Invariant Version Research Articles

The Asymptotic Expansion of the Spacetime Metric at the Event Horizon

AbstractHawking’s local rigidity theorem, proven in the smooth setting by Alexakis-Ionescu-Klainerman, says that the event horizon of any stationary non-extremal black hole is a non-degenerate Killing horizon. In this paper, we prove that the full asymptotic expansion of any smooth vacuum metric at a non-degenerate Killing horizon is determined by the geometry of the horizon. This gives a new perspective on the black hole uniqueness conjecture. In spacetime dimension 4, we also prove an existence theorem: Given any non-degenerate horizon geometry, Einstein’s vacuum equations can be solved to infinite order at the horizon in a unique way (up to isometry). The latter is a gauge invariant version of Moncrief’s classical existence result, without any restriction on the topology of the horizon. In the real analytic setting, the asymptotic expansion is shown to converge and we get well-posedness of this characteristic Cauchy problem.

3DReact: Geometric Deep Learning for Chemical Reactions.

Geometric deep learning models, which incorporate the relevant molecular symmetries within the neural network architecture, have considerably improved the accuracy and data efficiency of predictions of molecular properties. Building on this success, we introduce 3DReact, a geometric deep learning model to predict reaction properties from three-dimensional structures of reactants and products. We demonstrate that the invariant version of the model is sufficient for existing reaction data sets. We illustrate its competitive performance on the prediction of activation barriers on the GDB7-22-TS, Cyclo-23-TS, and Proparg-21-TS data sets in different atom-mapping regimes. We show that, compared to existing models for reaction property prediction, 3DReact offers a flexible framework that exploits atom-mapping information, if available, as well as geometries of reactants and products (in an invariant or equivariant fashion). Accordingly, it performs systematically well across different data sets, atom-mapping regimes, as well as both interpolation and extrapolation tasks.

Can the dark core of personality be measured briefly, multidimensionally, and invariantly? The D25 measure

The dark core of personality is a recent construct that encompasses the malevolent, antagonistic, and aversive side of human nature that represents the common inner core of dark personality traits. The aim of the present study is to develop a short, multidimensional, and invariant version of the dark core of personality D70 measure. Two samples of participants were used in the current study (N = 570, M = 35.42 years, SD = 11.41, range = 18–69; N = 242, M = 30.19 years, SD = 12.78, range = 16–77). The new D25 measure was developed, consisting of five factors – Callousness, Deceitfulness, Vindictiveness, Narcissistic Entitlement, and Sadism. All the structural models examined obtained adequate fits (1-factor, 5-factor, 5-factor bifactor, 5-factor 2nd order), but the 5-factor 2nd order model was considered the best option. Internal consistency/reliability, as measured by the alpha and omega coefficients and more traditional indicators, was mainly good. Construct validity (e.g., with moral disengagement, low self-control, and empathy measures) and criterion-related validity (e.g., with justice-involvement variables) were demonstrated. Scalar measurement invariance across gender and age was established, with male participants and younger participants scoring significantly higher than female participants and older participants. The findings provide support for the use of the D25 as a valid and reliable short multidimensional measure of the dark core of personality construct.

Revisiting the second Vassiliev (In)variant for polymer knots

Knots in open strands such as ropes, fibers, and polymers, cannot typically be described in the language of knot theory, which characterizes only closed curves in space. Simulations of open knotted polymer chains, often parameterized to DNA, typically perform a closure operation and calculate the Alexander polynomial to assign a knot topology. This is limited in scenarios where the topology is less well-defined, for example when the chain is in the process of untying or is strongly confined. Here, we use a discretized version of the Second Vassiliev Invariant for open chains to analyze Langevin Dynamics simulations of untying and strongly confined polymer chains. We demonstrate that the Vassiliev parameter can accurately and efficiently characterize the knotted state of polymers, providing additional information not captured by a single-closure Alexander calculation. We discuss its relative strengths and weaknesses compared to standard techniques, and argue that it is a useful and powerful tool for analyzing polymer knot simulations.

Group invariant variational principles

In this paper we introduce a group invariant version of the well-known Ekeland variational principle. To achieve this, we define the concept of convexity with respect to a group and establish a version of the theorem within this framework. Additionally, we present several consequences of the group invariant Ekeland variational principle, including Palais-Smale minimizing sequences, the Brønsted-Rockafellar theorem, and a characterization of the linear and continuous group invariant functionals space. Moreover, we provide an alternative proof of the Bishop-Phelps theorem and proofs for the group-invariant Hahn-Banach separating theorems. Finally, we discuss some implications and applications of these results.

Heat Kernel of Networks with Long-Range Interactions

The heat kernel associated with a discrete graph Laplacian is the basic solution to the heat diffusion equation of a strict graph or network. In addition, this kernel represents the heat transfer that occurs over time across the network edges. Its computation involves exponentiating the Laplacian eigensystem with respect to time. In this paper, we expand upon this concept by considering a novel network-theoretic approach developed in recent years, which involves defining the k-path Laplacian operator for networks. Prior studies have adopted the notion of integrating long-range interactions (LRI) in the transmission of “information” across the nodes and edges of the network. Various methods have been employed to consider long-range interactions. We explore here the incorporation of long-range interactions in network analysis through the use of Mellin and Laplace transforms applied to the k-path Laplacian matrix. The contribution of this paper is the computation of the heat kernel associated with the k-path Laplacian, called the generalized heat kernel (GHK), and its employment as the basis for extracting stable and useful novel versions of invariants for graph characterization. The results presented in this paper demonstrate that the use of LRI improves the results obtained with classical diffusion methods for networks characterization.

On the direct quantization of Proca gauge invariant field

In this study, we use the generalized integration constants method “GCI” [1,2] to quantize two fields with constraints. Indeed, Proca field, that describes a massive boson is studied first, based on a Lagrangian without gauge symmetry, in order to determine the necessary brackets for its canonical quantization. Then, another gauge invariant version of this field, obtained by adding an additional scalar field in the mass term, is quantized after fixing this gauge.”

Extension of monotone operators and Lipschitz maps invariant for a group of isometries

Abstract We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms, and we show that they always admit a maximal extension which preserves the same invariance. A similar result applies to Lipschitz maps in Hilbert spaces, thus providing an invariant version of Kirszbraun–Valentine extension theorem. We then provide a relevant application to the case of monotone operators in $L^{p}$ -spaces of random variables which are invariant with respect to measure-preserving isomorphisms, proving that they always admit maximal dissipative extensions which are still invariant by measure-preserving isomorphisms. We also show that such operators are law invariant, a much stronger property which is also inherited by their resolvents, the Moreau–Yosida approximations, and the associated semigroup of contractions. These results combine explicit representation formulae for the maximal extension of a monotone operator based on self-dual Lagrangians and a refined study of measure-preserving maps in standard Borel spaces endowed with a nonatomic measure, with applications to the approximation of arbitrary couplings between measures by sequences of maps.

Two-weight dyadic Hardy inequalities

We present various results concerning the two-weight Hardy inequality on infinite trees. Our main aim is to survey known characterizations (and proofs) for trace measures, as well as to provide some new ones. Also for some of the known characterizations we provide here new proofs. In particular, we obtain a new characterization in terms of a reverse Hölder inequality for trace measures, and one based on the well-known Muckenhoupt–Wheeden–Wolff inequality, of which we here give a new probabilistic proof. We provide a new direct proof for the so-called isocapacitary characterization and a new simple proof, based on a monotonicity argument, for the so-called mass-energy characterization. Furthermore, we introduce a conformally invariant version of the two-weight Hardy inequality, characterize the compactness of the Hardy operator, provide a list of open problems, and suggest some possible lines of future research.

Cross invariance, the Shapley value, and the Shapley–Shubik power index

In this paper we propose a simple axiom which, along with the axioms of additivity (transfer) and dummy player, characterizes the Shapley value (the Shapley–Shubik power index) on the domain of TU (simple) games. The new axiom, cross invariance, demands payoff invariance on symmetric players across “quasi-symmetric games,” that is, games where excluding null players, all players are symmetric. Additionally, we demonstrate that the axiom of additivity can be replaced by a new axiom called strong monotonicity, or it can be completely dropped if a stronger version of cross invariance is employed. We also show that the weighted Shapley values can be characterized using a weighted variant of cross invariance. Efficiency is derived rather than assumed in our characterizations. This fresh perspective contributes to a deeper understanding of the Shapley value and its applicability.

Graded group actions and generalized H-actions compatible with gradings

We introduce the notion of a graded group action on a graded algebra or, which is the same, a group action by graded pseudoautomorphisms. An algebra with such an action is a natural generalization of an algebra with a super- or a pseudoinvolution. We study groups of graded pseudoautomorphisms, show that the Jacobson radical of a group graded finite dimensional associative algebra A over a field of characteristic 0 is stable under graded pseudoautomorphisms, prove the invariant version of the Wedderburn–Artin Theorem and the analog of Amitsur's conjecture for the codimension growth of graded polynomial G-identities in such algebras A with a graded action of a group G.

Quantum Continuants, Quantum Rotundus and Triangulations of Annuli

We give enumerative interpretations of the polynomials arising as numerators and denominators of the $q$-deformed rational numbers introduced by Morier-Genoud and Ovsienko. The considered polynomials are quantum analogues of the classical continuants and of their cyclically invariant versions called rotundi. The combinatorial models involve triangulations of polygons and annuli. We prove that the quantum continuants are the coarea-generating functions of paths in a triangulated polygon and that the quantum rotundi are the (co)area-generating functions of closed loops on a triangulated annulus.

First order chemical reaction response in mixed convective Falkner-Skan Sutterby fluid with Cattaneo-Christov heat and mass flux model

The complex phenomenon of heat transfer plays a pivotal role in a broad spectrum of mechanical and industrial applications, ranging from space and nuclear reactor cooling to medicinal applications like magnetic drug targeting, energy production, heat conduction in biological tissues, among others. Historically, Fourier's law of heat conduction has served as a predictive model for heat transfer in numerous practical scenarios. However, this model's limitation is its generation of a parabolic energy expression, implying an immediate influence on the system by any initial disturbance. Numerous researchers have attempted to address this issue, leading to the Cattaneo's modification to Fourier's law, which introduces a relaxation time for heat flux, meaning the time needed to achieve stable heat conduction following the imposition of a temperature gradient. Christov later tweaked this model into a material invariant version, integrating the upper-connected derivative from the Oldroyd model. These models, now collectively termed as the Cattaneo-Christov (CC) model, form the core of the present study. In this study, we explore the mixed convective MHD Falkner-Skan Suttberby nanofluid flow towards a wedge surface, in the presence of a mixed convection. A two-phase nanofluid based C–C model is utilized to apply the nano-concept. The nonlinear equation system is solved using the bvp4c technique in MATLAB. The effects of various parameters including the magnetic field strength, wedge angle, nanoparticle volume fraction, and Prandtl number on the velocity and temperature profiles are discussed. The results show that the velocity increases with the magnetic parameter, wedge angle, and nanoparticle fraction, while the temperature decreases or increases depending on the suction or injection conditions. The Nusselt numbers and solutal transport rate are also reported for different parameter values. Overall, the study demonstrates that the non-Fourier C–C model yields significantly different solutal and heat transfer characteristics compared to the classical Fourier law. The results provide useful insights into the heat transfer behavior of nanofluids in stagnation point flows over geometries with complex shapes, which have practical applications in areas such as aerodynamics and thermal management of electronic devices.

A new Schwarz-Pick lemma at the boundary and rigidity of holomorphic maps

In this paper we establish several invariant boundary versions of the (infinitesimal) Schwarz-Pick lemma for conformal pseudometrics on the unit disk and for holomorphic selfmaps of strongly convex domains in CN in the spirit of the boundary Schwarz lemma of Burns–Krantz. Firstly, we focus on the case of the unit disk and prove a general boundary rigidity theorem for conformal pseudometrics with variable curvature. In its simplest cases this result already includes new types of boundary versions of the lemmas of Schwarz–Pick, Ahlfors–Schwarz and Nehari–Schwarz. The proof is based on a new Harnack–type inequality as well as a boundary Hopf lemma for conformal pseudometrics which extend earlier interior rigidity results of Golusin, Heins, Beardon, Minda and others. Secondly, we prove similar rigidity theorems for sequences of conformal pseudometrics, which even in the interior case appear to be new. For instance, a first sequential version of the strong form of Ahlfors' lemma is obtained. As an auxiliary tool we establish a Hurwitz–type result about preservation of zeros of sequences of conformal pseudometrics. Thirdly, we apply the one–dimensional sequential boundary rigidity results together with a variety of techniques from several complex variables to prove a boundary version of the Schwarz–Pick lemma for holomorphic maps of strongly convex domains in CN for N>1.

Modified gauge-unfixing formalism and gauge symmetries in the noncommutative chiral bosons theory

Abstract We use the gauge-unfixing (GU) formalism framework in a two-dimensional noncommutative chiral bosons (NCCB) model to disclose new hidden symmetries. That amounts to converting a second-class system to a first-class one without adding any extra degrees of freedom in phase space. The NCCB model has two second-class constraints —one of them turns out as a gauge symmetry generator while the other one, considered as a gauge-fixing condition, is disregarded in the converted gauge-invariant system. We show that it is possible to apply a conversion technique based on the GU formalism direct to the second-class variables present in the NCCB model, constructing deformed gauge-invariant GU variables, a procedure which we name here as modified GU formalism. For the canonical analysis in noncommutative phase space, we compute the deformed Dirac brackets between all original phase space variables. We obtain two different gauge-invariant versions for the NCCB system and, in each case, a GU Hamiltonian is derived satisfying a corresponding first-class algebra. Finally, the phase space partition function is presented for each case allowing for a consistent functional quantization for the obtained gauge-invariant NCCB.

Dose–response prediction for in-vitro drug combination datasets: a probabilistic approach

In this paper we propose PIICM, a probabilistic framework for dose–response prediction in high-throughput drug combination datasets. PIICM utilizes a permutation invariant version of the intrinsic co-regionalization model for multi-output Gaussian process regression, to predict dose–response surfaces in untested drug combination experiments. Coupled with an observation model that incorporates experimental uncertainty, PIICM is able to learn from noisily observed cell-viability measurements in settings where the underlying dose–response experiments are of varying quality, utilize different experimental designs, and the resulting training dataset is sparsely observed. We show that the model can accurately predict dose–response in held out experiments, and the resulting function captures relevant features indicating synergistic interaction between drugs.

Electric field-based quantization of the gauge invariant Proca theory

We consider the gauge invariant version of the Proca theory, where besides the real vector field there is also the real scalar field. We quantize the theory such that the commutator of the scalar field operator and the electric field operator is given by a predefined three-dimensional vector field, say $\cal E$ up to a global prefactor. This happens when the field operators of the gauge invariant Proca theory satisfy the proper gauge constraint. In particular, we show that $\cal E$ given by the classical Coulomb field leads to the Coulomb gauge constraint making the vector field operator divergenceless. We also show that physically unreadable gauge constraints can have a strikingly simple $\cal E$-representation in our formalism. This leads to the discussion of Debye, Yukawa, etc. gauges. In general terms, we explore the mapping between classical vector fields and gauge constraints imposed on the operators of the studied theory.

Wideband Spectrum Sensing Using Modulated Wideband Converter and Data Reduction Invariant Algorithms.

Wideband spectrum sensing is a challenging problem in the framework of cognitive radio and spectrum surveillance, mainly because of the high sampling rates required by standard approaches. In this paper, a compressed sensing approach was considered to solve this problem, relying on a sub-Nyquist or Xsampling scheme, known as a modulated wideband converter. First, the data reduction at its output is performed in order to enable a highly effective processing scheme for spectrum reconstruction. The impact of this data transformation on the behavior of the most popular sparse reconstruction algorithms is then analyzed. A new mathematical approach is proposed to demonstrate that greedy reconstruction algorithms, such as Orthogonal Matching Pursuit, are invariant with respect to the proposed data reduction. Relying on the same formalism, a data reduction invariant version of the LASSO (least absolute shrinkage and selection operator) reconstruction algorithm was also introduced. It is finally demonstrated that the proposed algorithm provides good reconstruction results in a wideband spectrum sensing scenario, using both synthetic and measured data.

Confinement in QCD: novelties

We report on recent progress in understanding confinement of colour in QCD as dual superconductivity of the vacuum. A gauge invariant version of the creation operator of monopoles is constructed whose vacuum expectation value is the order parameter. This order parameter is gauge-invariant from scratch, has no infrared divergences, is finite in the confined phase and vanishes in the deconfined phase. A natural explanation also emerges of why the electric field lines in the flux tubes keep no memory of the colour orientation of the condensing monopoles. A further by-product is that the order parameter can be traded with the two-point vacuum parallel correlator of the chromo-electric field.

Group Invariant Operators and Some Applications to Norm-Attaining Theory

In this paper, we study geometric properties of the set of group invariant continuous linear operators between Banach spaces. In particular, we present group invariant versions of the Hahn–Banach separation theorems and elementary properties of the invariant operators. This allows us to contextualize our main applications in the theory of norm-attaining operators; we establish group invariant versions of the properties \(\alpha \) of Schachermayer and \(\beta \) of Lindenstrauss, and present relevant results from this theory in this (much wider) setting. In particular, we generalize Bourgain’s result, which says that if X has the Radon–Nikodým property, then X has the G-Bishop–Phelps property for G-invariant operators whenever \(G \subseteq \mathcal {L}(X)\) is a compact group of isometries on X.