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Rota–Baxter operators on groups have been recently defined in [L. Guo, H. Lang and Y.  Sheng, Integration and geometrization of Rota–Baxter Lie algebras, Adv. Math. 387 (2021) 107834], and they share a close connection with skew braces, as demonstrated in [V. Bardakov and V. Gubarev, Rota–Baxter groups, skew left braces, and the Yang–Baxter equation, J. Algebra 587 (2022) 328–351]. In this paper, we classify all Rota–Baxter operators of weight 1 on the Heisenberg Lie algebra of dimension 3 up to the Jordan canonical form of a 2 × 2 matrix block by automorphisms of the Heisenberg Lie algebra. Using the fact that the exponential map from the Heisenberg Lie algebra to the Heisenberg group is bijective, we induce these operators on the Heisenberg group. Finally, we enumerate all skew left brace structures on the Heisenberg group induced by these Rota–Baxter operators. [ABSTRACT FROM AUTHOR]

Let k be a field of arbitrary characteristic, A be a domain, and K = frac(A). ThenAll exponential maps of k[3] are rigid, and we give a necessary and sufficient condition for the triangularity of δ ∈EXP(k[3]).If δ ∈EXP(A[3]) such that rank(δ) = rank(δK), then δ is rigid and we give a necessary and sufficient condition for the triangularity of δ.When k is of zero characteristic, (1) is due to [D. Daigle, A necessary and sufficient condition for triangulability of derivations of k[X,Y,Z], J. Pure Appl. Algebra 113(3) (1996) 297–305] and (2) is due to [M. K. Keshari and S. A. Lokhande, A note on rigidity and triangulability of a derivation, J. Commut. Algebra 6(1) (2014) 95–100]. [ABSTRACT FROM AUTHOR]

In this article we consider diffeomorphism groups of manifolds with smooth boundary. We show that the diffeomorphism groups of the manifold and its boundary fit into a short exact sequence which admits local sections. In other words, they form an infinite-dimensional fibre bundle. Manifolds with boundary are of interest in numerical analysis and with a view towards applications in machine learning we establish controllability results for families of vector fields. This generalises older results due to Agrachev and Caponigro in the boundary-less case. Our results show in particular that the diffeomorphism group of a manifold with smooth boundary is generated by the image of the exponential map. [ABSTRACT FROM AUTHOR]

The block trace and partial trace are closely related. However, the block determinant and partial determinant are quite different. We introduce a new determinant-like operation, the determinant-root, which has a partial operation that is more natural in the algebraic sense. The block determinant retains some of the algebraic properties of determinants, while the partial determinant-root retains the same algebraic properties for matrices which are Kronecker products. We investigate the equations $ \det (D(A)) = \det (A) $ det (D (A)) = det (A) , $ D(AB) = D(A)D(B) $ D (AB) = D (A) D (B) and $ D(e^A) = e^{T(A)} $ D (e A) = e T (A) , where D is the block/partial/partial-root determinant and T is the corresponding block/partial/partial-normalized trace. These results yield a characterization of non-linear preservers of determinants of Kronecker products. [ABSTRACT FROM AUTHOR]

It is well known that, over an algebraically closed field k of characteristic zero, for any three integers a , b , c ≥ 2 , any Pham-Brieskorn surface B (a , b , c) : = k [ X , Y , Z ] / (X a + Y b + Z c) is rigid when at most one of a , b , c is 2 and stably rigid when 1 a + 1 b + 1 c ≤ 1. In this paper we consider Pham-Brieskorn domains over an arbitrary field k of characteristic p ≥ 0 and give sufficient conditions on (a , b , c) for which any Pham-Brieskorn domain B (a , b , c) is rigid. This gives an alternative approach to showing that there does not exist any non-trivial exponential map on k [ X , Y , Z , T ] / (X m Y + T p r q + Z p e ) = k [ x , y , z , t ] , for m , q > 1 , p ∤ m q and e > r ≥ 1 , fixing y , a crucial result used in [10] to show that the Zariski Cancellation Problem (ZCP) does not hold for the affine 3-space in positive characteristic. We also provide a sufficient condition for B (a , b , c) to be stably rigid. Along the way we prove that for integers a , b , c ≥ 2 with g c d (a , b , c) = 1 and for F (Y) ∈ k [ Y ] , the ring k [ X , Y , Z ] / (X a Y b + Z c + F (Y)) is a rigid domain. [ABSTRACT FROM AUTHOR]

In this work, using some tools of Lie theory, we compute the Lie algebra of the Shearlet group regarding as a 3-fold semidirect product Lie group. As we will see, it is a 3-fold semidirect sum of Lie algebras. [ABSTRACT FROM AUTHOR]

Simulating the time evolution of a Hamiltonian system on a classical computer is hard—The computational power required to even describe a quantum system scales exponentially with the number of its constituents, let alone integrate its equations of motion. Hamiltonian simulation on a quantum machine is a possible solution to this challenge—Assuming that a quantum system composing of spin-½ particles can be manipulated at will, then it is tenable to engineer the interaction between those particles according to the one that is to be simulated, and thus predict the value of physical quantities by simply performing the appropriate measurements on the system. Establishing a linkage between the unitary operators described mathematically as a logic solution and the unitary operators recognizable as quantum circuits for execution, is therefore essential for algorithm design and circuit implementation. Most current techniques are fallible because of truncation errors or the stagnation at local solutions. This work offers an innovative avenue by tackling the Cartan decomposition with the notion of Lax dynamics. Within the integration errors that is controllable, this approach gives rise to a genuine unitary synthesis that not only is numerically feasible, but also can be utilized to gauge the quality of results produced by other means, and extend the knowledge to a wide range of applications. This paper aims at establishing the theoretic and algorithmic foundations by exploiting the geometric properties of Hamiltonian subalgebras and describing a common mechanism for deriving the Lax dynamics. [ABSTRACT FROM AUTHOR]

Geometric algebras of dimension n<6$$ n<6 $$ are becoming increasingly popular for the modeling of 3D and 3 + 1D geometry. With this increased popularity comes the need for efficient algorithms for common operations such as normalization, square roots, and exponential and logarithmic maps. The current work presents a signature‐agnostic analysis of these common operations in all geometric algebras of dimension n<6$$ n<6 $$ and gives efficient numerical implementations in the most popular algebras ℝ4,ℝ3,1,ℝ3,0,1$$ {\mathbb{R}}_4,{\mathbb{R}}_{3,1},{\mathbb{R}}_{3,0,1} $$, and ℝ4,1$$ {\mathbb{R}}_{4,1} $$, in the hopes of lowering the threshold for adoption of geometric algebra solutions by code maintainers. [ABSTRACT FROM AUTHOR]

Head pose estimation (HPE) is a challenging and critical research subject with a wide range of applications in areas such as driver monitoring, attention recognition, and human-computer interaction. However, there are two challenging problems in HPE, the first one is that in real application scenarios, occlusion is very common, which affects the accuracy of HPE to a great extent. The second is that most research works use Euler angles to represent the head pose, which may lead to problems in neural network optimization. To solve these problems, an adaptive occlusion hybrid second-order attention network model was proposed. First, facial landmarks were detected by the occlusion-aware module to generate heat maps reflecting the presence or absence of occlusion in the specific facial parts, thereby enhancing features in the non-occluded parts of the face and suppressing features in the occluded regions. Meanwhile, we designed a novel second-order information attention module to interact with spatial and channel information using second-order statistical information, such that the model learns the feature correlations of different facial parts while paying more attention to important channels and suppressing redundant ones to further reduce the effect of occlusion and excavate more powerful features. Furthermore, to avoid ambiguity in common head pose representation, we introduced an exponential map to represent the head pose and designed a prediction framework capable of capturing the geometry of the pose space. The results of the experiments showed that the proposed model was competitive with methods using depth information from the BIWI dataset and achieved obvious advantages on the challenging AFLW2000 dataset, with more robust performance under large poses and occlusion interference, and stronger robustness compared with other models. [ABSTRACT FROM AUTHOR]

In this paper, we have investigated the projective action of the Lie group SL(3;R) on the homogeneous space RP2. In particular, we have studied the action of the subgroups of SL(3;R) on the non-degenerate conics in the space RP2. Using the Iwasawa decomposition of SL(2;R), we demonstrate that the isotropy subgroup of the projective unit circle is isomorphic to PSL(2;R) under certain conditions. [ABSTRACT FROM AUTHOR]

We present the Julia package Manifolds.jl, providing a fast and easy-to-use library of Riemannian manifolds and Lie groups. This package enables working with data defined on a Riemannian manifold, such as the circle, the sphere, symmetric positive definite matrices, or one of the models for hyperbolic spaces. We introduce a common interface, available in ManifoldsBase.jl, with which new manifolds, applications, and algorithms can be implemented. We demonstrate the utility of Manifolds.jl using Bézier splines, an optimization task on manifolds, and principal component analysis on nonlinear data. In a benchmark, Manifolds.jl outperforms all comparable packages for low-dimensional manifolds in speed; over Python and Matlab packages, the improvement is often several orders of magnitude, while over C/C++ packages, the improvement is two-fold. For high-dimensional manifolds, it outperforms all packages except for Tensorflow-Riemopt, which is specifically tailored for high-dimensional manifolds. [ABSTRACT FROM AUTHOR]

The Heisenberg Lie Group is the most frequently used model for studying the representation theory of Lie groups. This Lie group is modular-noncompact and its Lie algebra is nilpotent. The elements of Heisenberg Lie group and algebra can be expressed in the form of matrices of size 3×3. Another specialty is also inherited by its three-dimensional Lie algebra and is called the Lie Heisenberg algebra. The Heisenberg Lie Group whose Lie Algebra is extended to the dimension 2n+1 is called the generalized Heisenberg Lie group and it is denoted by H whose Lie algebra is h_n. In this study, the surjectiveness of exponential mapping for H was studied with respect to h_n=⟨x ̅,y ̅,z ̅⟩ whose Lie bracket is given by [X_i,Y_i ]=Z. The purpose of this research is to prove the characterization of the Lie subgroup with respect to h_n. In this study, the results were obtained that if ⟨x ̅,y ̅ ⟩=:V⊆h_n a subspace and a set {e^(x_i ) e^(x_j ) ┤| x_i,x_j∈V }=:L⊆H then L=H and consequently Lie(L)≠V.

We approximate the front of a flat sub-Riemannian structure on the Engel distribution in a neighborhood of a non-subanalytic singularity by the front of a control system integrable in elementary functions. As a corollary, we find the asymptotics of the exponential map of a flat sub-Riemannian structure on the Engel distribution. [ABSTRACT FROM AUTHOR]

A subset B of a space X is said to be bounded (in X) if the restriction to B of every real-valued continuous function on X is bounded. A real-valued function on X is called bf-continuous if its restriction to each bounded subset of X has a continuous extension to the whole space X. bf-spaces are spaces such that bf-continuous functions are continuous. We take advantage to the exponential map in the realm of bf-spaces in order to study bf-extensions of bf-continuous functions. This allows us to improve several results concerning the distribution of the functor of the Dieudonné completion. We also prove that a relative version of the classical Glicksberg’s theorem characterizing the product of two pseudocompact spaces is valid for kr-spaces. In the last section we show that bf-hemibounded groups are Moscow spaces and, consequently, they are strong-PT-groups.

We prove a number of results concerning the Hausdorff and packing dimension of sets of points which escape (at least in average) to infinity at a given rate under non-autonomous iteration of exponential maps. In particular, we generalize the results proved by Sixsmith in 2016 and answer his question on annular itineraries for exponential maps. [ABSTRACT FROM AUTHOR]

Nonholonomic mechanics describes the motion of systems constrained by nonintegrable constraints. One of its most remarkable properties is that the derivation of the nonholonomic equations is not variational in nature. However, in this paper, we prove (Theorem 1.1) that for kinetic nonholonomic systems, the solutions starting from a fixed point q are true geodesics for a family of Riemannian metrics on the image submanifold M q nh of the nonholonomic exponential map. This implies a surprising result: the kinetic nonholonomic trajectories with starting point q, for sufficiently small times, minimize length in M q nh ! [ABSTRACT FROM AUTHOR]

A theory of local convexity for a second order differential equation (sode sode) on a Lie algebroid is developed. The particular case when the sode sode is homogeneous quadratic is extensively discussed. [ABSTRACT FROM AUTHOR]

We discuss the general Rodrigues problem and we give explicit determinant formulae for the coefficients when the eigenvalues of the matrix have double multiplicity (Theorem 5). When n = 4 explicit formulae and effective computations for the exponential map on the Lie group SO(4) are presented. [ABSTRACT FROM AUTHOR]

We show that a group of diffeomorphisms D on the open unit interval I, equipped with the topology of uniform convergence on any compact set of the derivatives at any order, is non-regular: the exponential map is not defined for some path of the Lie algebra. This result extends to the group of diffeomorphisms of finite dimensional, non-compact manifold M.

In this paper, we first consider the relationship between a polynomial ring B over a Noetherian domain R and the ring of invariants A of a G a -action on B , when A occurs as a retract of B. Next, we study retracts of a polynomial ring in general and address the questions of D.L. Costa raised in [5]. Finally, we examine the behaviour of ideals and certain properties of rings under retractions. [ABSTRACT FROM AUTHOR]

In this article, we prove an isoperimetric inequality for the harmonic mean of the first (n - 1) nonzero Steklov eigenvalues on bounded domains in n-dimensional hyperbolic space. Our approach to prove this result also gives a similar inequality for the first n nonzero Steklov eigenvalues on bounded domains in n-dimensional Euclidean space. [ABSTRACT FROM AUTHOR]

In this paper, we will construct a new type of non-landing exponential rays, each of whose accumulation sets is bounded, disjoint from the ray and homeomorphic to the closed topologist's sine curve. [ABSTRACT FROM AUTHOR]

During recent decades, G a -actions, especially certain invariants of G a -actions, have been important tools in the study of affine varieties. The G a -actions are usually studied through locally nilpotent derivations in characteristic zero and exponential maps (see Definition 1.1) in arbitrary characteristic. The "Makar-Limanov invariant" of locally nilpotent derivations played a pivotal role in solving the linearization conjecture in the 1990s, while invariants of exponential maps were central to N. Gupta's resolution of the Zariski cancellation problem in positive characteristic. In the study of locally nilpotent derivations on commutative algebras containing Q , Freudenburg and Moser-Jauslin (Mich Math J 62:227–258, (2013), Theorem 6.1) have introduced a new invariant called "rigid core" and used it to formulate an alternative version of Mason's theorem and to prove a well-known analogue of Fermat's last theorem for rational functions (Freudenburg and Moser-Jauslin (2013), Corollary 6.1). In this note, we consider the concept of the rigid core in the framework of exponential maps on commutative algebras over an algebraically closed field k of arbitrary characteristic. We observe that for any factorial k-domain B with tr.deg k (B) = 2 , the concept of rigid core coincides with the Makar-Limanov invariant. We also show that over any affine two-dimensional normal k-domain B, its rigid core is a stable invariant. [ABSTRACT FROM AUTHOR]

Image encryption (IE) has been essential for internet-based storing and transferring in recent years. Effective chaotic systems play a crucial role in IE schemes which widely depend on chaotic maps and keys. However, the existing chaotic maps suffer from low performance and narrow chaotic ranges, and they utilize casual key generation approaches rather than optimum keys. In this study, an IE scheme based on key optimization using particle swarm optimization (PSO) algorithm and a novel modular integrated logistic exponential (MILE) map is presented. The chaotic performance of the MILE map is validated across a comparison with the existing maps thorough measurements such as LE, SE, 0–1 Test, and PE with means values of 12.0000, 2.1866, 0.9981, and 0.9998, respectively. The key is optimized on a small part of the image to be encrypted instead of the whole image that is employed by the existing works. That is why the IE scheme is faster than those works. Afterward, the PSO-based IE scheme is undergone reliable crypto-analyses and attacks. It is also verified through a comparison with the existing IE schemes. The proposed IE scheme is the best owing to having exceptional mean values of key sensitivity 99.6117, variance 893.12, χ 2 222.4057, information entropy 7.9994, NPCR 99.6090, and UACI 33.4662. Thanks to the MILE map, the PSO-based IE scheme demonstrates excellent numerical and visual encryption results. [ABSTRACT FROM AUTHOR]

In this paper, first we establish the explicit relation between 3-derivations and 3- automorphisms of a Lie algebra using the differential and exponential map. More precisely, we show that the Lie algebra of 3-derivations is the Lie algebra of the Lie group of 3-automorphisms. Then we study the derivations and automorphisms of the standard embedding Lie algebra of a Lie triple system. We prove that derivations and automorphisms of a Lie triple system give rise to derivations and automorphisms of the corresponding standard embedding Lie algebra. Finally we compute the 3-derivations and 3-automorphisms of 3-dimensional real Lie algebras.

Let k be a field and let B be an affine normal domain over k. Let ϕ be a non-trivial exponential map on B and let A = B ϕ be the ring of ϕ -invariants. Since A is factorially closed in B, A = K ∩ B where K denotes the field of fractions of A. Hence A is a Krull domain. We investigate here a relation between the class group Cl (A) of A and the class group Cl (B) of B. In this direction, we give a sufficient condition for an injective group homomorphism from Cl (A) to Cl (B) . We also give an example to show that Cl (A) may not be realized as a subgroup of Cl (B) . [ABSTRACT FROM AUTHOR]

We compute the injectivity radius on particular subspaces of the Kendall shape space Σkm The space Σkm is useful for representing the shapes associated to collections of k vector columns in Rm. We determine also an upper bound of arc length parameter of specific curves lying in particular subspaces of the pre-shape sphere Skm Here, Σkm m is the quotient space of Skm modulo the special orthogonal group SO(m) acting on the left. [ABSTRACT FROM AUTHOR]

Devaney and Krych (Ergod Theory Dyn Syst 4:35–52, 1984) showed that for the exponential family λ e z , where 0 < λ < 1 / e , the Julia set consists of uncountably many pairwise disjoint simple curves tending to ∞ , which they called hairs. Viana proved that these hairs are smooth. Barański as well as Rottenfußer, Rückert, Rempe and Schleicher gave analogues of the result of Devaney and Krych for more general classes of functions. In contrast to Viana's result we construct in this article an entire function, where the Julia set consists of hairs, which are nowhere differentiable. [ABSTRACT FROM AUTHOR]

Let (X , *) {(X,*)} be a hypergroup and let w 0 {w_{0}} be a fixed measure on X. In this paper we study the two functional equations 〈 δ x * δ y * ω 0 , g 〉 + 〈 δ x * δ y ˇ * ω 0 , g 〉 = 2 ⁢ g ⁢ (x) ⁢ g ⁢ (y) , x , y ∈ X , \langle\delta_{x}*\delta_{y}*\omega_{0},g\rangle+\langle\delta_{x}*\delta_{% \check{y}}*\omega_{0},g\rangle=2g(x)g(y),\quad x,y\in X, and 〈 δ x * δ y ˇ * ω 0 , f 〉 - 〈 δ x * δ y * ω 0 , f 〉 = 2 ⁢ f ⁢ (x) ⁢ f ⁢ (y) , x , y ∈ X , \langle\delta_{x}*\delta_{\check{y}}*\omega_{0},f\rangle-\langle\delta_{x}*% \delta_{y}*\omega_{0},f\rangle=2f(x)f(y),\quad x,y\in X, where g , f : X → ℂ {g,f:X\to\mathbb{C}} are continuous and bounded functions to be determined. We express the solutions of the two functional equations in terms of multiplicative maps on (X , *) {(X,*)}. As an application we give the solution of the two functional equations on polynomial and Sturm–Liouville hypergroups. [ABSTRACT FROM AUTHOR]

In this work, we began to take forward kinematics of the Gough–Stewart (G-S) platform as an unconstrained optimization problem on the Lie group-structured manifold SE(3) instead of simply relaxing its intrinsic orthogonal constraint when algorithms are updated on six-dimensional local flat Euclidean space or adding extra unit norm constraint when orientation parts are parametrized by a unit quaternion. With this thought in mind, we construct two kinds of iterative problem-solving algorithms (Gauss–Newton (G-N) and Levenberg–Marquardt (L-M)) with mathematical tools from the Lie group and Lie algebra. Finally, a case study for a general G-S platform was carried out to compare these two kinds of algorithms on SE(3) with corresponding algorithms that updated on six-dimensional flat Euclidean space or seven-dimensional quaternion-based parametrization Euclidean space. Experiment results demonstrate that those algorithms on SE(3) behave better than others in convergence performance especially when the initial guess selection is near to branch solutions.

Exponential coordinates are widely used in simulation codes for flexible multibody systems based on a Lie group approach. The accurate and efficient evaluation of the exponential map, the tangent operator and its higher order derivatives is crucial. This paper presents a systematic derivation process based on the matrix series expansion of the exponential map and of the tangent operator. This approach is general as it can be applied to any matrix Lie group. For the Lie groups SO (3) and SE (3), the closed form expression of these operators can also be established and is summarized in the paper. It is shown that the closed form of the operators is affected by round-off errors for small rotation amplitudes, whereas the series form is affected by truncation errors at high rotation amplitudes. The computational efficiency of the two approaches is also discussed. A switching strategy between the closed form and the series form is then proposed to obtain an adjustable compromise between accuracy and computational cost. • Lie group operators for flexible multibody systems are developed. • The exponential map and its derivatives are given in truncated matrix series form. • The ill-conditioning of exponential map derivatives in closed form is analyzed. • Numerical errors and costs are analyzed for the exponential map and derivatives. • A switching strategy between the closed form and the series form is proposed. [ABSTRACT FROM AUTHOR]

Abstract The pseudo-unitary group U (p , q) of signature (p , q) is the group of matrices that preserve the indefinite pseudo-Euclidean metric on the vector space C p , q. The goal of this paper is to describe the set U s (p , p) of Hermitian, or, self-adjoint elements in U (p , p). [ABSTRACT FROM AUTHOR]

We investigate the connections between the differential-geometric properties of the exponential map from the space of real skew symmetric matrices onto the group of real special orthogonal matrices and the manifold of real orthogonal matrices equipped with the Riemannian structure induced by the Frobenius metric. [ABSTRACT FROM AUTHOR]

Computing geodesic distance on a mesh surface S efficiently and accurately is a central task in numerous computer graphics applications. In order to deal with high-resolution mesh surfaces, a lightweight preprocessing is a proper choice to make a balance between query accuracy and speed. In the preprocessing stage, we build a proximity graph G with regard to a set of sample points and keep the exact geodesic distance between any pair of nearby sample points. In the query stage, given two query points s and t , we augment the proximity graph G by adding s and t on-the-fly, and then use the shortest path between s and t on the augmented proximity graph to approximate the exact geodesic path between s and t . We establish an empirical relationship between the number of samples and expected accuracy (measured in relative error), which facilitates fast and accurate query of geodesic distance with a lightweight processing cost. We exhibit the uses of the new approach in two applications—real-time computation of discrete exponential map for texture mapping and interactive design of spline curves on surfaces. [ABSTRACT FROM AUTHOR]

Let L / K be an extension of complete discrete valuation fields, and assume that the residue field of K is perfect and of positive characteristic. The residue field of L is not assumed to be perfect. In this paper, we prove a formula for the Swan conductor of the image of a character χ ∈ H 1 ( K , Q / Z ) in H 1 ( L , Q / Z ) for χ sufficiently ramified. Further, we define generalizations ψ L / K ab and ψ L / K AS of the classical ψ -function and prove a formula for ψ L / K ab ( t ) for sufficiently large t ∈ R . [ABSTRACT FROM AUTHOR]

We introduce a Frechet Lie group structure on the Riordan group. We give a description of the corresponding Lie algebra as a vector space of infinite lower triangular matrices. We describe a natural linear action induced on the Frechet space K N by any element in the Lie algebra. We relate this to a certain family of bivariate linear partial differential equations. We obtain the solutions of such equations using one-parameter groups in the Riordan group. We show how a particular semidirect product decomposition in the Riordan group is reflected in the Lie algebra. We study the stabilizer of a formal power series under the action induced by Riordan matrices. We get stabilizers in the fraction field K ( ( x ) ) using bi-infinite representations. We provide some examples. The main tool to get our results is the paper [18] where the Riordan group was described using inverse sequences of groups of finite matrices. [ABSTRACT FROM AUTHOR]

Deep compositing is an important practical tool in creating digital imagery, but there has been little theoretical analysis of the underlying mathematical operators. Motivated by finding a simple formulation of the merging operation on OpenEXR-style deep images, we show that the Porter-Duff over function is the operator of a Lie group. In its corresponding Lie algebra, the splitting and mixing functions that OpenEXR deep merging requires have a particularly simple form. Working in the Lie algebra, we present a novel, simple proof of the uniqueness of the mixing function. The Lie group structure has many more applications, including new, correct resampling algorithms for volumetric images with alpha channels, and a deep image compression technique that outperforms that of OpenEXR. [ABSTRACT FROM AUTHOR]