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For a smooth affine algebra R of dimension d ≥ 3 over a field k and an invertible alternating matrix χ of rank 2 n , the group S p (χ) of invertible matrices of rank 2 n over R which are symplectic with respect to χ acts on the right on the set U m 2 n (R) of unimodular rows of length 2 n over R. In this paper, we prove that S p (χ) acts transitively on U m 2 n (R) if k is algebraically closed, d ! ∈ k × and 2 n ≥ d. [ABSTRACT FROM AUTHOR]

We study a family of completely integrable systems attached to Takiff algebras g N , extending Toda systems for split simple Lie algebras g. With respect to Darboux coordinates on coadjoint orbits O , the potentials of the hamiltonians are products of polynomial and exponential functions. Explicit general solutions for their equations of motion are obtained using differential operators called jet transformations. The classical integrable systems are then lifted to families of commuting operators in an enveloping algebra, quantizing the Poisson algebra of functions on O. These results are illustrated with concrete examples, including a 3-body problem based on sl (2) and an extension of soliton solutions for A ∞ to associated Takiff algebras. [ABSTRACT FROM AUTHOR]

Previously we showed that the tensor product of a weight module over a generalized Weyl algebra (GWA) with a weight module over another GWA is a weight module over a third GWA. In this paper we compute tensor products of simple and indecomposable weight modules over generalized Weyl algebras supported on a finite orbit. This allows us to give a complete presentation by generators and relations of the Grothendieck ring of the categories of weight modules over a tower of generalized Weyl algebras in this setting. We also obtain partial results about the split Grothendieck ring. We described the case of infinite orbits in previous work. [ABSTRACT FROM AUTHOR]

We show that the Nichols algebra of a simple Yetter-Drinfeld module over a projective special linear group over a finite field whose support is a semisimple orbit has infinite dimension, provided that the elements of the orbit are reducible; we obtain a similar result for all semisimple orbits in a finite symplectic group except in low rank. We prove that orbits of irreducible elements in the projective special linear groups of odd prime degree could not be treated with our methods. We conclude that any finite-dimensional pointed Hopf algebra H with group of group-like elements isomorphic to PSL n (q) (n ≥ 4) , PSL 3 (q) (q > 2) , or PSp 2 n (q) (n ≥ 3) , is isomorphic to a group algebra. [ABSTRACT FROM AUTHOR]

We study a class of Z -graded algebras introduced by Bell and Rogalski. Their construction generalizes in large part that of rank one generalized Weyl algebras (GWAs). We establish certain ring-theoretic properties of these algebras and study their connection to GWAs. We classify the simple weight modules in the infinite orbit case and provide a partial classification in the case of orbits of finite order. [ABSTRACT FROM AUTHOR]

We provide an explicit combinatorial description of highest weights of simple highest weight modules over the simple affine vertex algebra L κ (sl n + 1) with n ∈ N of admissible level κ. For admissible simple highest weight modules corresponding to the principal, subregular and maximal parabolic nilpotent orbits we give a realization using the Gelfand–Tsetlin theory, which also allows us to obtain a realization of certain classes of simple admissible sl 2 -induced modules in these orbits. In particular, simple admissible sl 2 -induced modules are fully classified for the principal nilpotent orbit. [ABSTRACT FROM AUTHOR]

Let G = 1 + A be a finite pattern group, where A is the associated pattern algebra over a finite field F q. For any monomial matrix T in A t , we explicitly construct an irreducible character Φ T of G , corresponding to T. Furthermore, if A is symmetric (in the sense of Definition 3.2), we can determine the largest character degree of 1 + A using this construction. We also propose a conjecture that an explicit one-to-one correspondence between irreducible representations and coadjoint orbits exists for any finite pattern groups. [ABSTRACT FROM AUTHOR]

Large rotating machinery is widely applied in many industrial applications and shaft orbit classification plays an important role in fault diagnosis of large rotating machinery. Therefore, we propose a novel raw data feature-based automatic and deterministic shaft orbit classification framework for large rotating machinery. The main idea of the proposed framework is that it directly uses the most discriminative geometric features of each shaft orbit type for classification. Different from existing supervised learning shaft orbit classification methods, it does not need any labeled data and it is a deterministic, quite simple, light, easy to implement. Extensive experiments of synchronous integrated period sampling and fixed sampling at different rotating speeds conducted on real and simulated datasets have shown the superiority of proposed framework. This study provides a new deterministic interpretable and explainable approach to support predictive maintenance of large rotating machinery in various industrial fields even when no labeled data is available. [Display omitted] • ADSOC is a raw data feature-based classification framework so that it does not need any labeled data. • ADSOC is invariant to shaft orbit translation, scale and rotation. • ADSOC model and results are easy to interpret and explain. • ADSOC is independent from both shaft orbit starting point and rotating direction, and the shaft orbit does not need to be closed. • ADSOC is a deterministic framework and it is quite simple, light and easy to implement. [ABSTRACT FROM AUTHOR]

We consider homoclinic solutions for Hamiltonian systems in symplectic Hilbert spaces and generalise spectral flow formulas that were proved by Pejsachowicz and the author in finite dimensions some years ago. Roughly speaking, our main theorem relates the spectra of infinite dimensional Hamiltonian systems under homoclinic boundary conditions to intersections of their stable and unstable spaces. Our proof has some interest in its own. Firstly, we extend a celebrated theorem by Cappell, Lee and Miller about the classical Maslov index in R 2 n to symplectic Hilbert spaces. Secondly, we generalise the classical index bundle for families of Fredholm operators of Atiyah and Jänich to unbounded operators for applying it to Hamiltonian systems under varying boundary conditions. Finally, we substantially make use of striking results by Abbondandolo and Majer to study Fredholm properties of infinite dimensional Hamiltonian systems. [ABSTRACT FROM AUTHOR]

Let n be a locally nilpotent infinite-dimensional Lie algebra over C. Let U (n) and S (n) be its respective universal enveloping algebra and symmetric algebra. Consider the Jacobson topology on the primitive spectrum of U (n) , and the Poisson topology on the primitive Poisson spectrum of S (n). We provide a homeomorphism between the corresponding topological spaces (at the level of points, it gives a bijection between the primitive ideals of U (n) and S (n)). We also show that all primitive ideals of S (n) from an open set in a properly chosen topology are generated by their intersections with the Poisson center. Under the assumption that n is a nil-Dynkin Lie algebra, we give two criteria for primitive ideals I (λ) ⊂ S (n) and J (λ) ⊂ U (n) , λ ∈ n ⁎ , to be nonzero. Most of these results generalize known facts about primitive and Poisson spectrum for finite-dimensional nilpotent Lie algebras (but note that for a finite-dimensional nilpotent Lie algebra all primitive ideals I (λ) , J (λ) are nonzero). [ABSTRACT FROM AUTHOR]

It is known that the Melnikov function method is equivalent to the averaging method for studying the number of limit cycles of planar analytic (or C ∞) near-Hamiltonian differential systems. In this paper, we study piecewise smooth near-integrable systems and establish the Melnikov function method and the averaging method for finding limit cycles. We also show the equivalence of the two methods even for systems in high dimensional space. Particularly, we obtain the formula of the second order Melnikov function for planar piecewise near-Hamiltonian systems. We finally provide an application example. [ABSTRACT FROM AUTHOR]

Given a finite group G and two unitary G -representations V and W , possible restrictions on topological degrees of equivariant maps between representation spheres S (V) and S (W) are usually expressed in a form of congruences modulo the greatest common divisor of lengths of orbits in S (V) (denoted by α (V)). Effective applications of these congruences is limited by answers to the following questions: (i) under which conditions, is α (V) > 1 ? and (ii) does there exist an equivariant map with the degree easy to calculate? In the present paper, we address both questions. We show that α (V) > 1 for each irreducible non-trivial C G -module if and only if G is solvable. This provides a new solvability criterion for finite groups. For non-solvable groups, we use 2-transitive actions to construct complex representations with non-trivial α -characteristic. Regarding the second question, we suggest a class of Norton algebras without 2-nilpotents giving rise to equivariant quadratic maps, which admit an explicit formula for the degree. [ABSTRACT FROM AUTHOR]

Abstract Let W m | n be the (finite) W -algebra attached to the principal nilpotent orbit in the general linear Lie superalgebra gl m | n (C). In this paper we study the Whittaker coinvariants functor , which is an exact functor from category O for gl m | n (C) to a certain category of finite-dimensional modules over W m | n. We show that this functor has properties similar to Soergel's functor V in the setting of category O for a semisimple Lie algebra. We also use it to compute the center of W m | n explicitly, and deduce consequences for the classification of blocks of O up to Morita/derived equivalence. [ABSTRACT FROM AUTHOR]

Abstract Let G be a group. The orbits of the natural action of Aut (G) on G are called "automorphism orbits" of G , and the number of automorphism orbits of G is denoted by ω (G). In this paper we prove that if G is an FC-group with finitely many automorphism orbits, then the derived subgroup G ′ is finite and G admits a decomposition G = Tor (G) × D , where Tor (G) is the torsion subgroup of G and D is a divisible characteristic subgroup of Z (G). We also show that if G is an infinite FC-group with ω (G) ⩽ 8 , then either G is soluble or G ≅ A 5 × H , where H is an infinite abelian group with ω (H) = 2. Moreover, we describe the structure of the infinite non-soluble FC-groups with at most eleven automorphism orbits. [ABSTRACT FROM AUTHOR]

Let C be a G -category for some group G . We show that a G -covering functor p : C ⟶ C / G induces G -precovering of their bounded derived categories, singularity categories and Gorenstein defect categories. Then we provide a Gorenstein version of Gabriel's theorem. Using this result we investigate the number of summands in a decomposition of the middle term of almost split sequences over monomial algebras. [ABSTRACT FROM AUTHOR]

An S U ( p , q ) -flag domain is an open orbit of the real Lie group S U ( p , q ) acting on the complex flag manifold associated to its complexification S L ( p + q , C ) . Any such flag domain contains certain compact complex submanifolds, called cycles, which encode much of the topological, complex geometric and representation theoretical properties of the flag domain. This article is concerned with the description of these cycles in homology using a specific type of Schubert varieties. They are defined by the condition that the fixed point of the Borel group in question is in the closed S U ( p , q ) -orbit in the ambient manifold. Equivalently, the Borel group contains the AN -factor of some Iwasawa decomposition. We consider the Schubert varieties of this type which are of complementary dimension to the cycles. It is known that if such a variety has non-empty intersection with a certain base cycle, then it does so transversally (in finitely many points). With the goal of understanding this duality, we describe these points of intersection in terms of flags as well as in terms of fixed points of a given maximal torus. The relevant Schubert varieties are described in terms of Weyl group elements. Much of our work is of an algorithmic nature, but, for example in the case of maximal parabolics, i.e. Grassmannians, formulas are derived. [ABSTRACT FROM AUTHOR]

This paper studies the orbital instability of standing waves for the Klein–Gordon–Schrödinger system in three space dimensions. By variational methods we first show the existence of ground states. Then we establish a Virial identity for this system, by which and a Virial theorem, we manage to prove that the standing waves we obtained are orbital instable as if the frequency ω is sufficiently small. Our results improve and complement some previous ones. [ABSTRACT FROM AUTHOR]

In this manuscript, we compute explicitly the Lusztig–Vogan bijection for local systems of some classical, special, nilpotent orbits. Using these results, we prove a conjecture of Achar and Sommers on regular functions of some covers of classical nilpotent orbits. [ABSTRACT FROM AUTHOR]

In this paper we establish the orbital stability of periodic waves related to the logarithmic Korteweg–de Vries equation. Our motivation is inspired in the recent work [3] , in which the authors established the well-posedness and the linear stability of Gaussian solitary waves. By using the approach put forward recently in [20] to construct a smooth branch of periodic waves as well as to get the spectral properties of the associated linearized operator, we apply the abstract theories in [13] and [25] to deduce the orbital stability of the periodic traveling waves in the energy space. [ABSTRACT FROM AUTHOR]

In this paper we study the period function of x ¨ = ( 1 + x ) p − ( 1 + x ) q , with p , q ∈ R and p > q . We prove three independent results. The first one establishes some regions in the parameter space where the corresponding center has a monotonous period function. This result extends the previous ones by Miyamoto and Yagasaki for the case q = 1 . The second one deals with the bifurcation of critical periodic orbits from the center. The third one is addressed to the critical periodic orbits that bifurcate from the period annulus of each one of the three isochronous centers in the family when perturbed by means of a one-parameter deformation. These three results, together with the ones that we obtained previously on the issue, lead us to propose a conjectural bifurcation diagram for the global behaviour of the period function of the family. [ABSTRACT FROM AUTHOR]

Let G be a real, reductive algebraic group, and let X be a homogeneous space for G with a non-zero invariant density. We give an explicit description of a Zariski open, dense subset of the asymptotics of the tempered support of L 2 ( X ) . Under additional hypotheses, this result remains true for vector bundle valued harmonic analysis on X . These results follow from an upper bound on the wave front set of an induced Lie group representation under a uniformity condition. [ABSTRACT FROM AUTHOR]

We call that a flow has the orbital shadowing property if for any ε > 0 there is d > 0 such that, for any d -pseudo orbit g ( t ) there exists an orbit Orb ( x ) satisfying d i s t H ( g ( t ) ‾ , Orb ( x ) ‾ ) < ε . In this paper, we show that the C 1 -interior of the set of 3-dimensional flows having the orbital shadowing property is contained in the set of Ω-stable 3-flows. [ABSTRACT FROM AUTHOR]

Let T : X → X be a hypercyclic operator of an infinite dimensional separable Banach space X . By modifying a construction of Grivaux, we will show that the T -orbit of a hypercyclic vector can intersect certain closed vector subspaces of X in many strange ways. Moreover, the set of visiting times of the orbit to the subspace can also be quite exotic, especially when the operator satisfies a stronger form of hypercyclicity. As a by-product, we will improve and/or supply new proofs to some of the recent results about subspace-hypercyclicity. [ABSTRACT FROM AUTHOR]

According to a theorem of Brieskorn and Slodowy, the intersection of the nilpotent cone of a simple Lie algebra with a transverse slice to the subregular nilpotent orbit is a simple surface singularity. At the opposite extremity of the poset of nilpotent orbits, the closure of the minimal nilpotent orbit is also an isolated symplectic singularity, called a minimal singularity. For classical Lie algebras, Kraft and Procesi showed that these two types of singularities suffice to describe all generic singularities of nilpotent orbit closures: specifically, any such singularity is either a simple surface singularity, a minimal singularity, or a union of two simple surface singularities of type A 2 k − 1 . In the present paper, we complete the picture by determining the generic singularities of all nilpotent orbit closures in exceptional Lie algebras (up to normalization in a few cases). We summarize the results in some graphs at the end of the paper. In most cases, we also obtain simple surface singularities or minimal singularities, though often with more complicated branching than occurs in the classical types. There are, however, six singularities that do not occur in the classical types. Three of these are unibranch non-normal singularities: an SL 2 ( C ) -variety whose normalization is A 2 , an Sp 4 ( C ) -variety whose normalization is A 4 , and a two-dimensional variety whose normalization is the simple surface singularity A 3 . In addition, there are three 4-dimensional isolated singularities each appearing once. We also study an intrinsic symmetry action on the singularities, extending Slodowy's work for the singularity of the nilpotent cone at a point in the subregular orbit. [ABSTRACT FROM AUTHOR]

For synchronous binary asteroid, the configuration of the spin-axes of the two asteroids being aligned and normal to the orbital plane is unstable in case of attitude instability. Focusing on the attitude stability of the secondary, there are four contributions of this study. (1) Stability analysis using the method of periodic orbit reveals the functional dependence of the attitude stability of the secondary on some system parameters — mass ratio of the binary, mutual orbit distance, orbit eccentricity, and the secondary's elongations. The stability results obtained for a wide range of system parameters help explain the significant lack of synchronous secondaries with equatorial elongations larger than 1.5 reported by Pravec et al. (2016) and also predict a lack of synchronous secondaries with equatorial elongations about 1.1. (2) Attitude unstable secondary is found to exhibit different excited rotation states which are sensitive to the orbit eccentricity: for small eccentricities the typical state is a small amplitude tumbling (non-principal axis rotation) which can be either regular or chaotic; as the eccentricity increases, the typical state is a chaotic rolling along its longest axis, i.e., the phenomenon known as barrel instability by Ćuk et al. (2021); for sufficiently large eccentricities, the synchronous state of the secondary may break within a short time interval. (3) Using the analytical approach of normal form of Hamiltonian system, the inherent mechanism behind the attitude instability is attributed to the secondary resonance between the fundamental frequencies of the principal axis rotation and the non-principal axis rotation. (4) A systematic comparison study is carried out between the full model adopted here with a finite mass ratio and the classic restricted model for planet–satellite case (Wisdom et al., 1984). The result shows that for binary asteroids with the secondary's elongation larger than 1.5, the full model is recommended. • Spatial spin-orbit coupling of binary asteroids is studied. • Attitude instability can happen while planar stability still holds. • Types of attitude instability are related with the orbit eccentricity. • The secondary resonance is responsible for the spatial instability. • Highly elongated secondaries (with Suruga as a candidate) can be attitude stable. [ABSTRACT FROM AUTHOR]

We introduce the notion of F - expansive measure by making the dynamical ball in [4] to depend on a given subset F of the set of all the reparametrizations H . We prove that these measures satisfy some interesting properties resembling the expansive ones. These include the equivalence with expansivity when F = H , the vanishing along the orbits, the absence of singularities in the support, the F -expansivity with respect to time t -maps, the invariance under equivalence and the characterization for suspensions. We also analyze the support of the F -expansive measures and prove that there exists a dense subset of measures (in the set of F -expansive measures) all of them with a common support. Finally, we extend to flows the recent result for homeomorphisms in [11] . [ABSTRACT FROM AUTHOR]

We study the regular function ring R ( O ) for all symplectic nilpotent orbits O with even column sizes. We begin by recalling the quantization model for all such orbits by Barbasch using unipotent representations. With this model, we express the multiplicities of fundamental representations appearing in R ( O ) by a parabolically induced module. Finally, we will use this formula to give a criterion on the normality of the Zariski closure O ‾ of O . [ABSTRACT FROM AUTHOR]

In this paper we start a classification of certain global integrals. First, we use the language of unipotent orbits to write down a family of global integrals. We then classify all those integrals which satisfy the dimension equation we set. After doing so, we check which of these integrals are global unipotent integrals. We do all this for groups of type A n , and using all this we derive a certain interesting conjecture about the length of these integrals. [ABSTRACT FROM AUTHOR]

• We perform a study of the 1-dim unstable and stable manifolds of L 3 in the RPC3BP for small values of the mass ratio. • In this work and its sequel we obtain an asymptotic formula for the distance between the manifolds of L 3. • This distance is exponentially small with respect to the mass ratio, thus classical perturbative methods do not apply. The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies, called the primaries. If the primaries perform circular motions and the massless body is coplanar with them, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In synodic coordinates, it is a two degrees of freedom Hamiltonian system with five critical points, L 1 ,.. , L 5 , called the Lagrange points. The Lagrange point L 3 is a saddle-center critical point which is collinear with the primaries and is located beyond the largest of the two. In this paper and its sequel [10] , we provide an asymptotic formula for the distance between the one dimensional stable and unstable invariant manifolds of L 3 when the ratio between the masses of the primaries μ is small. It implies that L 3 cannot have one-round homoclinic orbits. If the mass ratio μ is small, the hyperbolic eigenvalues are weaker than the elliptic ones by factor of order μ. This implies that the distance between the invariant manifolds is exponentially small with respect to μ and, therefore, the classical Poincaré–Melnikov method cannot be applied. In this first paper, we approximate the RPC3BP by an averaged integrable Hamiltonian system which possesses a saddle center with a homoclinic orbit and we analyze the complex singularities of its time parameterization. We also derive and study the inner equation associated to the original perturbed problem. The difference between certain solutions of the inner equation gives the leading term of the distance between the stable and unstable manifolds of L 3. In the sequel [10] we complete the proof of the asymptotic formula for the distance between the invariant manifolds. [ABSTRACT FROM AUTHOR]

The fusion orbit category F ‾ C (G) of a discrete group G over a collection C is the category whose objects are the subgroups H in C , and whose morphisms H → K are given by the G -maps G / H → G / K modulo the action of the centralizer group C G (H). We show that the higher limits over F ‾ C (G) can be computed using the hypercohomology spectral sequences coming from the Dwyer G -spaces for centralizer and normalizer decompositions for G. If G is the discrete group realizing a saturated fusion system F , then these hypercohomology spectral sequences give two spectral sequences that converge to the cohomology of the centric orbit category O c (F). This allows us to apply our results to the sharpness problem for the subgroup decomposition of a p -local finite group. We prove that the subgroup decomposition for every p -local finite group is sharp (over F -centric subgroups) if it is sharp for every p -local finite group with nontrivial center. We also show that for every p -local finite group (S , F , L) , the subgroup decomposition is sharp if and only if the normalizer decomposition is sharp. [ABSTRACT FROM AUTHOR]

We investigate the behaviour of two recent methods for the computation of preliminary orbits. These methods are based on the conservation laws of Kepler's problem, and enable the linkage of very short arcs of optical observations even when they are separated in time by a few years. Our analysis is performed using both synthetic and real data of 822 main belt asteroids. The differences between computed and true orbital elements have been analysed for the true linkages, as well as the occurrence of alternative solutions. Some metrics have been introduced to quantify the results, with the aim of discarding as many of the false linkages as possible and keeping the vast majority of true ones. These numerical experiments provide thresholds for the metrics which take advantage of the knowledge of the ground truth : the values of these thresholds can be used in normal operation mode, when we do not know the correct values of the orbital elements and whether the linkages are true or false. • Numerical sensitivity analysis of the KI methods. • Test of KI methods with large data sets: they are suitable to explore the ITF. • Indicators to estimate the quality of the preliminary solutions. [ABSTRACT FROM AUTHOR]

The article describes a new method for determining the orbit of a satellite of a distant body by five measured apparent relative positions. Objects can be components of a visual binary star. The method is based on old ideas proposed back in the 19th century. In 1883, T.N. Thiele theoretically deduced and proposed an interesting method for determining an orbit from three apparent relative positions for a given sector (areal) velocity of the body in apparent orbit. In this paper, this method is supplemented with analytical formulas for determining the sectoral velocity of a satellite of a star or planet in apparent orbit in five positions. Five relative positions are needed to accurately determine the sector velocity of a body in a visible orbit. Further, the parameters of the spatial orbit are determined by three positions. The semi-major axis of the orbit and the mean motion are determined independently. This makes it possible to determine the total mass of the system under study. The proposed method is first tested on a model example, and then its adequacy is shown for the problem of determining the orbit of a visual binary star ZZ Tau based on real observations. In this example, the method is tested, its adequacy is shown. We also consider the application of the method for determining the preliminary orbit of an asteroid satellite with the aim of subsequent refinement of the orbit based on collection of observations over a large time interval. The reliability of the method has yet to be tested on specific problems. For convenience, an online service has been developed on the Internet allowing to calculate the orbit from the reader's observations. • Determining the orbit of a satellite of a distant body from five measured positions. • Visual double star orbit. • Online server for determining orbits. [ABSTRACT FROM AUTHOR]

Given a polynomial of degree d over a number field, the image of the associated arboreal representation of the absolute Galois group of the field is a profinite group acting on the d -ary tree. Boston and Jones conjectured that for a quadratic polynomial, the image of such a representation contains a dense set of settled elements. Here an element is settled if it exhibits a certain pattern of growth of cycles at finite levels of the tree. In this paper, we prove the conjecture of Boston and Jones generically in the case when the quadratic polynomial has a strictly pre-periodic post-critical orbit of length 2, and provide new evidence that the conjecture holds for quadratic polynomials with strictly pre-periodic post-critical orbits of length at least 3. To prove our results, we introduce a new dynamical method, which uses the notions of a maximal torus and its Weyl group. These notions are analogous to the notions of maximal tori and Weyl groups in the theory of compact Lie groups, where they are fundamental. For profinite groups, maximal tori and Weyl groups contain the information about settled elements, and this is the foundation of our method. [ABSTRACT FROM AUTHOR]

A bi-Cayley graph Γ is a graph which admits a semiregular group H of automorphisms with two orbits. In this paper, the normalizer of H in the full automorphism group of Γ is determined. Applying this, a characterization of cubic edge-transitive graphs of order a 2-power is given. As byproducts, we answer a problem proposed in Godsil (1983) [16] regarding the existence of arc-regular non-normal Cayley graphs of order a 2-power, and construct the first known family of cubic semisymmetric graphs of order a 2-power. [ABSTRACT FROM AUTHOR]

Combining the usual energy functional with a higher-order conserved quantity originating from integrability theory, we show that the black soliton is a local minimizer of a quantity that is conserved along the flow of the cubic defocusing NLS equation in one space dimension. This unconstrained variational characterization gives an elementary proof of the orbital stability of the black soliton with respect to perturbations in H 2 ( R ) . [ABSTRACT FROM AUTHOR]

Periodic waves of the one-dimensional cubic defocusing NLS equation are considered. Using tools from integrability theory, these waves have been shown in [4] to be linearly stable and the Floquet–Bloch spectrum of the linearized operator has been explicitly computed. We combine here the first four conserved quantities of the NLS equation to give a direct proof that cnoidal periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. Our result is not restricted to the periodic waves of small amplitudes. [ABSTRACT FROM AUTHOR]

Consider a class of pairs ( g , g 1 ) , where g is a semisimple Lie algebra and g 1 is a subalgebra reductive in g , satisfying the following: For any Cartan subalgebra h 1 ⊆ g 1 there is a unique Cartan subalgebra h ⊆ g containing h 1 . Given such a pair ( g , g 1 ) and a Borel subalgebra b 1 ⊆ g 1 we study a (finite) set S Bor g ( b 1 ) of all Borel subalgebras b ⊆ g containing b 1 . In particular we point at the subclass of pairs ( g , g 1 ) when S Bor g ( b 1 ) is a singleton, for every Borel subalgebra b 1 ⊆ g 1 . As a consequence, for such pairs we relate the corresponding flag varieties B ( g ) and B ( g 1 ) . As an interesting class of pairs ( g , g 1 ) , for which we can apply our results on pairs of Borel subalgebras, we study in detail the pairs considered by R. Brylinski and B. Kostant related to shared orbit pairs. [ABSTRACT FROM AUTHOR]

For a semisimple Lie algebra over the complex numbers, Dynkin (1952) developed an algorithm to classify the regular semisimple subalgebras, up to conjugacy by the inner automorphism group. For a graded semisimple Lie algebra over the complex numbers, Vinberg (1979) showed that a classification of a certain type of regular subalgebras (called carrier algebras) yields a classification of the nilpotent orbits in a homogeneous component of that Lie algebra. Here we consider these problems for (graded) semisimple Lie algebras over the real numbers. First, we describe an algorithm to classify the regular semisimple subalgebras of a real semisimple Lie algebra. This also yields an algorithm for listing, up to conjugacy, the carrier algebras in a real graded semisimple real algebra. We then discuss what needs to be done to obtain a classification of the nilpotent orbits from that; such classifications have applications in differential geometry and theoretical physics. Our algorithms are implemented in the language of the computer algebra system GAP , using our package CoReLG ; we report on example computations. [ABSTRACT FROM AUTHOR]

A pair of Lie algebras ( g , g 1 ) will be called a Kostant pair if g is semisimple, g 1 is reductive in g and the restriction of the Killing form B g to g 1 is nondegenerate. We study the class of such (nonsymmetric) pairs and obtain some useful and new structural results. We study the structure of the normalizers N g ( g 1 ) , and as a consequence we obtain some corresponding worthy results about algebraic groups. In particular we consider an interesting case when g 1 is a distinguished sl 2 -subalgebra of g . Combined with the research due to V.L. Popov we observe that the notions of self-normalizing (reductive) subalgebras of a semisimple Lie algebra and projective self-dual algebraic subvarieties of the usual nilpotent cones are closely related. [ABSTRACT FROM AUTHOR]

Fourier coefficients of automorphic representations π of Sp 2 n ( A ) are attached to unipotent adjoint orbits in Sp 2 n ( F ) , where F is a number field and A is the ring of adeles of F . We prove that for a given π , all maximal unipotent orbits that give nonzero Fourier coefficients of π are special, and prove, under a well-acceptable assumption, that if π is cuspidal, then the stabilizer attached to each of those maximal unipotent orbits is F -anisotropic as algebraic group over F . These results strengthen, refine and extend the earlier work of Ginzburg, Rallis and Soudry on the subject. As a consequence, we obtain constraints on those maximal unipotent orbits if F is totally imaginary, further applications of which to the discrete spectrum with the Arthur classification will be considered in our future work. [ABSTRACT FROM AUTHOR]

In this paper, we study the global dynamics of a class of nonlinear Schrödinger equations using perturbative and non-perturbative methods. We prove the semi-global existence of solutions for initial conditions close to constant. That is, solutions will exist for all positive time or all negative time. The existence of an open set of initial data which limits to zero in both forward and backward time is also demonstrated. This result in turn forces the non-existence of any real-analytic conserved quantities. For the quadratic case, we prove the existence of two (infinite) families of nontrivial unstable equilibria and prove the existence of heteroclinic orbits limiting to the nontrivial equilibria in backward time and to zero in forward time. By a time reversal argument, we also obtain heteroclinic orbits limiting to the nontrivial equilibria in forward time and to zero in backward time. The proofs for the quadratic equation are computer-assisted and rely on three separate ingredients: an enclosure of a local unstable manifold at the equilibria, a rigorous integration of the flow (starting from the unstable manifold) and a proof that the solution enters a validated stable set (hence showing convergence to zero). [ABSTRACT FROM AUTHOR]

The Garsia–Haiman module is a bigraded S n -module whose Frobenius image is a Macdonald polynomial. The method of orbit harmonics promotes an S n -set X to a graded polynomial ring. The orbit harmonics can be applied to prove cyclic sieving phenomena which is a notion that encapsulates the fixed-point structure of finite cyclic group action on a finite set. By applying this idea to the Garsia–Haiman module, we provide cyclic sieving results regarding the enumeration of matrices that are invariant under certain cyclic row and column rotation and translation of entries. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a class of diffusive coupled logistic map with large strength is considered. Depending on the parameter of the logistic family and the strength of coupling, we show the existence of a forward invariant curve which determines, in some sense, the boundary of the set whose points have unbounded orbit. [Copyright &y& Elsevier]

Abstract: In this paper, finite type domains with hyperbolic orbit accumulation points are studied. We prove, in case of , it has to be a (global) pseudoconvex domain, after an assumption of boundary regularity. Moreover, one of the applications will realize the classification of domains within this class, precisely the domain is biholomorphic to one of the ellipsoids . This application generalizes [4] in which the boundary is assumed to be real analytic for the case of hyperbolic orbit accumulation points. [Copyright &y& Elsevier]

Abstract: Bifurcations of spatially nonhomogeneous periodic orbits and steady state solutions are rigorously proved for a reaction–diffusion system modeling predator–prey interaction. The existence of these patterned solutions shows the richness of the spatiotemporal dynamics such as oscillatory behavior and spatial patterns. [Copyright &y& Elsevier]

Abstract: A Frobenius group is a transitive but not regular permutation group such that only the identity element can fix two points. A finite Frobenius group can be expressed as with a nilpotent normal subgroup. A first-kind -Frobenius graph is a Cayley graph on with connection set an -orbit on generating , where is of even order or consists of involutions. We classify all 6-valent first-kind Frobenius circulant graphs such that the underlying kernel is cyclic. We give optimal gossiping and routing algorithms for such a circulant and compute its forwarding indices, Wiener indices and minimum gossip time. We also prove that its broadcasting time is equal to its diameter plus two or three. We prove that all 6-valent first-kind Frobenius circulants with cyclic kernels are Eisenstein–Jacobi graphs, the latter being Cayley graphs on quotient rings of the ring of Eisenstein–Jacobi integers. We also prove that larger Eisenstein–Jacobi graphs can be constructed from smaller ones as topological covers, and a similar result holds for 6-valent first-kind Frobenius circulants. As a corollary any Eisenstein–Jacobi graph with order congruent to 1 modulo 6 and underlying Eisenstein–Jacobi integer not an associate of a real integer, is a cover of a 6-valent first-kind Frobenius circulant. A distributed real-time computing architecture known as HARTS or hexagonal mesh is a special 6-valent first-kind Frobenius circulant. [Copyright &y& Elsevier]

Abstract: Induced representations were introduced and studied by F.G. Frobenius [8] for finite groups and developed by G.W. Mackey [22,23] for locally compact groups. We generalize the Mackey construction for infinite-dimensional groups. To do this, we construct some G-quasi-invariant measures on an appropriate completion of the initial space (since the Haar measure on G does not exist) and extend the representation S of the subgroup H to the representation of the corresponding completion . Kirillov's orbit method [9] describes all irreducible unitary representations of the finite-dimensional nilpotent group in terms of induced representations associated with orbits in coadjoint action of the group in a dual space of the Lie algebra . The induced representation defined in such a way allows us to start to develop an analog of the orbit method for the infinite-dimensional “nilpotent” group of infinite in both directions matrices. [Copyright &y& Elsevier]

Abstract: We prove that the stabiliser group of the Segre variety product in of three projective lines over a field has four orbits on singular points of , and that has five orbits on points of if is finite. [Copyright &y& Elsevier]

Abstract: In recent years many new definitions of shadowing, using a notion of ergodic (or average) pseudo-orbit were introduced. While, under the assumption of chain mixing, average pseudo-orbit usually can be equivalently replaced by pseudo-orbit in these definitions of shadowing, it is not completely clear when these shadowing properties (i.e. approximation of a pseudo-orbit by a real orbit on a sufficiently large set of indices) can or cannot occur. In this paper we analyze necessary and sufficient conditions for shadowing over a set with positive density (or syndetic). While we do not provide full characterization, a few relations to standard notions from topological dynamics are obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with a second order non-autonomous Hamiltonian system on time scales Under certain conditions, the existence and multiplicity of periodic solutions are obtained for this Hamiltonian system on time scales by using the saddle point theory, the least action principle as well as the three-critical-point theorem. In addition, the existence of homoclinic orbit is obtained as a limit of -periodic solutions of a given sequence of Hamiltonian system on time scales by means of the mountain pass theorem and the standard minimizing argument. [Copyright &y& Elsevier]