gravitational physics, it is pertinent to develop a systematic method for switching between the descriptions of physics relative to different choices of QRFs, which is valid in both fields. Here we continue with such a unifying approach, begun in [Quantum 4, 225 (2020)], whose key ingredient is a symmetry principle, which enforces physics to be relational. Thanks to gauge related redundancies, this leads to a perspective-neutral structure which contains all frame choices at once and via which frame perspectives can be consistently switched. Formulated in the language of constrained systems, the perspective-neutral structure is the constraint surface classically and the gauge invariant Hilbert space in the Dirac quantized theory. By contrast, a perspective relative to a specific frame corresponds to a gauge choice and the associated reduced phase and Hilbert space. QRF changes thus amount to a gauge transformation. We show that they take the form of `quantum coordinate changes'. We illustrate this in a general mechanical model, namely the relational N-body problem in 3D space with rotational and translational symmetry. This model is especially interesting because it features the Gribov problem so that globally valid gauge fixing conditions, and hence relational frame perspectives, are absent. The constraint surface is topologically non-trivial and foliated by 3-, 5- and 6-dimensional gauge orbits, where the lower dimensional orbits are a set of measure zero. The N-body problem also does not admit globally valid canonically conjugate pairs of Dirac observables. These challenges notwithstanding, we exhibit how one can construct the QRF transformations for the 3-body problem. Our construction also sheds new light on the generic inequivalence of Dirac and reduced quantization through its interplay with QRF perspectives.

A unified framework for visual domain adaptation with covariance matching

In the field of Machine Learning, it is widely acknowledged that training and test data should ideally stem from the same source and distribution. However, in the real world, this is not always feasible. Domain Adaptation (DA) techniques address this issue by adapting a classifier trained on annotated source domain data on unannotated target domain data while minimizing the impact of domain shift. Many recent DA approaches concentrate on learning a latent feature space that remains invariant to domain shift by mitigating various statistical and geometrical divergences. Although these methods have demonstrated effectiveness, they frequently overlook a crucial aspect of learning a domain-invariant discriminative latent feature space across diverse domains. To address this issue, we propose a novel framework called Unified Framework for Visual Domain Adaptation with Covariance Matching (UDACM) that aims to learn a domain invariant and class discriminative latent feature space by incorporating multiple objectives such as maximizing target domain variance, minimizing distribution and subspace divergence, performing manifold learning, preserving discriminative information of both the source and target domains, and measuring and matching covariance simultaneously. In our proposed framework, the measuring and matching of covariance play a crucial role in ensuring the learning of discriminative latent feature space by aligning the within-class and between-class covariance matrices of the source and target domains simultaneously. Detailed experiments demonstrate the outperformance of UDACM on baseline datasets such as CMU-PIE, Office+Caltech10, USPS+MNIST, VLCS, and Office–Home, compared to various established primitive, shallow, and deep domain adaptation methods on several image classification tasks.

Lorentz and gauge invariance of quantum space

Motivated by the generalized uncertainty principle, we derive a discrete picture of the space that respects Lorentz symmetry as well as gauge symmetry by setting an equivalency between the linear Generalized Uncertainty Principle (GUP) correction term and electromagnetic interaction term in the Dirac equation. We derived a wave function solution that satisfies this equivalency. This discreteness may explain the crystal and quasicrystal structures observed in nature at different energy scales.

A Carleson type measure and a family of Möbius invariant function spaces

For 0<s<1, let {zn} be a sequence in the open unit disk such that ∑n(1−|zn|2)sδzn is an s-Carleson measure. In this paper, we consider the connections between this s-Carleson measure and the theory of Möbius invariant F(p, p-2, s) spaces by the Volterra type operator, the reciprocal of a Blaschke product, and second order complex differential equations having a prescribed zero sequence.

Cones generated by a generalized fractional maximal function

The paper considers the space of generalized fractional-maximal function, constructed on the basis of a rearrangement-invariant space. Two types of cones generated by a nonincreasing rearrangement of a generalized fractional-maximal function and equipped with positive homogeneous functionals are constructed. The question of embedding the space of generalized fractional-maximal function in a rearrangement invariant space is investigated. This question reduces to the embedding of the considered cone in the corresponding rearrangement-invariant spaces. In addition, conditions for covering a cone generated by a generalized fractional-maximal function by the cone generated by generalized Riesz potentials are given. Cones from non-increasing rearrangements of generalized potentials were previously considered in the works of M. Goldman, E. Bakhtigareeva, G. Karshygina and others.

Explicit consistency conditions for fully symmetric cubature on the tetrahedron

A novel fully symmetric basis is derived for the S4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${S}_4$$\\end{document}-invariant polynomial space, by using symmetric polynomials and invariant theory. This new basis enables deriving explicitly the consistency conditions for non-overdeterminedness of moment equations in the case of fully symmetric cubature rules on the tetrahedron. Solving the corresponding linear integer programming problem, optimal and quasi-optimal rule structures are derived. Explicit formulas to calculate the estimated lower bounds in the number of integration points are also given. Additionally, the new basis is of practical interest in calculating specific cubature rules, since it allows decomposing the moment equations into a series of successively independent smaller subsystems, which can be exploited in designing more efficient solution methods. Solving the moment equations analytically we obtain several interesting new results.

Pseudodifferential operators with completely periodic symbols

Motivated by the recent paper of Boggiatto–Garello (J Pseudo-Differ Oper Appl 11:93–117, 2020) where a Gabor operator is regarded as pseudodifferential operator with symbol p(x,omega ) periodic on both the variables, we study the continuity and invertibility, on general time frequency invariant spaces, of pseudodifferential operators with completely periodic symbol and general tau quantization.

The pro-p-Iwahori invariants of the universal quotient representation for GL2(F)

Let [Formula: see text] be a finite extension of [Formula: see text]. Every supersingular mod [Formula: see text] representation of [Formula: see text] is a quotient of a certain universal module, say [Formula: see text] The space of pro-[Formula: see text]-Iwahori invariants of [Formula: see text] is instrumental in classifying the supersingular mod [Formula: see text] representations of [Formula: see text]. For [Formula: see text] the space of pro-[Formula: see text]-Iwahori invariants of [Formula: see text] has been determined by Schein and Hendel using the spherical Hecke algebra. An alternative way to study mod [Formula: see text] representations of [Formula: see text] is introduced by Anandavardhanan and Borisagar using the Iwahori–Hecke algebra. In this paper, we give a canonical isomorphism between the universal modules for [Formula: see text] obtained from the spherical approach and the Iwahori–Hecke approach. We also compute an explicit basis of the pro-[Formula: see text]-Iwahori invariant space of [Formula: see text] for [Formula: see text] using the Iwahori–Hecke approach. This method allows us to extend Hendel’s computation slightly in the range of [Formula: see text]-adic digits from [Formula: see text] to [Formula: see text]

Hofstadter Topology with Real Space Invariants and Reentrant Projective Symmetries.

Adding magnetic flux to a band structure breaks Bloch's theorem by realizing a projective representation of the translation group. The resulting Hofstadter spectrum encodes the nonperturbative response of the bands to flux. Depending on their topology, adding flux can enforce a bulk gap closing (a Hofstadter semimetal) or boundary state pumping (a Hofstadter topological insulator). In this Letter, we present a real space classification of these Hofstadter phases. We give topological indices in terms of symmetry-protected real space invariants, which reveal the bulk and boundary responses of fragile topological states to flux. In fact, we find that the flux periodicity in tight-binding models causes the symmetries which are broken by the magnetic field to reenter at strong flux where they form projective point group representations. We completely classify the reentrant projective point groups and find that the Schur multipliers which define them are Arahanov-Bohm phases calculated along the bonds of the crystal. We find that a nontrivial Schur multiplier is enough to predict and protect the Hofstadter response with only zero-flux topology.

On quarkonium masses in 3D non-commutative space

We modify the calculation of quarkonium masses using the radial WKB and Pekeris-type approximations for the case of three-dimensional, rotationally invariant non-commutative space. We obtain corrections to the charmonium ({text {c}}bar{{text {c}}}), bottomonium ({text {b}}bar{{text {b}}}) and bottom-charmed meson ({text {c}}bar{{text {b}}}) masses due to the discrete short distance structure of the space introduced by the space non-commutativity. For the fundamental length at the Planck scale we obtain relative correction at the order of 10^{-39}, and taking into account the current experimental data, we obtain the upper bound at the order of {10^{-18},{{text {m}}}} for the fundamental length scale of space.

Preservation of Quadratic Invariants by Semiexplicit Symplectic Integrators for Nonseparable Hamiltonian Systems

We prove that the recently developed semiexplicit symplectic integrators for non-separable Hamiltonian systems preserve any linear and quadratic invariants possessed by the Hamiltonian systems. This is in addition to being symmetric and symplectic as shown in our previous work; hence it shares the crucial structure-preserving properties with some of the well-known symplectic Runge--Kutta methods such as the Gauss--Legendre methods. The proof follows two steps: First we show how the extended Hamiltonian system proposed by Pihajoki inherits linear and quadratic invariants in the extended phase space from the original Hamiltonian system. Then we show that this inheritance in turn implies that our integrator preserves the original linear and quadratic invariants in the original phase space. We also analyze preservation/non-preservation of these invariants by Tao's extended Hamiltonian system and the extended phase space integrators of Pihajoki and Tao. The paper concludes with numerical demonstrations of our results using a simple test case and a system of point vortices.

Gradient continuity for p(x)-Laplacian systems under minimal conditions on the exponent

We consider solutions of p(x)-Laplacian systems with coefficients and we show that their gradient is continuous provided that the variable exponent has distributional gradient belonging to the Lorentz-Zygmund space Ln,1log⁡L and that the gradient of the coefficient belongs to the Lorentz space Ln,1. The result is new since the use of the sharp Sobolev embedding in rearrangement invariant spaces does not ensure the unique (up to now) known assumption for such result, namely the log-Dini continuity of p(⋅) and the plain Dini continuity of the coefficient. Our approach relies on perturbation arguments and allows to slightly improve results in dimension two even for the case where p(⋅) is constant.

DI-NIDS: Domain invariant network intrusion detection system

The performance of machine learning based network intrusion detection systems (NIDSs) severely degrades when deployed on a network with significantly different feature distributions from the ones of the training dataset. In various applications, such as computer vision, domain adaptation techniques have been successful in mitigating the gap between the distributions of the training and test data. In the case of network intrusion detection however, the state-of-the-art domain adaptation approaches have had limited success. According to recent studies, as well as our own results, the performance of an NIDS considerably deteriorates when the ‘unseen’ test dataset does not follow the training dataset distribution. In order to enhance the generalisability of machine learning based network intrusion detection systems, we propose to extract domain invariant features using adversarial domain adaptation from multiple network domains, and then apply an unsupervised technique for recognising abnormalities, i.e., intrusions. More specifically, we train a domain adversarial neural network on labelled source domains, extract the domain invariant features, and train a One-Class SVM (OSVM) model to detect anomalies. At test time, we feedforward the unlabelled test data to the feature extractor network to project it into a domain invariant space, and then apply OSVM on the extracted features to achieve our final goal of detecting intrusions. Our extensive experiments on the NIDS benchmark datasets of NFv2-CIC-2018 and NFv2-UNSW-NB15 show that our proposed setup demonstrates superior cross-domain performance in comparison to the previous approaches.

Approximation by linear combinations of translates in invariant Banach spaces of tempered distributions via Tauberian conditions

This paper describes an approximation theoretic approach to the problem of completeness of a set of translates of a “Tauberian generator”, which is an integrable function whose Fourier transform does not vanish. This is achieved by the construction of finite rank operators, whose range is contained in the linear span of the translates of such a generator, and which allow uniform approximation of the identity operator over compact sets of certain Banach spaces (B,‖⋅‖B). The key assumption is availability of a double module structure on (B,‖⋅‖B), meaning the availability of sufficiently many smoothing operators (via convolution) and also pointwise multipliers, allowing localization of its elements. This structure is shared by a wide variety of function spaces and allows us to make explicit use of the Riesz–Kolmogorov Theorem characterizing compact subsets in such Banach spaces.The construction of these operators is universal with respect to large families of such Banach spaces, i.e. they do not depend on any further information concerning the particular Banach space. As a corollary we conclude that the linear span of the set of the translates of such a Tauberian generator is dense in any such space (B,‖⋅‖B). Our work has been inspired by a completeness result of V. Katsnelson which was formulated in the context of specific Hilbert spaces within this family and Gaussian generators.