Journal articles on the topic 'Non-smooth differential geometry'

This paper provides two new applications of conservation laws in biology. The first is the application of the van der Waals fluid formalism for action potentials. The second is the application of the conservation laws of differential geometry (Gauss–Codazzi equations) to produce non-smooth surfaces representing Endoplasmic Reticulum sheets.

Information geometry concerns the study of a dual structure (g,∇,∇*) upon a smooth manifold M. Such a geometry is totally encoded within a potential function usually referred to as a divergence or contrast function of (g,∇,∇*). Even though infinitely many divergences induce on M the same dual structure, when the manifold is dually flat, a canonical divergence is well defined and was originally introduced by Amari and Nagaoka. In this pedagogical paper, we present explicit non-trivial differential geometry-based proofs concerning the canonical divergence for a special type of dually flat manifold represented by an arbitrary 1-dimensional path γ. Highlighting the geometric structure of such a particular canonical divergence, our study could suggest a way to select a general canonical divergence by using the information from a general dual structure in a minimal way.

Information geometry concerns the study of a dual structure (g,∇,∇*) upon a smooth manifold M. Such a geometry is totally encoded within a potential function usually referred to as a divergence or contrast function of (g,∇,∇*). Even though infinitely many divergences induce on M the same dual structure, when the manifold is dually flat, a canonical divergence is well defined and was originally introduced by Amari and Nagaoka. In this pedagogical paper, we present explicit non-trivial differential geometry-based proofs concerning the canonical divergence for a special type of dually flat manifold represented by an arbitrary 1-dimensional path γ. Highlighting the geometric structure of such a particular canonical divergence, our study could suggest a way to select a general canonical divergence by using the information from a general dual structure in a minimal way.

Information geometry concerns the study of a dual structure (g,∇,∇*) upon a smooth manifold M. Such a geometry is totally encoded within a potential function usually referred to as a divergence or contrast function of (g,∇,∇*). Even though infinitely many divergences induce on M the same dual structure, when the manifold is dually flat, a canonical divergence is well defined and was originally introduced by Amari and Nagaoka. In this pedagogical paper, we present explicit non-trivial differential geometry-based proofs concerning the canonical divergence for a special type of dually flat manifold represented by an arbitrary 1-dimensional path γ. Highlighting the geometric structure of such a particular canonical divergence, our study could suggest a way to select a general canonical divergence by using the information from a general dual structure in a minimal way.

This paper relates to differential geometry, and the research technique is based on G. F. Laptev’s method of extensions and envelopments, which generalizes E. Cartan’s method of moving frame and exterior forms. We consider a smooth m-dimensional manifold, its tangent and cotangent spaces, as well as the second-order frames and coframes on this manifold. Using the perturbation of the exterior derivative and ordinary diffe­ren­tial, mappings are introduced that enable us to construct non-sym­met­rical second-order frames and coframes on a smooth manifold. It is shown that the extension of the second order tangent space to a smooth m-dimen­sional manifold is carried out by adding the vertical vectors to the linear frame bundle over the manifold to the second order tangent vectors to this manifold. A deformed external differential is widely used, which is a differen­tial, i. e., its reapplication vanishes. We introduce a deformed external dif­ferential being a differential along the curves on the manifold, i. e., its re­peated application along the curves on the manifold gives zero.

We develop the geometric two-scale convergence on forms in order to describe the homogenization of partial differential equations with random variables on non-flat domain. We prove the compactness theorem and some two-scale behaviours for differential forms. For its applications, we investigate the limiting equations of the n-dimensional Maxwell equations with random coefficients, with given initial and boundary conditions, where are symmetric positive-definite matrices for x ∈ M, and M is an n-dimensional compact oriented Riemannian manifold with smooth boundary. The limiting system of n-dimensional Maxwell equations turns out to be degenerate and it is proven to be well-posed. The homogenized coefficients affected by the geometry of the domain are presented, and compared with the homogenized coefficient of the second order elliptic equation. We present the convergence theorem in order to explain the convergence of the solutions of Maxwell system as a parabolic partial differential equation.

This paper builds on the theory of nonlinear generalized functions begun in Nigsch & Vickers (Nigsch, Vickers 2021 Proc. R. Soc. A 20200640 ( doi:10.1098/rspa.2020.0640 )) and extends this to a diffeomorphism-invariant nonlinear theory of generalized tensor fields with the sheaf property. The generalized Lie derivative is introduced and shown to commute with the embedding of distributional tensor fields and the generalized covariant derivative commutes with the embedding at the level of association. The concept of a generalized metric is introduced and used to develop a non-smooth theory of differential geometry. It is shown that the embedding of a continuous metric results in a generalized metric with well-defined connection and curvature and that for C 2 metrics the embedding preserves the curvature at the level of association. Finally, we consider an example of a conical metric outside the Geroch–Traschen class and show that the curvature is associated to a delta function.

Abstract The aim of this survey is to present the main important techniques and tools from variational analysis used for first and second order dynamical systems of implicit type for solving monotone inclusions and non-smooth optimization problems. The differential equations are expressed by means of the resolvent (in case of a maximally monotone set valued operator) or the proximal operator for non-smooth functions. The asymptotic analysis of the trajectories generated relies on Lyapunov theory, where the appropriate energy functional plays a decisive role. While the most part of the paper is related to monotone inclusions and convex optimization problems in the variational case, we present also results for dynamical systems for solving non-convex optimization problems, where the Kurdyka-Łojasiewicz property is used.

AbstractWe study spaces with a cuspidal (or horn-like) singularity embedded in a smooth Riemannian manifold and analyze the geodesics in these spaces which start at the singularity. This provides a basis for understanding the intrinsic geometry of such spaces near the singularity. We show that these geodesics combine to naturally define an exponential map based at the singularity, but that the behavior of this map can deviate strongly from the behavior of the exponential map based at a smooth point or at a conical singularity: While it is always surjective near the singularity, it may be discontinuous and non-injective on any neighborhood of the singularity. The precise behavior of the exponential map is determined by a function on the link of the singularity which is an invariant of the induced metric. Our methods are based on the Hamiltonian system of geodesic differential equations and on techniques of singular analysis. The results are proved in the more general natural setting of manifolds with boundary carrying a so-called cuspidal metric.

In the deformation of layered materials such as geological strata, or stacks of paper, mechanical properties compete with the geometry of layering. Smooth, rounded corners lead to voids between the layers, while close packing of the layers results in geometrically induced curvature singularities. When voids are penalized by external pressure, the system is forced to trade off these competing effects, leading to sometimes striking periodic patterns. In this paper, we construct a simple model of geometrically nonlinear multi-layered structures under axial loading and pressure confinement, with non-interpenetration conditions separating the layers. Energy minimizers are characterized as solutions of a set of fourth-order nonlinear differential equations with contact-force Lagrange multipliers, or equivalently of a fourth-order free-boundary problem. We numerically investigate the solutions of this free-boundary problem and compare them with the periodic solutions observed experimentally.

In this paper, we study a system of linear equations that define the Lie algebra of differentiations DerA of an arbitrary finite-dimensional linear algebra over a field. A system of equations is obtained, which is satisfied by the components of an arbitrary differentiation with respect to a fixed basis of algebra A. This system is a system of linear homogeneous equa­tions. The law of transformation of the matrix of this system is proved. The invariance of the rank of the matrix of this system in the transition to a new basis in algebra is proved. Next, we consider the possibility of ap­plying the obtained results in differential geometry when estimating the dimensions of groups of affine transformations from above. As an exam­ple, the method of I. P. Egorov is given for studying the dimensions of Lie algebras of affine vector fields on smooth manifolds equipped with linear connections having non-zero torsion tensor fields.

AbstractThe asymptotic properties of solutions of the non-linear eigenvalue problem, associated with the homogeneoud Dirichlet problem forare investigated. Here f and g are smooth functions of position in a finite plane region with a smooth boundary. The results for the positive solution are well established, but knowledge of other branches of solutions is scarce. Here positive solutions are pieced together across lines partitioning the domain, and variational arguments are framed, as an effective means of locating the lines, so that the composite function is everywhere a solution of *. Heuristic arguments suggest strongly that there is a close relationship between the nodal lines of * and certain classes of weighted geodesic lines defined by the classical variational problem for the functionalwhich provides an effective basis for computation. Some results are proved but others remain conjectures. Analogous results are proved for the associated ordinary differential equation. The geometry of the solutions is surprisingly restricted when the coefficients are spatially variable. The arguments are extended to a class of reactive, diffusive systems. It is possible to predict the pattern of domains of different outcomes in terms of properties of the surface on which the reactions occur, without a knowledge of the chemical kinetics. The results appear to provide a basis for stringent testing of the postulated role of reactive-diffusive mechanisms in the formation of complex patterns in biological species.

For a strictly inviscid barotropic flow with conservative body forces, the Helmholtz vorticity theorem shows that material or Lagrangian surfaces which are vortex surfaces at time t = 0 remain so for t > 0. In this study, a systematic methodology is developed for constructing smooth scalar fields φ(x, y, z, t = 0) for Taylor–Green and Kida–Pelz velocity fields, which, at t = 0, satisfy ω·∇φ = 0. We refer to such fields as vortex-surface fields. Then, for some constant C, iso-surfaces φ = C define vortex surfaces. It is shown that, given the vorticity, our definition of a vortex-surface field admits non-uniqueness, and this is presently resolved numerically using an optimization approach. Additionally, relations between vortex-surface fields and the classical Clebsch representation are discussed for flows with zero helicity. Equations describing the evolution of vortex-surface fields are then obtained for both inviscid and viscous incompressible flows. Both uniqueness and the distinction separating the evolution of vortex-surface fields and Lagrangian fields are discussed. By tracking φ as a Lagrangian field in slightly viscous flows, we show that the well-defined evolution of Lagrangian surfaces that are initially vortex surfaces can be a good approximation to vortex surfaces at later times prior to vortex reconnection. In the evolution of such Lagrangian fields, we observe that initially blob-like vortex surfaces are progressively stretched to sheet-like shapes so that neighbouring portions approach each other, with subsequent rolling up of structures near the interface, which reveals more information on dynamics than the iso-surfaces of vorticity magnitude. The non-local geometry in the evolution is quantified by two differential geometry properties. Rolled-up local shapes are found in the Lagrangian structures that were initially vortex surfaces close to the time of vortex reconnection. It is hypothesized that this is related to the formation of the very high vorticity regions.

Second-order Ordinary Differential Equations (ODEs) with discontinuous forcing have numerous applications in engineering and computational sciences. The analysis of the solution spaces of non-homogeneous ODEs is difficult due to the complexities in multidimensional systems, with multiple discontinuous variables present in forcing functions. Numerical solutions are often prone to failures in the presence of discontinuities. Algebraic decompositions are employed for analysis in such cases, assuming that regularities exist, operators are present in Banach (solution) spaces, and there is finite measurability. This paper proposes a generalized, finite-dimensional algebraic analysis of the solution spaces of second-order ODEs equipped with periodic Dirac delta forcing. The proposed algebraic analysis establishes the conditions for the convergence of responses within the solution spaces without requiring relative smoothness of the forcing functions. The Lipschitz regularizations and Lebesgue measurability are not considered as preconditions maintaining generality. The analysis shows that smooth and locally finite responses can be admitted in an exponentially stable solution space. The numerical analysis of the solution spaces is computed based on combinatorial changes in coefficients. It exhibits a set of locally uniform responses in the solution spaces. In contrast, the global response profiles show localized as well as oriented instabilities at specific neighborhoods in the solution spaces. Furthermore, the bands of the expansions–contractions of the stable response profiles are observable within the solution spaces depending upon the values of the coefficients and time intervals. The application aspects and distinguishing properties of the proposed approaches are outlined in brief.

The present paper concentrates on the numerical modelling of masonry vaults, adopting a type of analysis first developed at the University of Parma for applied mechanics, based on the use of non-smooth dynamics software, through a Differential Variational Inequalities (DVI) formulation specifically developed for the 3D discrete elements method. It allows to follow large displacements and the opening and closure of cracks in dynamic field, typical of the masonry vaulted structures. Once the modelling instrument was calibrated, thanks to the comparison with the recurrent damage mechanisms previously analysed, it was also applied to foresee the behavior of the same structure with different actions and with different types of strengthening. The development of damage mechanisms, both in quasi-static cases (for insufficient lateral confinement or for possible soil settlements) and in the occurrence of seismic events, make this type of structures very difficult to be modelled precisely with other methods. Given the three-dimensional CAD model of a vault modeled with a great number of masonry units with specific positions and pattern, the method proved to be able to reproduce the behavior of the structure under both static and seismic loads, showing the mechanism of collapse, the network of contact forces, the displacements and other useful data. The aim is to inspect the possible influences in the structural behavior given by the discrete geometry and the changes in the mechanisms development given by different strengthening interventions. Once the modeling instrument will be calibrated, also through the comparison with real cases and with the results obtained through limit analysis, it will be possible to adopt it as a base also for the prevision of the future behavior of the vaults subjected to strengthening, avoiding uncalibrated and uncritical applications of materials based more on trends rather than on a thorough analysis for the specific case.

In this paper, we consider a surrogate modeling approach using a data-driven nonparametric likelihood function constructed on a manifold on which the data lie (or to which they are close). The proposed method represents the likelihood function using a spectral expansion formulation known as the kernel embedding of the conditional distribution. To respect the geometry of the data, we employ this spectral expansion using a set of data-driven basis functions obtained from the diffusion maps algorithm. The theoretical error estimate suggests that the error bound of the approximate data-driven likelihood function is independent of the variance of the basis functions, which allows us to determine the amount of training data for accurate likelihood function estimations. Supporting numerical results to demonstrate the robustness of the data-driven likelihood functions for parameter estimation are given on instructive examples involving stochastic and deterministic differential equations. When the dimension of the data manifold is strictly less than the dimension of the ambient space, we found that the proposed approach (which does not require the knowledge of the data manifold) is superior compared to likelihood functions constructed using standard parametric basis functions defined on the ambient coordinates. In an example where the data manifold is not smooth and unknown, the proposed method is more robust compared to an existing polynomial chaos surrogate model which assumes a parametric likelihood, the non-intrusive spectral projection. In fact, the estimation accuracy is comparable to direct MCMC estimates with only eight likelihood function evaluations that can be done offline as opposed to 4000 sequential function evaluations, whenever direct MCMC can be performed. A robust accurate estimation is also found using a likelihood function trained on statistical averages of the chaotic 40-dimensional Lorenz-96 model on a wide parameter domain.

We implement a differential-geometric approach to normal forms for contracting measurable cocycles to $\operatorname{Diff}^{q}(\mathbb{R}^{n},\mathbf{0})$, $q\geq 2$. We obtain resonance polynomial normal forms for the contracting cocycle and its centralizer, via $C^{q}$ changes of coordinates. These are interpreted as non-stationary invariant differential-geometric structures. We also consider the case of contracted foliations in a manifold, and obtain $C^{q}$ homogeneous structures on leaves for an action of the group of subresonance polynomial diffeomorphisms together with translations.

Inverse reconstruction from images is a central problem in many scientific and engineering disciplines. Recent progress on differentiable rendering has led to methods that can efficiently differentiate the full process of image formation with respect to millions of parameters to solve such problems via gradient-based optimization. At the same time, the availability of cheap derivatives does not necessarily make an inverse problem easy to solve. Mesh-based representations remain a particular source of irritation: an adverse gradient step involving vertex positions could turn parts of the mesh inside-out, introduce numerous local self-intersections, or lead to inadequate usage of the vertex budget due to distortion. These types of issues are often irrecoverable in the sense that subsequent optimization steps will further exacerbate them. In other words, the optimization lacks robustness due to an objective function with substantial non-convexity. Such robustness issues are commonly mitigated by imposing additional regularization, typically in the form of Laplacian energies that quantify and improve the smoothness of the current iterate. However, regularization introduces its own set of problems: solutions must now compromise between solving the problem and being smooth. Furthermore, gradient steps involving a Laplacian energy resemble Jacobi's iterative method for solving linear equations that is known for its exceptionally slow convergence. We propose a simple and practical alternative that casts differentiable rendering into the framework of preconditioned gradient descent. Our pre-conditioner biases gradient steps towards smooth solutions without requiring the final solution to be smooth. In contrast to Jacobi-style iteration, each gradient step propagates information among all variables, enabling convergence using fewer and larger steps. Our method is not restricted to meshes and can also accelerate the reconstruction of other representations, where smooth solutions are generally expected. We demonstrate its superior performance in the context of geometric optimization and texture reconstruction.

We study some global geometric properties of a static Lorentzian manifold Λ embedded in a differentiable manifold M, with possibly non-smooth boundary ∂Λ. We prove a variational principle for geodesics in static manifolds, and using this principle we establish the existence of geodesics that do not touch ∂Λ and that join two fixed points of Λ. The results are obtained under a suitable completeness assumption for Λ that generalizes the property of global hyperbolicity, and a weak convexity assumption on ∂Λ. Moreover, under a non-triviality assumption on the topology of Λ, we also get a multiplicity result for geodesics in Λ joining two fixed points.

Entropic Dynamics (ED) is a framework in which Quantum Mechanics is derived as an application of entropic methods of inference. In ED the dynamics of the probability distribution is driven by entropy subject to constraints that are codified into a quantity later identified as the phase of the wave function. The central challenge is to specify how those constraints are themselves updated. In this paper we review and extend the ED framework in several directions. A new version of ED is introduced in which particles follow smooth differentiable Brownian trajectories (as opposed to non-differentiable Brownian paths). To construct ED we make use of the fact that the space of probabilities and phases has a natural symplectic structure (i.e., it is a phase space with Hamiltonian flows and Poisson brackets). Then, using an argument based on information geometry, a metric structure is introduced. It is shown that the ED that preserves the symplectic and metric structures—which is a Hamilton-Killing flow in phase space—is the linear Schrödinger equation. These developments allow us to discuss why wave functions are complex and the connections between the superposition principle, the single-valuedness of wave functions, and the quantization of electric charges. Finally, it is observed that Hilbert spaces are not necessary ingredients in this construction. They are a clever but merely optional trick that turns out to be convenient for practical calculations.

We propose and study nonlinear mathematical models describing the intracellular time dynamics of viral RNA accumulation for positive-sense single-stranded RNA viruses. Our models consider different replication modes ranging between two extremes represented by the geometric replication (GR) and the linear stamping machine replication (SMR). We first analyse a model that quantitatively reproduced experimental data for the accumulation dynamics of both polarities of turnip mosaic potyvirus RNAs. We identify a non-degenerate transcritical bifurcation governing the extinction of both strands depending on three key parameters: the mode of replication ( α ), the replication rate ( r ) and the degradation rate ( δ ) of viral strands. Our results indicate that the bifurcation associated with α generically takes place when the replication mode is closer to the SMR, thus suggesting that GR may provide viral strands with an increased robustness against degradation. This transcritical bifurcation, which is responsible for the switching from an active to an absorbing regime, suggests a smooth (i.e. second-order), absorbing-state phase transition. Finally, we also analyse a simplified model that only incorporates asymmetry in replication tied to differential replication modes.

In this paper we define a set of radii called thickness for simple closed curves denoted by K, which are assumed to be differentiable. These radii capture a balanced view between the geometric and the topological properties of these curves. One can think of these radii as representing the thickness of a rope in space and of K as the core of the rope. Great care is taken to define our radii in order to gain freedom from small pieces with large curvature in the curve. Intuitively, this means that we tend to allow the surface of the ropes that represent the knots to deform into a non smooth surface. But as long as the radius of the rope is less than the thickness so defined, the surface of the rope will remain a two manifold and the rope (as a solid torus) can be deformed onto K via strong deformation retract. In this paper we explore basic properties of these thicknesses and discuss the relationship amongst them.

A special case of a fundamental result of Loewner and Horn [Trans. Amer. Math. Soc. 136 (1969), pp. 269–286] says that given an integer n ⩾ 1 n \geqslant 1 , if the entrywise application of a smooth function f : ( 0 , ∞ ) → R f : (0,\infty ) \to \mathbb {R} preserves the set of n × n n \times n positive semidefinite matrices with positive entries, then f f and its first n − 1 n-1 derivatives are non-negative on ( 0 , ∞ ) (0,\infty ) . In a recent joint work with Belton–Guillot–Putinar [J. Eur. Math. Soc., in press], we proved a stronger version, and used it to strengthen the Schoenberg–Rudin characterization of dimension-free positivity preservers [Duke Math. J. 26 (1959), pp. 617–622; Duke Math. J. 9 (1942), pp. 96–108]. In recent works with Belton–Guillot–Putinar [Adv. Math. 298 (2016), pp. 325–368] and with Tao [Amer. J. Math. 143 (2021), pp. 1863-1929] we used local, real-analytic versions at the origin of the Horn–Loewner condition, and discovered unexpected connections between entrywise polynomials preserving positivity and Schur polynomials. In this paper, we unify these two stories via a Master Theorem (Theorem A) which (i) simultaneously unifies and extends all of the aforementioned variants; and (ii) proves the positivity of the first n n nonzero Taylor coefficients at individual points rather than on all of ( 0 , ∞ ) (0,\infty ) . A key step in the proof is a new determinantal / symmetric function calculation (Theorem B), which shows that Schur polynomials arise naturally from considering arbitrary entrywise maps that are sufficiently differentiable. Of independent interest may be the following application to symmetric function theory: we extend the Schur function expansion of Cauchy’s (1841) determinant (whose matrix entries are geometric series 1 / ( 1 − u j v k ) 1 / (1 - u_j v_k) ), as well as of a determinant of Frobenius [J. Reine Angew. Math. 93 (1882), pp. 53–68] (whose matrix entries are a sum of two geometric series), to arbitrary power series, and over all commutative rings.

Abstract. Non target high resolution organic structural spectroscopy of marine dissolved organic matter (DOM) isolated on 27 November 2008 by means of solid phase extraction (SPE) from four different depths in the South Atlantic Ocean off the Angola coast (3.1° E; −17.7° S; Angola basin) provided molecular level information of complex unknowns with unprecedented coverage and resolution. The sampling was intended to represent major characteristic oceanic regimes of general significance: 5 m (FISH; near surface photic zone), 48 m (FMAX; fluorescence maximum), 200 m (upper mesopelagic zone) and 5446 m (30 m above ground). 800 MHz proton (1H) nuclear magnetic resonance (NMR) 1H NMR, spectra were least affected by fast and differential transverse NMR relaxation and produced at first similar looking, rather smooth bulk NMR envelopes reflecting intrinsic averaging from massive signal overlap. Visibly resolved NMR signatures were most abundant in surface DOM but contributed at most a few percent to the total 1H NMR integral and were mainly limited to unsaturated and singly oxygenated carbon chemical environments. The relative abundance and variance of resolved signatures between samples was maximal in the aromatic region; in particular, the aromatic resolved NMR signature of the deep ocean sample at 5446 m was considerably different from that of all other samples. When scaled to equal total NMR integral, 1H NMR spectra of the four marine DOM samples revealed considerable variance in abundance for all major chemical environments across the entire range of chemical shift. Abundance of singly oxygenated CH units and acetate derivatives declined from surface to depth whereas aliphatics and carboxyl-rich alicyclic molecules (CRAM) derived molecules increased in abundance. Surface DOM contained a remarkably lesser abundance of methyl esters than all other marine DOM, likely a consequence of photodegradation from direct exposure to sunlight. All DOM showed similar overall 13C NMR resonance envelopes typical of an intricate mixture of natural organic matter with noticeable peaks of anomerics and C-aromatics carbon whereas oxygenated aromatics and ketones were of too low abundance to result in noticeable humps at the S/N ratio provided. Integration according to major substructure regimes revealed continual increase of carboxylic acids and ketones from surface to deep marine DOM, reflecting a progressive oxygenation of marine DOM, with concomitant decline of carbohydrate-related substructures. Isolation of marine DOM by means of SPE likely discriminated against carbohydrates but produced materials with beneficial NMR relaxation properties: a substantial fraction of dissolved organic molecules present allowed the acquisition of two-dimensional NMR spectra with exceptional resolution. JRES, COSY and HMBC NMR spectra were capable to depict resolved molecular signatures of compounds exceeding a certain minimum abundance. Here, JRES spectra suffered from limited resolution whereas HMBC spectra were constrained because of limited S/N ratio. Hence, COSY NMR spectra appeared best suited to depict organic complexity in marine DOM. The intensity and number of COSY cross peaks was found maximal for sample FMAX and conformed to about 1500 molecules recognizable in variable abundance. Surface DOM (FISH) produced a slightly (~25%) lesser number of cross peaks with remarkable positional accordance to FMAX (~80% conforming COSY cross peaks were found in FISH and FMAX). With increasing water depth, progressive attenuation of COSY cross peaks was caused by fast transverse NMR relaxation of yet unknown origin. However, most of the faint COSY cross peak positions of deep water DOM conformed to those observed in the surface DOM, suggesting the presence of a numerous set of identical molecules throughout the entire ocean column even if the investigated water masses belonged to different oceanic regimes and currents. Aliphatic chemical environments of methylene (CH2) and methyl (CH3) in marine DOM were nicely discriminated in DEPT HSQC NMR spectra. Classical methyl groups terminating aliphatic chains represented only ~15% of total methyl in all marine DOM investigated. Chemical shift anisotropy from carbonyl derivatives (i.e. most likely carboxylic acids) displaced aliphatic methyl 1H NMR resonances up to δH ~1.6 ppm, indicative of alicyclic geometry which furnishes more numerous short range connectivities for any given atom pairs. A noticeable fraction of methyl (~2%) was bound to olefinic carbon. The comparatively large abundance of methyl ethers in surface marine DOM contrasted with DOM of freshwater and soil origin. The chemical diversity of carbohydrates as indicated by H2CO-groups (δC ~ 62 ± 2 ppm) and anomerics (δC ~ 102 ± 7 ppm) exceeded that of freshwater and soil DOM considerably. HSQC NMR spectra were best suited to identify chemical environments of methin carbon (CH) and enabled discrimination of olefinic and aromatic cross peaks (δC > 110 ppm) and those of doubly oxygenated carbon (δC < 110 ppm). The abundance of olefinic protons exceeded that of aromatic protons; comparison of relative HSQC cross peak integrals indicated larger abundance of olefinic carbon than aromatic carbon in all marine DOM as well. A considerable fraction of olefins seemed isolated and likely sterically constrained as judged from small nJHH couplings associated with those olefins. High S/N ratio and fair resolution of TOCSY and HSQC cross peaks enabled unprecedented depiction of sp2-hybridized carbon chemical environments in marine DOM with discrimination of isolated and conjugated olefins as well as α, β-unsaturated double bonds. However, contributions from five-membered heterocycles (furan, pyrrol and thiophene derivatives) even if very unlikely from given elemental C/N and C/S ratios and upfield proton NMR chemical shift (δH < 6.5 ppm) could not yet been ruled out entirely. In addition to classical aromatic DOM, like benzene derivatives and phenols, six-membered nitrogen heterocycles were found prominent contributors to the downfield region of proton chemical shift (δH > 8 ppm). Specifically, a rather confined HSQC cross peak at δH/δC = 8.2/164 ppm indicated a limited set of nitrogen heterocycles with several nitrogen atoms in analogy to RNA derivatives present in all four marine DOM. Appreciable amounts of extended HSQC and TOCSY cross peaks derived from various key polycyclic aromatic hydrocarbon substructures suggested the presence of previously proposed but NMR invisible thermogenic organic matter (TMOC) in marine DOM at all water depths. Eventually, olefinic unsaturation in marine DOM will be more directly traceable to ultimate biogenic precursors than aromatic unsaturation of which a substantial fraction originates from an aged material which from the beginning was subjected to complex and less specific biogeochemical reactions like thermal decomposition. The variance in molecular mass as indicated from Fourier transform ion cyclotron resonance (FTICR) mass spectra was limited and could not satisfactorily explain the observed disparity in NMR transverse relaxation of the four marine DOM samples. Likewise, the presence of metal ions in isolated marine DOM remained near constant or declined from surface to depth for important paramagnetic ions like Mn, Cr, Fe, Co, Ni and Cu. Iron in particular, a strong complexing paramagnetic ion, was found most abundant by a considerable margin in surface (FISH) marine DOM for which well resolved COSY cross peaks were observed. Hence, facile relationships between metal content of isolated DOM (which does not reflect authentic marine DOM metal content) and transverse NMR relaxation were not observed. High field (12 T) negative electrospray ionization FTICR mass spectra showed at first view rather conforming mass spectra for all four DOM samples with abundant CHO, CHNO, CHOS and CHNOS molecular series with slightly increasing numbers of mass peaks from surface to bottom DOM and similar fractions (~50%) of assigned molecular compositions throughout all DOM samples. The average mass increased from surface to bottom DOM by about 10 Dalton. The limited variance of FTICR mass spectra probably resulted from a rather inherent conformity of marine DOM at the mandatory level of intrinsic averaging provided by FTICR mass spectrometry, when many isomers unavoidably project on single nominal mass peaks. In addition, averaging from ion suppression added to the accordance observed. The proportion of CHO and CHNO molecular series increased from surface to depth whereas CHOS and especially CHNOS molecular series markedly declined. The abundance of certain aromatic CHOS compounds declined with water depth. For future studies, COSY NMR spectra appear best suited to assess organic molecular complexity of marine DOM and to define individual DOM molecules of yet unknown structure and function. Non-target organic structural spectroscopy at the level demonstrated here covered nearly all carbon present in marine DOM. The exhaustive characterization of complex unknowns in \\mbox{marine} DOM will reveal a meaningful assessment of individual marine biogeosignatures which carry the holistic memory of the oceanic water masses (Koch et al., 2011).

AbstractBased on a recent extension theorem for reflexive differential forms, that is, regular differential forms defined on the smooth locus of a possibly singular variety, we study the geometry and cohomology of sheaves of reflexive differentials.First, we generalise the extension theorem to holomorphic forms on locally algebraic complex spaces. We investigate the (non-)existence of reflexive pluri-differentials on singular rationally connected varieties, using a semistability analysis with respect to movable curve classes. The necessary foundational material concerning this stability notion is developed in an appendix to the paper. Moreover, we prove that Kodaira–Akizuki–Nakano vanishing for sheaves of reflexive differentials holds in certain extreme cases, and that it fails in general. Finally, topological and Hodge-theoretic properties of reflexive differentials are explored.

The effects of a wavy wall on the stability of a hypersonic boundary layer on a flared cone are investigated by detailed experimental measurements and direct numerical simulations. The non-contact optical measurement method of focused laser differential interferometry is used to measure the disturbance development within the wavy region. The measurement results show that the second mode for the wavy wall is suppressed significantly compared with the smooth wall, and that multiple disturbances at low frequencies appear within the wavy region. Numerical corroboration against experimental measurements reveals good quantitative agreement. It is found that the disturbances at $f=360$ kHz on the wavy wall are suppressed appreciably, which are very significant on the smooth wall. And the disturbances at $f=140$ kHz and $f=260$ kHz develop within the wavy region, and increase considerably. Also, the disturbances achieve a significant increase over the first half of a wavy trough and become more stable over the second half of a wavy trough. The physical mechanism is found to be due to the change in wall geometry and is attributed to the spatially modulated mean flow. The disturbance growth rate is closely related to the level of the mean-flow distortion.

In this article, we study compactifications of homogeneous spaces coming from equivariant, open embeddings into a generalized flag manifold $G/P$ . The key to this approach is that in each case $G/P$ is the homogeneous model for a parabolic geometry; the theory of such geometries provides a large supply of geometric tools and invariant differential operators that can be used for this study. A classical theorem of Wolf shows that any involutive automorphism of a semisimple Lie group $G$ with fixed point group $H$ gives rise to a large family of such compactifications of homogeneous spaces of $H$ . Most examples of (classical) Riemannian symmetric spaces as well as many non-symmetric examples arise in this way. A specific feature of the approach is that any compactification of that type comes with the notion of ‘curved analog’ to which the tools we develop also apply. The model example of this is a general Poincaré–Einstein manifold forming the curved analog of the conformal compactification of hyperbolic space. In the first part of the article, we derive general tools for the analysis of such compactifications. In the second part, we analyze two families of examples in detail, which in particular contain compactifications of the symmetric spaces $\mathit{SL}(n,\mathbb{R})/\mathit{SO}(p,n-p)$ and $\mathit{SO}(n,\mathbb{C})/\mathit{SO}(n)$ . We describe the decomposition of the compactification into orbits, show how orbit closures can be described as the zero sets of smooth solutions to certain invariant differential operators and prove a local slice theorem around each orbit in these examples.

We study the geodesics and pre-geodesics of a smooth manifold with smooth pseudo riemannian metric which changes bilinear type (i. e. the signature changes) on a hypersurface. We classify all geodesics and pre-geodesics that cross the hypersurface of type change transversely. We then apply these results to the eikonal partial differential equation to find geometric conditions for the local existence or non-existence of smooth, transverse solutions.

Abstract This paper considers the problem of representing a sufficiently smooth non-linear dynamical [system] as a structured potential-driven system. The proposed method is based on a decomposition of a differential one-form associated to a given vector field into its exact and anti-exact components, and into its co -exact and anti-coexact components. The decomposition method, based on the Hodge decomposition theorem, is rendered constructive by introducing a dual operator to the standard homotopy operator. The dual operator inverts locally the co-differential operator, and is used in the present paper to identify the symplectic structure of the dynamics. Applications of the proposed approach to gradient systems, Hamiltonian systems and generalized Hamiltonian systems are given to illustrate the proposed approach. Finally, integrability conditions for generalized Hamiltonian systems are established using the proposed decomposition.

Abstract Given a smooth Hermitian vector bundle $\mathcal{E}$ over a closed Riemannian manifold $(M,g)$, we study generic properties of unitary connections $\nabla ^{\mathcal{E}}$ on the vector bundle $\mathcal{E}$. First of all, we show that twisted conformal Killing tensors (CKTs) are generically trivial when $\dim (M) \geq 3$, answering an open question of Guillarmou–Paternain–Salo–Uhlmann [ 14]. In negative curvature, it is known that the existence of twisted CKTs is the only obstruction to solving exactly the twisted cohomological equations, which may appear in various geometric problems such as the study of transparent connections. The main result of this paper says that these equations can be generically solved. As a by-product, we also obtain that the induced connection $\nabla ^{\textrm{End}({\operatorname{{\mathcal{E}}}})}$ on the endomorphism bundle $\textrm{End}({\operatorname{{\mathcal{E}}}})$ has generically trivial CKTs as long as $(M,g)$ has no nontrivial CKTs on its trivial line bundle. Eventually, we show that, under the additional assumption that $(M,g)$ is Anosov (i.e., the geodesic flow is Anosov on the unit tangent bundle), the connections are generically opaque, namely that generically there are no non-trivial subbundles of $\mathcal{E}$ that are preserved by parallel transport along geodesics. The proofs rely on the introduction of a new microlocal property for (pseudo)differential operators called operators of uniform divergence type, and on perturbative arguments from spectral theory (especially on the theory of Pollicott–Ruelle resonances in the Anosov case).