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A new intrinsically-relativistic kinetic mechanism for generation of non-isotropic relativistic kinetic equilibria in collisionless N-body systems is pointed out. The theory is developed in the framework of the covariant Vlasov statistical description. The new effect is based on the constraints placed by the conservation laws of neutral single-particle dynamics in prescribed background curved-spacetimes demonstrating existence of Killing tensors. As an illustration, the particular case of the Kerr space-time admitting the so-called Carter constant for the particle geodesic motion is considered. The general functional form of the equilibrium kinetic distribution function (KDF) is determined and an explicit realization in terms of Gaussian-like distributions is provided. It is shown that, due to the Carter constant, these equilibrium KDFs exhibit an anisotropic phase-space functional dependence in terms of the single-particle 4-velocity components, giving rise to corresponding non-isotropic continuum fluid fields. The qualitative properties of the equilibrium stress-energy tensor associated with these systems are discussed, with a particular emphasis on the related occurrence of temperature anisotropy effects. The theory is susceptible of astrophysical applications, including in particular the statistical properties of dark matter halos around stellar-mass or galactic-center black holes., 14 pages

Feature matching is a fundamental technique in remote sensing image processing. This article proposes a new formulation of affine covariant feature matching for remote sensing images, where we suggest matching features by matching two sets of triplets. Compared with previous works, the formulation exploits the whole feature frame rather than the 2-D location to reject outliers. Besides, we also develop a new latent variable model to combine the feature frame and the SIFT ratio values, to enhance the convergence speed and success rate in challenging cases. We evaluate our model on three challenging datasets in terms of both qualitative and quantitative experiments. We also study the robustness to outliers since remote sensing images are typically affected by mismatches. The results demonstrate that the proposed method provides excellent matching performance with satisfying runtime and shows good robustness to outliers.

The kinetic description of relativistic plasmas in the presence of time-varying and spatially non-uniform electromagnetic fields is a fundamental theoretical issue both in astrophysics and plasma physics. This refers, in particular, to the treatment of collisionless and strongly-magnetized plasmas in the presence of intense radiation sources. In this paper the problem is investigated in the framework of a covariant gyrokinetic treatment for Vlasov-Maxwell equilibria. The existence of a new class of kinetic equilibria is pointed out, which occur for spatially-symmetric systems. These equilibria are shown to exist in the presence of non-uniform background EM fields and curved space-time. In the non-relativistic limit this feature permits the determination of kinetic equilibria even for plasmas in which particle energy is not conserved due to the occurrence of explicitly time-dependent EM fields. Finally, absolute stability criteria are established which apply in the case of infinitesimal symmetric perturbations that can be either externally or internally produced., Comment: 8 pages

In this paper, we obtain an anisotropic Bianchi type-II space–time with dark matter and modified holographic Ricci dark energy in the scale-covariant theory of gravitation. Exact solutions of the field equations are obtained by assuming (i) a negative constant value of the deceleration parameter, (ii) the component [Formula: see text] of the shear tensor [Formula: see text] is proportional to the mean Hubble parameter, and (iii) the gauge function [Formula: see text] is proportional to a power function of the average scale factor. We have also discussed some important physical aspects of the model, which is in agreement with the modern cosmological observations.

The properties of the system of eigenvectors of the annihilation operator are investigated for the case of non-classical commutation relations between the operators of creation and annihilation. The existence of a measure on the set of eigenvalues of the annihilation operator is proved, with respect to which coherent states form a generalized crowded system. Formulas of covariant symbols of vectors and operators in the representation of generalized coherent states are obtained. It is shown that, similarly to the classical boson case, generalized coherent states are states with minimal uncertainty.

We show that the one- and two-loop $\beta$-functions of the closed, bosonic string can be written in a manifestly O($D$,$D$)-covariant form. Based on this result, we prove that 1) Poisson-Lie symmetric $\sigma$-models are two-loop renormalisable and 2) their $\beta$-functions are invariant under Poisson-Lie T-duality. Moreover, we identify a distinguished scheme in which Poisson-Lie symmetry is manifest. It simplifies the calculation of two-loop $\beta$-functions significantly and thereby provides a powerful new tool to advance into the quantum regime of integrable $\sigma$-models and generalised T-dualities. As an illustrating example, we present the two-loop $\beta$-functions of the integrable $\lambda$- and $\eta$-deformation., Comment: 44 pages, 2 tables

All $$\mathrm{SL}(n)$$ contravariant vector valuations on polytopes in $${\mathbb {R}}^n$$ are completely classified without any additional assumptions. The facet vector is defined. It turns out to be the unique class of such valuations for $$n\ge 3$$ . In dimension two, the classification corresponds to the known case of $$SL (2)$$ covariant valuations.

The construction of integrity bases for invariant and covariant polynomials built from a set of three dimensional vectors under the $${{\mathrm {SO(3)}}}$$ and $${{\mathrm {O(3)}}}$$ symmetries is presented. This paper is a follow-up to our previous work that dealt with a set of two dimensional vectors under the action of the $${{\mathrm {SO(2)}}}$$ and $${{\mathrm {O(2)}}}$$ groups (Dhont and Zhilinskii in J Phys A Math Theor 46:455202, 2013). The expressions of the Molien generating functions as one rational function are a useful guide to build integrity bases for the rings of invariants and the free modules of covariants. The structure of the non-free modules of covariants is more complex. In this case, we write the Molien generating function as a sum of rational functions and show that its symbolic interpretation leads to the concept of generalized integrity basis. The integrity bases and generalized integrity bases for $${\mathrm {O(3)}}$$ are deduced from the $${\mathrm {SO(3)}}$$ ones. The results are useful in quantum chemistry to describe the potential energy or multipole moment hypersurfaces of molecules. In particular, the generalized integrity bases that are required for the description of the electric and magnetic quadrupole moment hypersurfaces of tetratomic molecules are given for the first time.

In a recent article an infinite set of balance equations has been proposed to modelize polyatomic gases with rotational and vibrational modes in the non-relativistic context. To obtain particular cases, it has been truncated to obtain a model with 7 or 15 moments. Here the following objectives are pursued: 1) to obtain the relativistic counterpart of this model which, at the non-relativistic limit, gives the same balance equations as in the known classical case; 2) to obtain the previous result for the model with an arbitrary but fixed number of moments, 3) to obtain the closure of the resulting relativistic model so that all the functions appearing in the balance equations are expressed in terms of the independent variables. To achieve these goals, the following methods are used: 1) The Entropy Principle is imposed. As a result is obtained that the closure is determined up to a single 4-vectorial function usually called 4-potential. 2) To determine this last function, a more restrictive principle is imposed, namely the Maximum Entropy Principle (MEP). 3) Since all the functions involved must be expressed in the covariant form, so as not to depend on the observer, the Representation Theorems are used. Findings of this article are exactly the goals outlined earlier. They are clearly novelty because they had never been achieved before. They can be considered also improvements because, if the aforementioned arbitrary number of moments is restricted to 16, the present work coincide with that already known in literature. Doi: 10.28991/HIJ-2021-02-03-04 Full Text: PDF

We calculated a subclass of four nonleptonic two-body weak decays of the double charm baryon ground states Ξcc++. Nonleptonic decays can be dividedinto two group: factorizable and nonfactorizable decays. The first one can be easily calculated from first principles. Therefore, its good example understands all pros and cons of a model. We focused on a weak two-body nonleptonic decay consists only fromthe factorizing contribution precluding a contamination from W-exchange. We use the covariant confined quark model previously developed by us to calculate the various helicity amplitudes which describe the dynamics of the transition induced by the Cabibbo-favored effective currents. Achievement of CCQM is that only size parameter Λvarying can describe all tree diagrams of nonleptonic decay. We then proceed to calculate the rates of the decay. The rates, branching ratios and helicity amplitude were calculated using dimensionless invariant form factors. Also, we calculated leptonic constant for scalar and vector mesons which has good agreement with experimental data. There isn’t any experimental data about the decay so we waiting for new experimental observation in the heavy baryon sector.

Starting from the notion of linear and nonlinear transformations, affine and functional-nonlinear mappings of coordinates and coordinate systems, geometrical and kinematical invariants along linear or nonlinear transformations their coordinates from one to other coordinate system are pointed out. In a curvilinear coordinate system, coordinates of a geometrical or kinematical point are not equal as coordinates of its corresponding position vector. Expressions of basic vectors of tangent space of kinematical point vector positions in generalized curvilinear coordinate systems for the cases of orthogonal and nonorthogonal curvilinear coordinate systems are derived. Examples of expressions of basic vectors of tangent space of kinematical point vector position in polar-cylindrical, spherical, parabolic-cylindrical and three-dimensional-three-parabolic system of curvilinear orthogonal coordinates are presented. Next, expressions of change of basic vectors of tangent space of kinematical point vector position with time, also, are done. Geometrical (physical), covariant and contra-variant coordinates of position vector of a kinetic mass point, in a coordinate system determined by basic vectors of tangent space of this kinetic point vector position, in generalized curvilinear coordinate systems, are pointed out and determined. Original expressions of angular velocity and velocity of dilatations of basic vectors of tangent space of kinetic point vector position, in generalized curvilinear coordinate systems as well as in series of special orthogonal curvilinear coordinate system are derived by author of this paper and presented.

In this article, we propose two Banach-type fixed point theorems on bipolar metric spaces. More specifically, we look at covariant maps between bipolar metric spaces and consider iterates of the map involved. We also propose a generalization of the Banach fixed point result via Caristi-type arguments.

The cosmology of the fully α′-corrected duality-invariant action for the Neveu-Schwarz sector of string theory is revisited, with special emphasis on its coupling to matter sources. The role of the duality covariant pressure and dilatonic charge of the matter sector is explored in various contexts, from the low-curvature regime to non-perturbative solutions in α′. We comment on how an infinite tower of α′ corrections allows for fixed-dilaton de Sitter solutions, even in vacuum. We further investigate the necessary conditions for accelerated expansion in the Einstein frame, as well as for non-singular bounces that could resolve the big bang singularity. In particular, explicit examples are constructed, which show that the tower of α′ corrections may support an Einstein-frame non-singular cosmological bouncing background, even when the matter sector respects the null energy condition.

The main objective of the paper is to study the three-dimensional fractional Fourier Mellin transforms (3DFRFMT), their basic properties and applicability due to mainly use in the radar system, reconstruction of grayscale images, in the detection of the human face, etc. Only the fractional Fourier transform is based on time-frequency distribution, whereas only the fractional Mellin transform is on scale covariant transformation. Both transforms can discover action in the definite assortment. The fractional Fourier transform is applicable for controlling the range of shift, whereas the fractional Mellin transform is accustomed to managing the range of rotation and scaling of the function. So, combining both transformations, we get an elegant expression for 3DFRFMT, which can be used in several fields. The paper introduces the concept of three-dimensional fractional Fourier Mellin transforms and their applications. Modulation property is the most useful concept in the signal system, radar technology, pattern reorganization, and many more in the integral transform. Parseval's identity applies to the conservation of energy in the universe. Thus we establish the modulation theorem, Parseval's theorem, scaling theorem, analytic theorem for three-dimensional fractional Fourier Mellin transform. We also give some examples of three-dimensional fractional Fourier-Mellin transform on some functions. Finally, we provide three-dimensional fractional Fourier-Mellin transform applications for solving homogeneous and non-homogeneous Mboctara partial differential equations that we can apply with advantages to solve the different types of problems in signal processing systems. The transform is beneficial in a maritime strategy as a co-realtor to control moments in any specific three-dimensional space. The concept is the most powerful tool to deal with any information system problems. After obtaining the generalization, we can explore many more ideas in applying three-dimensional fractional Fourier-Mellin transformations in many real word problems.

Given an (irreducible) Mobius covariant net $\mathcal A$, we prove a Bisognano-Wichmann theorem for its categorical extension $\mathscr E^{\textrm{d}}$ associated to the braided $C^*$-tensor category $\textrm{Rep}^{\textrm{d}}(\mathcal A)$ of dualizable (more precisely "dualized") Mobius covariant $\mathcal A$-modules. As a closely related result, we prove a (modified) Bisognano-Wichmann theorem for any (possibly) non-local extension of $\mathcal A$ obtained by a $C^*$-Frobenius algebra $Q$ in $\textrm{Rep}^{\textrm{d}}(\mathcal A)$. As an application, we discuss the relation between the domains of modular operators and the preclosedness of certain unbounded operators in $\mathscr E^{\textrm{d}}$., 44 pages, to appear in Annales Henri Poincar\'e. A new section added (Sec. 3), notation changed, Sec. 5 (previously Sec. 4) rewritten, the proofs of the main results of Sec. 6 and Sec. 7 are rewritten

A generalization of the Choi–Jamiolkowski isomorphism for completely positive maps between operator algebras is introduced. Particular emphasis is placed on the case of normal unital completely positive maps defined between von Neumann algebras. This generalization is applied especially to the study of maps which are covariant under actions of a symmetry group. We highlight with the example of, e.g., phase-shift-covariant quantum channels, the ease of this method in particular in the case of a compact symmetry group. We also discuss the case of channels which are covariant under actions of the Euclidean group of rigid motions in three dimensions.

We analyze a novel approach to gauging rigid higher derivative (higher spin) symmetries of free relativistic actions defined on flat spacetime, building on the formalism originally developed by Bonora et al. and Bekaert et al. in their studies of linear coupling of matter fields to an infinite tower of higher spin fields. The off-shell definition is based on fields defined on a $2d$-dimensional master space equipped with a symplectic structure, where the infinite dimensional Lie algebra of gauge transformations is given by the Moyal commutator. Using this algebra we construct well-defined weakly non-local actions, both in the gauge and the matter sector, by mimicking the Yang-Mills procedure. The theory allows for a description in terms of an infinite tower of higher spin spacetime fields only on-shell. Interestingly, Euclidean theory allows for such a description also off-shell. Owing to its formal similarity to non-commutative field theories, the formalism allows for the introduction of a covariant potential which plays the role of the generalised vielbein. This covariant formulation uncovers the existence of other phases and shows that the theory can be written in a matrix model form. The symmetries of the theory are analyzed and conserved currents are explicitly constructed. By studying the spin-2 sector we show that the emergent geometry is closely related to teleparallel geometry, in the sense that the induced linear connection is opposite to Weitzenb\"{o}ck's., Comment: 61 pages, 2 figures; v2: 56 pages, paper shortened, old section 6 removed, results and conclusions unchanged, version accepted for publishing in JHEP

We consider the natural $$\mathrm {SO}(3,\Bbbk )$$ linear representation, $$\Bbbk =\mathbb {C}$$ or $$\mathbb {R}$$ , on the $$\Bbbk$$ vector space $${\mathsf {V}}=n\mathbb {S}^{2}(\mathbb {R}^3)$$ of n second-order symmetric tensors, the associated invariant field $$\Bbbk ({\mathsf {V}})^{\mathrm {SO}(3,\Bbbk )}$$ being known to be a purely transcendental extension in the complex case. We give an explicit tensorial form of a minimal generating set of the field of invariants, in both the complex and the real cases, showing that the invariant field is also a purely transcendental extension in the real case. The present results rely on some octahedral polynomial invariants obtained from Clebsch–Gordan projectors defined by a fourth-order octahedral covariant. Thanks to Cartan’s map, we also obtain a minimal set of generators for the $$\mathrm {SL}(2,\mathbb {C})$$ -rational invariant field of n binary quartics.

Covariant Lyapunov vectors or CLVs span the expanding and contracting directions of perturbations along trajectories in a chaotic dynamical system. Due to efficient algorithms to compute them that only utilize trajectory information, they have been widely applied across scientific disciplines, principally for sensitivity analysis and predictions under uncertainty. In this paper, we develop a numerical method to compute the directional derivatives of the first CLV along its own direction; the norm of this derivative is also the curvature of one-dimensional unstable manifolds. Similar to the computation of CLVs, the present method for their derivatives is iterative and analogously uses the second-order derivative of the chaotic map along trajectories, in addition to the Jacobian. We validate the new method on a super-contracting Smale–Williams Solenoid attractor. We also demonstrate the algorithm on several other examples including smoothly perturbed Arnold Cat maps, and the Lorenz’63 attractor, obtaining visualizations of the curvature of each attractor. Furthermore, we reveal a fundamental connection of the derivation of the CLV self-derivative computation with an efficient computation of linear response of chaotic systems.

In a conformally flat three-dimensional spacetime, the linearised higher-spin Cotton tensor $\mathfrak{C}_{\alpha(n)}(h)$ is the unique conserved conformal current which is a gauge-invariant descendant of the conformal gauge prepotential $h_{\alpha(n)}$. The explicit form of $\mathfrak{C}_{\alpha(n)}(h)$ is well known in Minkowski space. Here we solve the problem of extending the Minkowskian result to the case of anti-de Sitter (AdS) space and derive a closed-form expression for $\mathfrak{C}_{\alpha(n)}(h)$ in terms of the AdS Lorentz covariant derivatives. It is shown that every conformal higher-spin action $S_{\text{CS}}^{(n)}[h]\propto \int\text{d}^3x\, e \, h^{\alpha(n)}\mathfrak{C}_{\alpha(n)}(h) $ factorises into a product of $(n-1)$ first-order operators that are associated with the spin-$n/2$ partially massless AdS values. Our findings greatly facilitate the on-shell analysis of massive higher-spin gauge-invariant actions in AdS$_3$. The main results are extended to the case of $\mathcal{N}=1$ AdS supersymmetry. In particular, we derive simple expressions for the higher-spin super-Cotton tensors in AdS$_3$., Comment: 44 pages; V2: comments added and typos corrected; V3: published version

Background/Objectives: With reference to the theory of general relativity, the rotating metrics can be designed with mass function ˆM (u; r) . Methods/Statistical analysis: The methods/analysis adapted is the theoretical and mathematical analysis on the theory of general relativity. Findings: The line element can be found out with the help of mass function ˆM(u; r) following the find of the covariant complex null tetrad vectors for the rotating metrics. Then, the NP spin — coefficients, the Ricci scalars, and the Weyl scalars for the rotating metrics can be found out. The expanded form of the mass function ˆM(u; r) with (a=0) can be shown from Wang and Wu (1999). From the expended form of the mass function ˆM(u; r)with (a = 0) , the NP coefficients can be derived. With the help of the scalar k, the surface gravity of the black hole is derived. Novelty/Applications: The findings of covariant complex null tetrad vectors for the rotating metrics, the NP spin — coefficients, the Ricci scalars, and the Weyl scalars are new analysis towards the theory of general relativity. Specific applications are the deep studies on the black hole and its surface gravity. It can be concluded that generally the rotating metric possesses a geodesic (k =e = 0), actually shear free (s = 0) , purely expanding (ˆq ( = 0) and a non-zero twistw 2 = 0) null vector la (Chandrasekhar, 1983). With the help of a scalar k, on a horizon of a Black hole, the surface gravity of the black hole is derived. Keywords: The Mass Function; NP Spin – Coefficients; The Ricci Scalars; The Weyl Scalars; The Surface Gravity of the Black Hole

We have considered the generalized version of chiral schwinger model with the Lorentz covariant masslike term for gauge field with the choice a − r2 = 0. We carry out the quantization by the canonical Dirac method of both the gauge-invariant and non-invariant version of this model to determine the phase space structure. Therefore we have shown that the gauge invariant theory has the same physical spectrum as that of the original gauge noninvariant formulation.

Several possible definiions of strains in a general shell theory of I.N. Vekua – A.A. Amosov type are considered. The higher-order shell model is definedon a two-dimensional manifold within a set of fieldvariables of the firstkind determined by the expansion factors of the spatial vector fieldof the translation. Two base vector systems are introduced, the firs one so-called concomitant corresponds to the cotangent fibrtion of the modelling surface while the other is defind on a surface equidistant to the modelling one. The distortion appears as a two-point tensor referred to both base systems after covariant differentiationof the translation vector feld. Thus, two main definition of the strain tensor become possible, the firstone referred to the main basis whereas the second to the concomitant one. Some possible simplificationsof these tensors are considered, and the interrelation between the general theory of A.A. Amosov type and the classical ones is shown.

There are well-known criteria on the potential and field-space geometry for determining if slow-roll, slow-turn, multi-field inflation is possible. However, even though it has been a topic of much recent interest, slow-roll, rapid-turn inflation only has such criteria in the restriction to two fields. In this work, we generalize the two-field, rapid-turn inflationary attractor to an arbitrary number of fields. We quantify a limit, which we dub extreme turning, in which rapid-turn solutions may be found efficiently and develop methods to do so. In particular, simple results arise when the covariant Hessian of the potential has an eigenvector in close alignment with the gradient -- a situation we find to be common and we prove generic in two-field hyperbolic geometries. We verify our methods on several known rapid-turn models and search two type-IIA constructions for rapid-turn trajectories. For the first time, we are able to efficiently search for these solutions and even exclude slow-roll, rapid-turn inflation from one potential., 30 pages, 8 figures. v2: added missed references

The aim of this paper is to compare the Cartier and covariant Dieudonne modules of connected p-divisible groups over perfect fields of positive characteristic p. In fact, we construct an explicit, canonical and functorial isomorphism and its inverse from the Dieudonne module to the Cartier module. The fact that the Cartier and (covariant) Dieudonne modules of a p-divisible formal Lie group over a perfect field are isomorphic is attributed to W. Messing, but a written proof does not exist in the literature ( [1, §5] ). We also give an application of this explicit isomorphism.

A neural network (NN) approach was recently developed to construct accurate quasidiabatic Hamiltonians for two-state systems with conical intersections. Here, we derive the transformation properties of elements of 3 × 3 quasidiabatic Hamiltonians based on a valence bond (VB) model and extend the NN-based method to accurately diabatize the three lowest electronic singlet states of H3+, which exhibit the avoided crossing between the ground and first excited states and the conical intersection between the first and second excited states for equilateral triangle configurations (D3h). The current NN framework uses fundamental invariants (FIs) as the input vector and appropriate symmetry-adapted functions called covariant basis to account for the special symmetry of complete nuclear permutational inversion (CNPI). The resulting diabatic potential energy matrix (DPEM) can reproduce the ab initio adiabatic energies, energy gradients, and derivative couplings between adjacent states as well as the particular symmetry. The accuracy of DPEM is further validated by full-dimensional quantum dynamics calculations. The flexibility of the FI-NN approach based on the VB model shows great potential to resolve diabatization problems for many extended and multistate systems.

LetTbe a countably abelian, totally sub-projective random vari-able. It is well known thatC′′is naturally generic, contra-n-dimensional,meromorphic and contra-Hamilton. We show that there exists a com-pact and differentiablep-adic subring. Unfortunately, we cannot as-sume that every globally connected path is local. It is well known that ̄Pis homeomorphic to Ω.

Matching images of a scene with high precision is a core component in real time applications such as image-based navigation of unmanned aerial vehicles, terminal guidance of missiles. The aim of the research discussed in this paper is to develop a scene matching algorithm of high accuracy, robust to the variations between the images compared with scope for parallelism. After a study of state-of-the-art scene matching algorithms, a local feature-based approach was chosen to meet the stated requirements. The paper discusses the techniques for modules of the algorithm chosen and the methodology implemented after a metric study with a dynamic threshold based nearest neighbour distance ratio matching strategy (DT-NNDR). The metrics and the proposed strategy were tested for the matching of planar images from Affine covariant features group, unmanned aerial vehicle videos from a national defence agency, and satellite images from Google. The performance of strategy was compared with the existing NN-DR strategy with an experimental optimal threshold. The overall precision of the proposed strategy is 88%, making it suitable for applications which require high value of true positive.

A (closed) dynamical system is a notion of how things can be, together with a notion of how they may change given how they are. The idea and mathematics of closed dynamical systems has proven incredibly useful in those sciences that can isolate their object of study from its environment. But many changing situations in the world cannot be meaningfully isolated from their environment - a cell will die if it is removed from everything beyond its walls. To study systems that interact with their environment, and to design such systems in a modular way, we need a robust theory of open dynamical systems. In this extended abstract, we put forward a general definition of open dynamical system. We define two general sorts of morphisms between these systems: covariant morphisms which include trajectories, steady states, and periodic orbits; and contravariant morphisms which allow for plugging variables of some systems into parameters of other systems. We define an indexed double category of open dynamical systems indexed by their interface and use a double Grothendieck construction to construct a double category of open dynamical systems. In our main theorem, we construct covariantly representable indexed double functors from the indexed double category of dynamical systems to an indexed double category of spans. This shows that all covariantly representable structures of dynamical systems - including trajectories, steady states, and periodic orbits - compose according to the laws of matrix arithmetic., Comment: In Proceedings ACT 2020, arXiv:2101.07888

Starting from a complete family for the unit sphere S n in the complex n-space C n (whose elements are coherent states attached to the Barut–Girardello space), we obtain an asymptotic expansion for the associated Berezin transform. The proof involves the computation of the asymptotic behaviour of functions in the complete family. Furthermore, in an analogous manner (slightly weaker) we obtain the asymptotic expansion of the covariant symbol of a pseudo-differential operator on L 2 ( S n ).

We study vertex-like operators built from the Brownian loop soup in the limit as the loop soup intensity tends to infinity. More precisely, following Camia, Gandolfi and Kleban (Nuclear Physics B 902, 2016), we take a Brownian loop soup in a planar domain and assign a random sign to each loop. We then consider random fields defined by taking, at every point of the domain, the exponential of a purely imaginary constant times the sum of the signs associated to the loops that wind around that point. As smaller loops are included in the count, that sum diverges logarithmically with the diameter of the loops, but we show that a suitable renormalization procedure allows to define the fields in an appropriate Sobolev space. Subsequently, we let the intensity of the loop soup tend to infinity and prove that these vertex-like fields tend to a conformally covariant random field which can be expressed as an explicit functional of the imaginary Gaussian multiplicative chaos with covariance kernel given by the Brownian loop measure. Besides using properties of the Brownian loop soup and the Brownian loop measure, a main tool in our analysis is an explicit Wiener-It\^{o} chaos expansion of linear functionals of vertex-like fields., Comment: 67 pages, no figures. V2: We have expanded the introduction, improved Theorem 1, added a new appendix (A.6), corrected several typos, made other minor changes

Over the past decades, the relativistic density functional theory has been greatly developed and widely applied to investigate a variety of nuclear phenomena. In this paper, we briefly review the concept of covariant density functional theory in nuclear physics with a few latest applications in describing nuclear ground-state and excitation properties as well as nuclear dynamics. Moreover, attempts to build a microscopic and universal density functional are also discussed in terms of the successful fully self-consistent relativistic Brueckner–Hartree–Fock calculations.

We elaborate an algorithm to optimally control the actuators of robot manipulators. By regarding the robot kinetic energy as a metric, the dynamic model is formulated using tensors in a Riemannian ...

A complete classification of continuous $$\text {SL}(n)$$ covariant vector-valued valuations on $$L^{p}({\mathbb {R}}^{n},|x|dx)$$ is obtained without any homogeneity assumptions. The moment vector is shown to be essentially the only such valuation.

It is analyzed in detail the covariance of the integral forms of the Maxwell Equations. Different forms of writing the integral Maxwell equations in explicitly covariant form are analyzed. First we show how one of these ways can be obtained from the differential Maxwell equations, both from their usual three vector form as from their covariant four vector form. Then we discuss how this covariant integral Maxwell equations can be obtained from usual integral Maxwell equations. It is emphasized the necessity to write the usual integral equations without time derivatives. The integrations regions in the three-dimensional space and time are identified with parts of the same hyper-surface in four-dimensional space-time. This point is carefully analyzed. Later we discuss other forms of the integral Maxwell equations. We show how these new versions can be expressed in an explicitly covariant form.

The cosmological implications of the Covariant Canonical Gauge Theory of Gravity (CCGG) are investigated. CCGG is a Palatini theory derived from first principles using the canonical transformation formalism in the covariant Hamiltonian formulation. The Einstein-Hilbert theory is thereby extended by a quadratic Riemann-Cartan term in the Lagrangian. Moreover, the requirement of covariant conservation of the stress-energy tensor leads to necessary presence of torsion. In the Friedman universe that promotes the cosmological constant to a time-dependent function, and gives rise to a geometrical correction with the EOS of dark radiation. The resulting cosmology, compatible with the $\Lambda$CDM parameter set, encompasses bounce and bang scenarios with graceful exits into the late dark energy era. Testing those scenarios against low-z observations shows that CCGG is a viable theory., Comment: Contribution to the International Workshop IWARA on Astronomy and Relativistic Astrophysics, September 2020. Accepted for publication in the Astronomische Nachrichten 2021; 8 pages, 6 figures

Color image restoration is one of the important tasks in color image processing. Covariant differentiation has been applied to handle vector bundles arising from color images in the red, green, blu...

A new relativistically covariant approach is discussed for the derivation of local conservation theorems for homogeneous anisotropic and, in particular, dispersive media. We start from a three-dimensional operator equation for the electric field and obtain mainly by coordinate-invariant methods the results basically expressed by the slowly varying amplitudes of the electric field. Apart from local energy and momentum conservation formulated by the energy-momentum conservation, we find a local conservation theorem for the action which is more general and which is the only one which remains also true for inhomogeneous media.

Any particular classical system and its quantum version are normally viewed as separate formulations that are strictly distinct. Our goal is to overcome the two separate languages and create a smooth and common procedure that provides a clear and continuous passage between the conventional distinction of either a strictly classical or a strictly quantized state. While path integration, among other procedures, provides an alternative route to connect classical and quantum expressions, it normally involves complicated, model-dependent, integrations. Our alternative procedures involve only model-independent procedures, and use more natural and straightforward integrations that are universal in kind. To introduce the basic procedures our presentation begins with familiar methods that are limited to basic, conventional, canonical quantum mechanical examples. In the final sections we illustrate how alternative quantization procedures, e.g., spin and affine quantizations, can also have smooth paths between classical and quantum stories, and with a few brief remarks, can also lead to similar stories for non-renormalizable covariant scalar fields as well as quantum gravity., 12 pages; A smooth passage between classical to quantum or vice versa; several errors fixed; Figure added

The problem of real bremsstrahlung calculation is considered using the modern methods of regularization of divergencies. In particular, we calculate soft photon bremsstrahlung in the most general form using the method of dimensional regularization of infrared divergences. The general calculation algorithm of hard photon bremsstrahlung is described. It is shown that the contribution of hard bremsstrahlung can be separated into the finite and divergent parts. The divergent part can be factorized with the contribution of the initial process in the Born approximation. It is shown that a good choice of kinematic variables makes an analytic covariant calculation of the divergent part of the hard bremsstrahlung possible. In a particular case, an algorithm for determining the kinematic constraints on the invariants is described. A numerical analysis of the radiative corrections for gauge bosons production processes in the case of electron-photon collisions is performed. It is discovered that the contribution of the finite part of bremsstrahlung at high collision energies reaches 20 per cent and must be taken into account in calculations of radiative corrections. The results obtained can be used in various calculations, including covariant ones, performed in the context of confirmation of the Standard Model theoretical predictions or searching for manifestations of alternative gauge models.

In this paper, we investigate covariant and contravariant symbols of operators generated by a representation of the integer group $\mathbb{Z}$. Then we describe some properties (Existence, Uniqueness, Boundedness, Compactnessi and Finite rank) of these operators and reformulated some know results in terms of wavelet transform (covariant and contravariant symbols).

We introduce a new class of first-order partial differential equations (coined as covariantly equipped systems of equations) invariant with respect to (pseudo)orthogonal changes of Cartesian coordinates in a (pseudo)Euclidean space. We propose a procedure for reducing the Cauchy problem for a system of equations to a Cauchy problem for a covariantly equipped system of equations (the covariant equipping procedure). We prove that the covariant equipping procedure can be applied to the Cauchy problem for Maxwell’s equations.

We initiate a study of finite temperature transport in gapless and strongly coupled quantum theories with charge and dipole conservation using gauge-gravity duality. In a model with non-dynamical gravity, the bulk fields of our model include a suitable mixed-rank tensor which encodes the boundary multipole symmetry. We describe how such a theory can arise at low energies in a theory with a covariant bulk action. Studying response functions at zero density, we find that charge relaxes via a fourth-order subdiffusion equation, consistent with a recently-developed field-theoretic framework., 14+5 pages, 1+0 figure; v2: added references. v3: added discussions

We critically revisit the definition of thermal equilibrium, in its operational formulation, provided by standard thermodynamics. We show that it refers to experimental conditions which break the covariance of the theory at a fundamental level and that, therefore, it cannot be applied to the case of moving bodies. We propose an extension of this definition which is manifestly covariant and can be applied to the study of isolated systems in special relativity. The zeroth law of thermodynamics is, then, proven to establish an equivalence relation among bodies which have not only the same temperature, but also the same center of mass four-velocity., 15 pages, 0 figures

How does subatomic matter organize itself? Neutron stars are cosmic laboratories uniquely poised to answer this fundamental question that lies at the heart of nuclear science. Newly commissioned rare isotope facilities, telescopes operating across the entire electromagnetic spectrum, and ever more sensitive gravitational wave detectors will probe the properties of neutron-rich matter with unprecedented precision over an enormous range of densities. Yet, a coordinated effort between observation, experiment, and theoretical research is of paramount importance for realizing the full potential of these investments. Theoretical nuclear physics provides valuable insights into the properties of neutron-rich matter in regimes that are not presently accessible to experiment or observation. In particular, nuclear density functional theory is likely the only tractable framework that can bridge the entire nuclear landscape by connecting finite nuclei to neutron stars. This compelling connection is the main scope of the present review., 23 pages, 5 figures. When citing this paper, please use the following: Yang J, Piekarewicz J. Covariant Density Functional Theory in Nuclear Physics and Astrophysics. Annual Review of Nuclear and Particle Science Volume 70: Submitted. DOI: 10.1146/annurev-nucl-101918-023608

A classification of $${\text {SL}}(n)$$ and translation covariant Minkowski valuations on log-concave functions is established. The moment vector and the recently introduced level set body of log-concave functions are characterized. Furthermore, analogs of the Euler characteristic and volume are characterized as $${\text {SL}}(n)$$ and translation invariant valuations on log-concave functions.

We present a four-vector analogue Uμ ≡ (U0, U) = (|V′|, V) that can be directly constructed from the velocity of a projectile. Here, V and V′ are the velocities of a projectile at heights h + H and h, respectively. In the non-relativistic regime, the mathematical structure of this four-vector analogue has an exact correspondence to the relativistic counterpart four-velocity uμ = γ(1, β)c of a massive particle. Based on this observation, we illustrate the design of an introductory laboratory experiment to investigate the Lorentz invariance and its covariant nature by measuring the velocity of a projectile. The experiment may help students to acquire a concrete perspective of Lorentz covariance through their own measurements and analyses of a free-fall motion.

We define the path coalgebra and Gabriel quiver constructions as functors between the category of $k$-quivers and the category of pointed $k$-coalgebras, for $k$ a field. We define a congruence relation on the coalgebra side, show that the functors above respect this relation, and prove that the induced Gabriel $k$-quiver functor is left adjoint to the corresponding path coalgebra functor. We dualize, obtaining adjoint pairs of functors (contravariant and covariant) for pseudocompact algebras. Using these tools we describe precisely to what extent presentations of coalgebras and algebras in terms of path objects are unique, giving an application to homogeneous algebras., 18 pages, some small changes and clarifications