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A preconditioner is proposed for Laplace exterior boundary value problems on multi-screens. To achieve this, the quotient-space boundary element method and operator preconditioning are combined. For a fairly general subclass of multi-screens, it is shown that this approach paves the way for block diagonal Calderón preconditioners which achieve a spectral condition number that grows only logarithmically with decreasing mesh size, just as in the case of simple screens. Since the resulting scheme contains many more degrees of freedom than strictly required, strategies are presented to remove almost all redundancy without significant loss of effectiveness of the preconditioner. The performance of this method is verified by providing representative numerical results. Further numerical experiments suggest that these results can be extended to a much wider class of multi-screens that cover essentially all geometries encountered in practice, leading to a significantly reduced simulation cost. [ABSTRACT FROM AUTHOR]

We study the shape of a string hanging under its weight and subjected in opposite direction by a force due to a soap film adhered to it. These curves are critical points of an energy E which involves the gravitational potential of the string and the surface area of the soap film. We show that these curves also are critical points of the functional γ ↦ ∫ γ κ + σ , where κ is the curvature of the curve γ and σ is a constant. This gives a relationship between these shapes and the classical Euler's elastics. Finally we extend the problem assuming that the weight of the soap film is relevant on the physical system. [ABSTRACT FROM AUTHOR]

Let M be an oriented three-dimensional Riemannian manifold of constant sectional curvature k = 0 , 1 , - 1 and let S O M be its direct orthonormal frame bundle (direct refers to positive orientation), which may be thought of as the set of all positions of a small body in M. Given λ ∈ R , there is a three-dimensional distribution D λ on S O M accounting for infinitesimal rototranslations of constant pitch λ . When λ ≠ k 2 , there is a canonical sub-Riemannian structure on D λ . We present a geometric characterization of its geodesics, using a previous Lie theoretical description. For k = 0 , - 1 , we compute the sub-Riemannian length spectrum of S O M , D λ in terms of the complex length spectrum of M (given by the lengths and the holonomies of the periodic geodesics) when M has positive injectivity radius. In particular, for two complex length isospectral closed hyperbolic 3-manifolds (even if they are not isometric), the associated sub-Riemannian metrics on their direct orthonormal bundles are length isospectral. [ABSTRACT FROM AUTHOR]

Branched coverings boast a rich history, ranging from the ramification of Riemann surfaces to the realization of 3-manifolds as coverings branched over knots and spanning both geometric topology and algebraic geometry. This work delves into branched coverings “à la Fox” of (G, X)-manifolds, encompassing three main avenues: Firstly, we introduce a comprehensive class of singular (G, X)-manifolds, elucidating elementary theory paired with illustrative examples to showcase its efficacy and universality. Secondly, building on Montesinos’ work, we revisit and augment the prevailing knowledge, formulating a Galois theory tailored for such branched coverings. This includes a detailed portrayal of the fiber above branching points. Lastly, we identify local attributes that guarantee the existence of developing maps for singular (G, X)-manifolds within the branched coverings framework. Notably, we pinpoint conditions that ensure the existence of developing maps for these singular manifolds. This research proves especially pertinent for non-metric singular (G, X)-manifolds like those of Lorentzian or projective nature, as discussed by Barbot, Bonsante, Suhyoung Choi, Danciger, Seppi, Schlenker, and the author, among others. While examples are sprinkled throughout, a standout application presented is a uniformization theorem “à la Mess” for singular locally Minkowski manifolds exhibiting BTZ-like singularities. [ABSTRACT FROM AUTHOR]

We prove that a holomorphic projective connection on a complex projective threefold is either flat, or it is a translation invariant holomorphic projective connection on an abelian threefold. In the second case, a generic translation invariant holomorphic affine connection on the abelian variety is not projectively flat. We also prove that a simply connected compact complex threefold with trivial canonical line bundle does not admit any holomorphic projective connection. [ABSTRACT FROM AUTHOR]

We study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of a compact connected Lie group U with Lie algebra $\mathfrak {u}$ extends holomorphically to an action of the complexified group $U^{\mathbb {C}}$ and that the U-action on Z is Hamiltonian. If $G\subset U^{\mathbb {C}}$ is compatible, there exists a gradient map $\mu _{\mathfrak p}:X \longrightarrow \mathfrak p$ where $\mathfrak g=\mathfrak k \oplus \mathfrak p$ is a Cartan decomposition of $\mathfrak g$. In this paper, we describe compact orbits of parabolic subgroups of G in terms of the gradient map $\mu _{\mathfrak p}$. [ABSTRACT FROM AUTHOR]

In this paper a new approach to construct cooperative behavior in dynamic multicriteria games with asymmetric players is presented. To obtain noncooperative and cooperative equilibria the ideas of multi-objective optimization and game theory are combined. To construct a multicriteria Nash equilibrium the bargaining solution is adopted. To design a multicriteria cooperative equilibrium a compromise scheme that guarantees the fulfillment of rationality conditions is applied. The concept of dynamic stability is adopted for dynamic multicriteria games, and the time-consistent payoff distribution procedure is presented. A dynamic bi-criteria bioresource management problem with different discount factors is considered. The strategies and payoffs of players are obtained under cooperative and noncooperative behavior. [ABSTRACT FROM AUTHOR]

We prove a uniqueness result of weak solutions to the nD (n ⩾ 3) Cauchy problem of a Keller-Segel-Navier-Stokes system with a logistic term. [ABSTRACT FROM AUTHOR]

In this paper we deal with the class C of decomposable solvable Lie groups having dimension six. We determine those Lie groups in C and their subgroups which are the multiplication groups Mult(L) and the inner mapping groups Inn(L) for three-dimensional connected simply connected topological loops L. This result completes the classification of the at most 6-dimensional solvable multiplication Lie groups of the loops L. Moreover, we obtain that every at most 3-dimensional connected topological proper loop having a solvable Lie group of dimension at most six as its multiplication group is centrally nilpotent of class two. [ABSTRACT FROM AUTHOR]

Stoichiometric producer–grazer models are nonsmooth due to the Liebig’s Law of Minimum and can generate new dynamics such as bistability for producer–grazer interactions. Environmental noises can be extremely important and change dynamical behaviors of a stoichiometric producer–grazer model. In this paper, we consider a stochastically forced producer–grazer model and study the phenomena of noise-induced state switching between two stochastic attractors in the bistable zone. Namely, there is a frequent random hopping of phase trajectories between attracting basins of the attractors. In addition, by applying the stochastic sensitivity function technique, we construct the confidence ellipse and confidence band to find the configurational arrangement of equilibria and a limit cycle, respectively. [ABSTRACT FROM AUTHOR]

In this paper we will discuss the orbits of the fixed-point group on the tori of the generalized symmetric spaces of SL 2 (k) where k is a finite field. Specifically, we will provide a characterization and classification of the maximal k-split and k-anisotropic tori. [ABSTRACT FROM AUTHOR]

We show that simply connected Riemannian homogeneous spaces of compact semisimple Lie groups with polar isotropy actions are symmetric, generalizing results of Fabio Podestà and the third named author. Without assuming compactness, we give a classification of Riemannian homogeneous spaces of semisimple Lie groups whose linear isotropy representations are polar. We show for various such spaces that they do not have polar isotropy actions. Moreover, we prove that Heisenberg groups and non-symmetric Damek–Ricci spaces have non-polar isotropy actions. [ABSTRACT FROM AUTHOR]

A flag domain of a real from G 0 of a complex semismiple Lie group G is an open G 0 -orbit D in a (compact) G-flag manifold. In the usual way one reduces to the case where G 0 is simple. It is known that if D possesses non-constant holomorphic functions, then it is the product of a compact flag manifold and a Hermitian symmetric bounded domain. This pseudoconvex case is rare in the geography of flag domains. Here it is shown that otherwise, i.e., when O (D) ≅ C , the flag domain D is pseudoconcave. In a rather general setting the degree of the pseudoconcavity is estimated in terms of root invariants. This estimate is explicitly computed for domains in certain Grassmannians. [ABSTRACT FROM AUTHOR]

Recognised as one of the hallmarks of cancer, local cancer cell invasion is a complex multiscale process that combines the secretion of matrix-degrading enzymes with a series of altered key cell processes (such as abnormal cell proliferation and changes in cell–cell and cell–matrix adhesion leading to enhanced migration) to degrade important components of the surrounding extracellular matrix (ECM) and this way spread further in the human tissue. In order to gain a deeper understanding of the invasion process, we pay special attention to the interacting dynamics between the cancer cell population and various constituents of the surrounding tumour microenvironment. To that end, we consider the key role that ECM plays within the human body tissue, and in particular we focus on the special contribution of its fibrous proteins components, such as collagen and fibronectin, which play an important part in cell proliferation and migration. In this work, we consider the two-scale dynamic cross-talk between cancer cells and a two-component ECM (consisting of both a fibre and a non-fibre phase). To that end, we incorporate the interlinked two-scale dynamics of cell–ECM interactions within the tumour support that contributes simultaneously both to cell adhesion and to the dynamic rearrangement and restructuring of the ECM fibres. Furthermore, this is embedded within a multiscale moving boundary approach for the invading cancer cell population, in the presence of cell adhesion at the tissue scale and cell-scale fibre redistribution activity and leading edge matrix-degrading enzyme molecular proteolytic processes. The overall modelling framework will be accompanied by computational results that will explore the impact on cancer invasion patterns of different levels of cell adhesion in conjunction with the continuous ECM fibres rearrangement. [ABSTRACT FROM AUTHOR]

Sensing and reciprocating cellular systems (SARs) are important for the operation of many biological systems. Production in interferon (IFN) SARs is achieved through activation of the Jak-Stat pathway, and downstream upregulation of IFN regulatory factor (IRF)-7 and IFN transcription, but the role that high- and low-affinity IFNs play in this process remains unclear. We present a comparative between a minimal spatio-temporal partial differential equation model and a novel spatio-structural-temporal (SST) model for the consideration of receptor, binding, and metabolic aspects of SAR behaviour. Using the SST framework, we simulate single- and multi-cluster paradigms of IFN communication. Simulations reveal a cyclic process between the binding of IFN to the receptor, and the consequent increase in metabolism, decreasing the propensity for binding due to the internal feedback mechanism. One observes the effect of heterogeneity between cellular clusters, allowing them to individualise and increase local production, and within clusters, where we observe ‘subpopular quiescence’; a process whereby intra-cluster subpopulations reduce their binding and metabolism such that other such subpopulations may augment their production. Finally, we observe the ability for low-affinity IFN to communicate a long range signal, where high affinity cannot, and the breakdown of this relationship through the introduction of cell motility. Biological systems may utilise cell motility where environments are unrestrictive and may use fixed system, with low-affinity communication, where a localised response is desirable. [ABSTRACT FROM AUTHOR]

Abstract: Let G2(2) be the non‐compact connected simple Lie group of type G2 over R, and let M be a connected analytic complete pseudo‐Riemannian manifold that admits an isometric G2(2)‐action with a dense orbit. For the case dim ( M ) ≤ 21, we provide a full description of the manifold M, its geometry and its G2(2)‐action. The latter are always given in terms of a Lie group geometry related to G2(2), and in one case M is essentially the quotient of SO0(3, 4) by a lattice. [ABSTRACT FROM AUTHOR]

In his seminal work [13] , R. Palais extended a substantial part of the theory of compact transformation groups to the case of proper actions of locally compact groups. Here we extend to proper actions some other important results well known for compact group actions. In particular, we prove that if H is a compact subgroup of a locally compact group G and S is a small (in the sense of Palais) H -slice in a proper G -space, then the action map G × S → G ( S ) is open. This is applied to prove that the slicing map f S : G ( S ) → G / H is continuous and open, which provides an external characterization of a slice. Also an equivariant extension theorem is proved for proper actions. As an application, we give a short proof of the compactness of the Banach–Mazur compacta. [ABSTRACT FROM AUTHOR]

Spatio-temporal models have long been used to describe biological systems of cancer, but it has not been until very recently that increased attention has been paid to structural dynamics of the interaction between cancer populations and the molecular mechanisms associated with local invasion. One system that is of particular interest is that of the urokinase plasminogen activator (uPA) wherein uPA binds uPA receptors on the cancer cell surface, allowing plasminogen to be cleaved into plasmin, which degrades the extracellular matrix and this way leads to enhanced cancer cell migration. In this paper, we develop a novel numerical approach and associated analysis for spatio-structuro-temporal modelling of the uPA system for up to two-spatial and two-structural dimensions. This is accompanied by analytical exploration of the numerical techniques used in simulating this system, with special consideration being given to the proof of stability within numerical regimes encapsulating a central differences approach to approximating numerical gradients. The stability analysis performed here reveals instabilities induced by the coupling of the structural binding and proliferative processes. The numerical results expound how the uPA system aids the tumour in invading the local stroma, whilst the inhibitor to this system may impede this behaviour and encourage a more sporadic pattern of invasion. [ABSTRACT FROM AUTHOR]

We give a geometric characterization of certain hypersurfaces of cohomogeneity one in the complex projective and hyperbolic planes. We also obtain some partial classifications of austere hypersurfaces and of Levi-flat hypersurfaces with constant mean curvature in these spaces. [ABSTRACT FROM AUTHOR]

We classify and explicitly describe homomorphisms of Verma modules for conformal Galilei algebras $${\mathfrak {cga}}_\ell (d,\mathbb {C})$$ with $$d=1$$ for any integer value $$\ell \in {\mathbb {N}}$$ . The homomorphisms are uniquely determined by singular vectors as solutions of certain differential operators of flag type and identified with specific polynomials arising as coefficients in the expansion of a parametric family of symmetric polynomials into power sum symmetric polynomials. [ABSTRACT FROM AUTHOR]

In this paper we extend previous results regarding SL2(k) over any finite fieldkby investigating the structure of the symmetric spaces for the family of special linear groups SLn(k) for any integern>2. Specifically, we discuss the generalized and extended symmetric spaces of SLn(k) for all conjugacy classes of involutions over a finite field of odd or even characteristic. We characterize the structure of these spaces and provide an explicit difference set in cases where the two spaces are not equal. [ABSTRACT FROM PUBLISHER]

We classify polar actions on complex hyperbolic spaces up to orbit equivalence. [ABSTRACT FROM AUTHOR]

We consider a Keller-Segel-Euler model and prove a regularity criterion of the local strong solutions in a 3D bounded domain Ω. [ABSTRACT FROM AUTHOR]