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In the present paper a generalized Kählerian space is considered, as a generalized Riemannian space GR N with almost complex structure F i h that is covariantly constant with respect to the first and the second kind of covariant derivative. In the general case of a holomorphically projective mapping f of two non-symmetric generalized Kählerian spaces GK N and G K ‾ N it is impossible to obtain a generalization of the holomorphically projective curvature tensor. In the present paper we study the case when GK N and G K ‾ N have the same torsion in corresponding points. Such a mapping we call “equitorsion mapping”. We obtain quantities H P W θ ( θ = 1 , ⋯ , 5 ) , that are generalizations of the holomorphically projective tensor, i.e. they are invariants based on f . Among H P W θ only H P W 5 is a tensor. Using another five linearly independent curvature tensors, we can prove that there exist three holomorphically projective tensors. [ABSTRACT FROM AUTHOR]

RGB-D cameras have been attracting increasing researches for solving traditional problems in the domain of computer vision and robotics. Among the existing local features, most are proposed for the color channel or depth channel separately, while little attention has been paid to designing new composite features based on the physical characteristics. In this work, we propose a novel perspective invariant feature transform (PIFT) for RGB-D images. We integrate the color and depth information together making full use of the intrinsic characteristics of the two types of information to enhance the robustness and adaptability to large spatial variations of local appearance. The depth information is used to project the feature patch to its tangent plane to make it consistent with different views. It also helps to filter out the “fake keypoints” which are unstable in 3D space. Binary descriptors are then generated in the feature patches using a color coding method. Experiments on publicly available RGB-D datasets show that the proposed method has the best precision and the second best recall rate comparing against state-of-the-art local features, when applied to feature matching with large spatial variations. [ABSTRACT FROM AUTHOR]

We give a description of the field of rational natural differential invariants for a class of nonlinear differential operators of order k ≥ 2 on an n -dimensional manifold, n ≥ 2 , and apply the results to the equivalence problem of such operators. [ABSTRACT FROM AUTHOR]

Editorial of Papers Published in 2020-2021 in the Mathematics and Symmetry/Asymmetry Section This editorial is a short review of papers accepted in the Mathematics and Symmetry/Asymmetry section in 2020-2021 about the symmetry methods. In [[3]], a geometrical formulation for adjoint-symmetries as one-forms is studied for general partial differential equations (PDEs), which provides a dual counterpart of the geometrical meaning of symmetries as tangent vector fields on the solution space of a PDE. [Extracted from the article]

The article by L. Petrova [[7]] indicates the importance of considering some aspects of Hamiltonian systems in mathematical and physical from a differential-geometric point of view. The Special Issue "Symmetry of Hamiltonian Systems: Classical and Quantum Aspects" is addressed to mathematical physicists wanting to find some fresh views on results and perspectives in symmetry analysis of a wide class of Hamiltonian systems featuring their many applications in modern classical and quantum theory. They present a detailed Lie-algebraic construction of a new two-parametric hierarchy of commuting to each other Monge-type Hamiltonian vector fields jointly with a specially constructed reciprocal transformation, producing a Frobenius manifold potential function in terms of the solutions of these Monge-type Hamiltonian systems. [Extracted from the article]

As a unitary quantum walk with infinitely many internal degrees of freedom, the quantum walk in terms of quantum Bernoulli noise (recently introduced by Wang and Ye) shows a rather classical asymptotic behavior, which is quite different from the case of the usual quantum walks with a finite number of internal degrees of freedom. In this paper, we further examine the structure of the walk. By using the Fourier transform on the state space of the walk, we obtain a formula that links the moments of the walk’s probability distributions directly with annihilation and creation operators on Bernoulli functionals. We also prove some other results on the structure of the walk. Finally, as an application of these results, we establish a quantum central limit theorem for the annihilation and creation operators themselves. [ABSTRACT FROM AUTHOR]

Symmetries and the corresponding algebras of differential invariants of viscid fluids on a sphere are given. Their dependence on thermodynamical states of media is studied, and a classification of thermodynamical states is given. [ABSTRACT FROM AUTHOR]

Nowadays, wavelets have been widely used in various fields of science and technology. Meanwhile, the wavelet transforms and the generation of new Mother wavelets are noteworthy. In this paper, we generate new Mother wavelets and analyze the differential equations by using of their corresponding wavelet transforms. This method by Mother wavelets and the corresponding wavelet transforms produces analytical solutions for PDEs and ODEs. [ABSTRACT FROM AUTHOR]

We consider the a f f ( n | 1 ) − module structure on the spaces of differential bilinear operators acting on the superspaces of weighted densities. We classify a f f ( n | 1 ) − invariant binary differential operators acting on the spaces of weighted densities. This result allows us to compute the first a f f ( n | 1 ) − relative differential cohomology of K ( n ) with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. [ABSTRACT FROM AUTHOR]

We study geometric and algebraic approaches to classification problems of differential equations. We consider the so-called Lie problem: provide the point classification of ODEs y" = F(x, y). In the first part of the paper we consider the case of smooth right-hand side F. The symmetry group for such equations has infinite dimension, so classical constructions from the theory of differential invariants do not work. Nevertheless, we compute the algebra of differential invariants and obtain a criterion for the local equivalence of two ODEs y" = F(x, y). In the second part of the paper we develop a new approach to the study of subgroups in the Cremona group. Namely, we consider class of differential equations y" = F(x, y) with rational right hand sides and its symmetry group. This group is a subgroup in the Cremona group of birational automorphisms of C², which makes it possible to apply for their study methods of differential invariants and geometric theory of differential equations. Also, using algebraic methods in the theory of differential equations we obtain a global classification for such equations instead of local classifications for such problems provided by Lie, Tresse and others. [ABSTRACT FROM AUTHOR]

Scale selection methods based on local extrema over scale of scale-normalized derivatives have been primarily developed to be applied sparsely-at image points where the magnitude of a scale- normalized differential expression additionally assumes local extrema over the domain where the data are defined. This paper presents a methodology for performing dense scale selection, so that hypotheses about local characteristic scales in images, temporal signals, and video can be computed at every image point and every time moment. A critical problem when designing mechanisms for dense scale selection is that the scale at which scale-normalized differential entities assume local extrema over scale can be strongly dependent on the local order of the locally dominant differential structure. To address this problem, we propose a methodology where local extrema over scale are detected of a quasi quadrature measure involving scale-space derivatives up to order two and pro- pose two independent mechanisms to reduce the phase dependency of the local scale estimates by (i) introducing a second layer of postsmoothing prior to the detection of local extrema over scale, and (ii) performing local phase compensation based on a model of the phase dependency of the local scale estimates depending on the relative strengths between first- and second-order differential structures. This general methodology is applied over three types of domains: (i) spatial images, (ii) temporal signals, and (iii) spatio-temporal video. Experiments demonstrate that the proposed methodology leads to intuitively reasonable results with local scale estimates that re ect variations in the characteristic scales of locally dominant structures over space and time. [ABSTRACT FROM AUTHOR]

In this paper we study the dynamical behavior of the differential equation x ″ + a x + − b x − = f ( t ) , where x + = max ⁡ { x , 0 } , x − = max ⁡ { − x , 0 } , a and b are two different positive constants, f ( t ) is a real analytic almost periodic function. For this purpose, firstly, we have to establish some variants of the invariant curve theorem of planar almost periodic mappings, which was proved recently by the authors (see [11] ). Then we will discuss the existence of almost periodic solutions and the boundedness of all solutions for the above asymmetric oscillation. [ABSTRACT FROM AUTHOR]

We obtain minimal dimension matrix representations for each of the Lie algebras of dimensions five, six, seven and eight obtained by Turkowski that have a non-trivial Levi decomposition. The key technique involves using the invariant subspaces associated to a particular representation of a semi-simple Lie algebra to help in the construction of the radical in the putative Levi decomposition. [ABSTRACT FROM AUTHOR]

The sequence space 𝐵BVσ was introduced and studied by Mursaleen [Houston J. Math. 9, 505-509 (1983; Zbl 0542.40003)]. The main aim of this paper is to study some new double sequence spaces of invariant means defined by ideal and modulus function. Furthermore, we also study several properties relevant to topological structures and inclusion relations between these spaces. [ABSTRACT FROM AUTHOR]

We consider feed-forward and auto-regulation feed-forward neural (weighted) coupled cell networks. In feed-forward neural networks, cells are arranged in layers such that the cells of the first layer have empty input set and cells of each other layer receive only inputs from cells of the previous layer. An auto-regulation feed-forward neural coupled cell network is a feed-forward neural network where additionally some cells of the first layer have auto-regulation, that is, they have a self-loop. Given a network structure, a robust pattern of synchrony is a space defined in terms of equalities of cell coordinates that is flow-invariant for any coupled cell system (with additive input structure) associated with the network. In this paper, we describe the robust patterns of synchrony for feed-forward and auto-regulation feed-forward neural networks. Regarding feed-forward neural networks, we show that only cells in the same layer can synchronize. On the other hand, in the presence of auto-regulation, we prove that cells in different layers can synchronize in a robust way and we give a characterization of the possible patterns of synchrony that can occur for auto-regulation feed-forward neural networks. [ABSTRACT FROM AUTHOR]

Algebras of symmetries and the corresponding algebras of differential invariants for 3D-flows of viscid newtonian fluids are given. Their dependence on thermodynamical states of media is studied. [ABSTRACT FROM AUTHOR]

Starting from the pseudo- $${\mathcal {B}}_0$$ gauge solution for marginal deformations in OSFT, we analytically compute the relation between the perturbative deformation parameter $$\tilde{\lambda }$$ in the solution and the BCFT marginal parameter $$\lambda $$ , up to fifth order, by evaluating the Ellwood invariants. We observe that the microscopic reason why $$\tilde{\lambda }$$ and $$\lambda $$ are different is that the OSFT propagator renormalizes contact-term divergences differently from the contour deformation used in BCFT. [ABSTRACT FROM AUTHOR]

The subtle interplay between local and global charges for topological semimetals exactly parallels that for singular vector fields. Part of this story is the relationship between cohomological semimetal invariants, Euler structures, and ambiguities in the connections between Weyl points. Dually, a topological semimetal can be represented by Euler chains from which its surface Fermi arc connectivity can be deduced. These dual pictures, and the link to topological invariants of insulators, are organised using geometric exact sequences. We go beyond Dirac-type Hamiltonians and introduce new classes of semimetals whose local charges are subtle Atiyah-Dupont-Thomas invariants globally constrained by the Kervaire semicharacteristic, leading to the prediction of torsion Fermi arcs. [ABSTRACT FROM AUTHOR]

A point classification of ordinary differential equations of the form y″ = F( x, y) is considered. The algebra of differential invariants of the action of the point symmetry pseudogroup on the right-hand sides of equations of the form y″ = F( x, y) is calculated, and Lie's problem on the point equivalence of such equations is solved. [ABSTRACT FROM AUTHOR]

We present a complete classification and the construction of Mp(2n+2,ℝ)-equivariant differential operators acting on the principal series representations, associated with the contact projective geometry on ℝℙ2n+1 and induced from the irreducible Mp(2n,ℝ)-submodules of the Segal-Shale-Weil representation twisted by a one-parameter family of characters. The proof is based on the classification of homomorphisms of generalized Verma modules for the Segal-Shale-Weil representation twisted by a oneparameter family of characters, together with a generalization of the well-known duality between homomorphisms of generalized Verma modules and equivariant differential operators in the category of inducing smooth admissible modules. [ABSTRACT FROM AUTHOR]

We propose an invariant three-point curvature approximation for plane curves based on the arc of a parabolic sector, and we analyze how close this approximation is to the true curvature of the curve. We compare our results with those obtained with other invariant three-point curvature approximations. Finally, an application is discussed. [ABSTRACT FROM AUTHOR]

Multi-view texture classification is a very challenging task since the view-point variation often leads to the inconsistent local texton patterns. Existing studies focus on the extraction of scale, rotation or affine invariant representations by some specially designed invariant measurements or local descriptors. Differently, in this paper, we propose another framework for multi-view texture classification. A number of synthetic images are hierarchically created to enlarge the training dataset to cover the possible variations of different view-points. Then, a classifier based on random forests is trained based on these synthetic images. In the classification stage, we also create synthetic images for each testing image, and the synthetic images are classified with the pre-trained classifier. The final decision for this testing image is made by the majority voting of the classification results of all these synthetic images. The classification performance is evaluated on the UIUC texture dataset. Our method achieves the classification rate of 99.21 %, which is higher than most of the state-of-the-arts. [ABSTRACT FROM AUTHOR]

We study a geometry associated with rank 3 distributions in dimension 8, whose symbol algebra is constant and has a simple Lie algebra sp ( 3 , R ) as Tanaka prolongation. We restrict our considerations to only those distributions that are defined in terms of a systems of ODEs of the form z ˙ i j = ∂ 2 f ( x ˙ 1 , x ˙ 2 ) ∂ x ˙ i ∂ x ˙ j , i ≤ j = 1 , 2 . For them we built the full system of local differential invariants, by solving an equivalence problem à la Cartan, in the spirit of his 1910's five variable paper. The considered geometry is a parabolic geometry, and we show that its main invariant – the harmonic curvature – is a certain quintic. In the case when this quintic is maximally degenerate but nonzero, we use Cartan's reduction procedure and reduce the EDS governing the invariants to 11, 10 and 9 dimensions. As a byproduct all homogeneous models having maximally degenerate harmonic curvature quintic are found. They have symmetry algebras of dimension 11 (a unique structure), 10 (a 1-parameter family of nonequivalent structures) or 9 (precisely two nonequivalent structures). [ABSTRACT FROM AUTHOR]

Differential invariants and their corresponding canonical forms for systems of three 2nd-order ODEs possessing three-dimensional Lie algebras are constructed. Their extension up to kth-order system of three 2nd-order ODEs is presented. Furthermore singularity in invariant structure for the canonical forms is investigated. In addition integrability of these canonical forms is discussed. Illustrative physical examples from mechanics of system of particles are provided. [ABSTRACT FROM AUTHOR]

We classify two-by-two traceless Hamiltonians depending smoothly on a threedimensional Bloch wavenumber and having a band crossing at the origin of the wavenumber space. Recently these Hamiltonians attract much interest among researchers in the condensed matter field since they are found to be effective Hamiltonians describing the band structure of the exotic materials such as Weyl semimetals. In this classification, we regard two such Hamiltonians as equivalent if there are appropriate special unitary transformation of degree 2 and diffeomorphism in the wavenumber space fixing the origin such that one of the Hamiltonians transforms to the other. Based on the equivalence relation, we obtain a complete list of classes up to codimension 7. For each Hamiltonian in the list, we calculate multiplicity and Chern number [D. J. Thouless et al., Phys. Rev. Lett. 49, 405 (1982); M. V. Berry, Proc. R. Soc. A 392, 45 (1983); and B. Simon, Phys. Rev. Lett. 51, 2167 (1983)], which are invariant under an arbitrary smooth deformation of the Hamiltonian. We also construct a universal unfolding for each Hamiltonian and demonstrate how they can be used for bifurcation analysis of band crossings. [ABSTRACT FROM AUTHOR]

We consider a three-dimensional coupled cubic nonlinear Schrödinger system appearing in nonlinear optics. If (P, Q) is a ground state solution, we show that for any initial data (u0, v0) inH1(ℝ3)× H1(ℝ3) satisfying M (u0, v0)A(u0, v0)

The binary Darboux transformation method is applied to the coupled Sasa-Satsuma equations, which can be used to describe the propagation dynamics of femtosecond vector solitons in the birefringent fibers with third-order dispersion, self-steepening, and stimulated Raman scattering higher-order effects. An N-fold iterative formula of the resulting binary Darboux transformation is presented in terms of the quasideterminants. Via the simplest case of this formula, a few of illustrative explicit solutions to the coupled Sasa-Satsuma equations are generated from vanishing and non-vanishing backgrounds, which include the breathers, single- and double-hump bright vector solitons, and antidark vector solitons. [ABSTRACT FROM AUTHOR]

In this paper, the classical Galois theory to the H*-Galois case is developed. Let H be a semisimple and cosemisimple Hopf algebra over a field k, A a left H-module algebra, and A/ A a right H*-Galois extension. The authors prove that, if A is a separable k-algebra, then for any right coideal subalgebra B of H, the B-invariants A = { a ∈ A | b · a = ε( b) a, ∀ b ∈ B} is a separable k-algebra. They also establish a Galois connection between right coideal subalgebras of H and separable subalgebras of A containing A as in the classical case. The results are applied to the case H = ( kG)* for a finite group G to get a Galois 1-1 correspondence. [ABSTRACT FROM AUTHOR]

We introduce new classes of solutions to the three-dimensional Navier–Stokes equations in the whole and half-spaces that add rotational correction to self-similar and discretely self-similar solutions. We construct forward solutions in these new classes for arbitrarily large initial data inon the whole and half-spaces. As a special case, this gives a construction of self-similar and discretely self-similar solutions on the half-space. We also comment on the backward case. [ABSTRACT FROM AUTHOR]

In this paper, local geometry of curves in the symplectic Grassmannian homogeneous space Sp ( 4 , R ) / ( Sp ( 2 , R ) × Sp ( 2 , R ) ) and its connection with that of the pseudo-hyperbolic space H 2 , 2 are studied. The group-based Serret–Frenet equations and the associated Maurer–Cartan differential invariants for the Grassmannian curves are obtained by using the equivariant moving frame method. The Grassmannian natural frame is also constructed by a gauge transformation from the Serret–Frenet frame, relating to the hyperbolic natural frame by the local Lie group isomorphism. Using the natural frames, invariant curve flows in the Grassmannian and the hyperbolic spaces are studied. It is shown that certain intrinsic curve flows induce the bi-Hamiltonian integrable matrix mKdV equation on the Maurer–Cartan differential invariants. [ABSTRACT FROM AUTHOR]

Synchronization has been studied extensively in the context of weakly coupled oscillators using the so-called phase response curve (PRC) which measures how a change of the phase of an oscillator is affected by a small perturbation. This approach was based upon the work of Malkin, and it has been extended to relaxation oscillators. Namely, synchronization conditions were established under the weak coupling assumption, leading to a criterion for the existence of synchronous solutions of weakly coupled relaxation oscillators. Previous analysis relies on the fact that the slow nullcline does not intersect the fast nullcline near one of its fold points, where canard solutions can arise. In the present study we use numerical continuation techniques to solve the adjoint equations and we show that synchronization properties of canard cycles are different than those of classical relaxation cycles. In particular, we highlight a new special role of the maximal canard in separating two distinct synchronization regimes: the Hopf regime and the relaxation regime. Phase plane analysis of slow–fast oscillators undergoing a canard explosion provides an explanation for this change of synchronization properties across the maximal canard. [ABSTRACT FROM AUTHOR]

The complexity of many biological processes, which, thanks to the development of many fields of science, becomes for us more and more obvious, makes these processes extremely interesting for further analysis. In this paper a quantitative model of the process of macrophage differentiation, which is essential for many phenomena occurring in the human body, is proposed and analyzed. The model is expressed in the language of Petri net theory on the basis of one of the three hypotheses concerning macrophage differentiation existing in the literature. The performed analysis allowed to find an importance of individual factors in the studied phenomenon. [ABSTRACT FROM AUTHOR]

Let G be a graph of order n and λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n be the eigenvalues of G . The energy of G is defined as E ( G ) = ∑ k = 1 n | λ k | . A well-known result regarding the energy of graphs is the Coulson integral formula, which defines the relationship between the energy and the characteristic polynomial of graphs. Let μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n = 0 be the Laplacian eigenvalues of G . The general Laplacian energy-like invariant of G , denoted by L E L α ( G ) , is defined as ∑ μ k ≠ 0 μ k α when μ 1 ≠ 0 , and 0 when μ 1 = 0 , where α is a real number. In this study, we give some Coulson-type integral formulas for the general Laplacian energy-like invariant of graphs in the case where α is a rational number. Based on this result, we also give some Coulson-type integral formulas for the general energy and general Laplacian energy of graphs in the case where α is a rational number. We also show that our formulas hold when α is an irrational number where 0 < | α | < 1 , whereas they do not hold when | α | > 1 . [ABSTRACT FROM AUTHOR]

We compute the quotient of the self-duality equation for conformal metrics by the action of the diffeomorphism group. We also determine Hilbert polynomial, counting the number of independent scalar differential invariants depending on the jet-order, and the corresponding Poincaré function. We describe the field of rational differential invariants separating generic orbits of the diffeomorphism pseudogroup action, resolving the local recognition problem for self-dual conformal structures. [ABSTRACT FROM AUTHOR]

The goal and the main result of the paper is to provide a complete description of the field of rational differential invariants of one class of second order ordinary differential equations with scalar control parameter with respect to Lie pseudo-group of local feedback transformations. In particular, considered class describes behavior of conservative mechanical systems. We construct the class of rational differential invariants that separate regular orbits. It is well known that differential invariants form algebra with respect to the operation of addition and multiplication (Alekseevskij et al. 1991) [20] . In our case, constructed rational differential operators form a field (in algebraic sense). Rational differential invariants were studied by Rosenlicht (1956, 1963) [25,26] , Kruglikov and Lychagin (2011) [24] . [ABSTRACT FROM AUTHOR]

We compute the Hilbert polynomial and the Poincaré function counting the number of fixed jet-order differential invariants of conformal metric structures modulo local diffeomorphisms, and we describe the field of rational differential invariants separating generic orbits of the diffeomorphism pseudogroup action. This resolves the local recognition problem for conformal structures. [ABSTRACT FROM AUTHOR]

In the present paper we continue the project of systematic construction of invariant differential operators on the example of representations of the conformal algebra induced from the maximal cuspidal parabolic. [ABSTRACT FROM AUTHOR]

Motivated by the challenge of defining twisted quantum field theories in the context of higher categories, we develop a general framework for lax and oplax transformations and their higher analogs between strong ( ∞ , n ) -functors. We construct a double ( ∞ , n ) -category built out of the target ( ∞ , n ) -category governing the desired diagrammatics. We define (op)lax transformations as functors into parts thereof, and an (op)lax twisted field theory to be a symmetric monoidal (op)lax natural transformation between field theories. We verify that lax trivially-twisted relative field theories are the same as absolute field theories. As a second application, we extend the higher Morita category of E d -algebras in a symmetric monoidal ( ∞ , n ) -category C to an ( ∞ , n + d ) -category using the higher morphisms in C . [ABSTRACT FROM AUTHOR]

In this paper, we study invariants of linear differential operators with respect to algebraic Lie pseudogroups. Then we use these invariants and the principle of n-invariants to get normal forms (or models) of the differential operators and solve the equivalence problem for actions of algebraic Lie pseudogroups. As a running example of application of the methods, we use the pseudogroup of local symplectomorphisms. [ABSTRACT FROM AUTHOR]