Infinite Regularization Research Articles

Global martingale and pathwise solutions and infinite regularity of invariant measures for a stochastic modified Swift–Hohenberg equation

We consider a 2D stochastic modified Swift–Hohenberg equations with multiplicative noise and periodic boundary. First, we establish the existence of local and global martingale and pathwise solutions in the regular Sobolev space for each . Associated with the unique global pathwise solution, we obtain a Markovian transition semigroup. Then, we show the existence of invariant measures and ergodic invariant measures for this Markovian semigroup on . At last, we improve the regularity of the obtained invariant measures to . With appropriate conditions on the diffusion coefficient, we can deduce the infinite regularity of the invariant measures, which was conjectured by Glatt-Holtz et al in their situation (2014 J. Math. Phys. 55 277–304).

Globally Existing Solutions to the Problem of Dirichlet for the Fractional 3D Poisson Equation

A general approach to solving the Dirichlet problem, both for bounded 3D domains and for their unbounded complements, in terms of the fractional (3D) Poisson equation, is presented. Lauren Schwartz class solutions are sought for tempered distributions. The solutions found are represented by a formula that contains the volume Riesz potential and the one-layer potential, the latter depending on the boundary data. Infinite regularity of fractional harmonic functions, analogous to the infinite smoothness of the classical harmonic functions, is also proved in the respective domain, no matter what the boundary conditions are. Other properties of the solutions, that are presumably of interest to mathematical physics, are also investigated. In particular, an intrinsic decay property, valid far from the common boundary, is shown.

Azimuthal equatorial flows in spherical coordinates with discontinuous stratification

We are concerned here with an exact solution to the governing equations for geophysical fluid dynamics in spherical coordinates, which incorporates discontinuous fluid stratification. This solution represents a steady, purely azimuthal equatorial flow with an associated free surface and an interface separating two fluid regions, each of which has its own continuous distribution of density. However, the two density functions do not match along the interface. Following the derivation of the solution, we demonstrate that there is a well-defined relationship between the imposed pressure at the free surface and the resulting distortion of the surface's shape. Moreover, imposing the continuity of the pressure along the interface generates an equation that describes (implicitly) the shape of the interface. Interestingly, it turns out that the interface defining function has infinite regularity.

Smooth linearization under nonuniform hyperbolicity

For any sufficiently small perturbation of a tempered exponential dichotomy, we obtain appropriate versions of the Grobman–Hartman theorem and of the Sternberg theorems both for finite and infinite regularity, thus providing, respectively, topological and smooth conjugacies. The constructions are heavily based on the existence of normal forms, with the resonances expressed in terms of the connected components of the nonuniform spectrum, which is a tempered version of the Sacker–Sell spectrum. In order to obtain specific tempered bounds for the derivatives up to a certain order, we first construct smooth stable and unstable invariant manifolds for any sufficiently small perturbation of a tempered exponential dichotomy. We also make several preparations of the dynamics so that the linear part has a block form, the nonlinear part has no terms up to a given order, and the stable and unstable manifolds coincide, respectively, with the stable and unstable spaces. The conjugacies are then constructed via the homotopy method.

Modules of infinite regularity over commutative graded rings

In this work, we prove that if a graded, commutative algebra $R$ over a field $k$ is not Koszul then, denoting by $\mathfrak{m}$ the maximal homogeneous ideal, the nonzero modules of the form $\mathfrak{m} M$ have infinite Castelnuovo-Mumford regularity. We also prove that over complete intersections which are not Koszul, a nonzero direct summand of a syzygy of $k$ has infinite regularity. Finally, we relate the vanishing of the graded deviations of $R$ to having a nonzero direct summand of a syzygy of $k$ of finite regularity.

Gibbs Measures Over Locally Tree-Like Graphs and Percolative Entropy Over Infinite Regular Trees

Consider a statistical physical model on the $d$-regular infinite tree $T_{d}$ described by a set of interactions $\Phi$. Let $\{G_{n}\}$ be a sequence of finite graphs with vertex sets $V_n$ that locally converge to $T_{d}$. From $\Phi$ one can construct a sequence of corresponding models on the graphs $G_n$. Let $\{\mu_n\}$ be the resulting Gibbs measures. Here we assume that $\{\mu_{n}\}$ converges to some limiting Gibbs measure $\mu$ on $T_{d}$ in the local weak$^*$ sense, and study the consequences of this convergence for the specific entropies $|V_n|^{-1}H(\mu_n)$. We show that the limit supremum of $|V_n|^{-1}H(\mu_n)$ is bounded above by the \emph{percolative entropy} $H_{perc}(\mu)$, a function of $\mu$ itself, and that $|V_n|^{-1}H(\mu_n)$ actually converges to $H_{perc}(\mu)$ in case $\Phi$ exhibits strong spatial mixing on $T_d$. We discuss a few examples of well-known models for which the latter result holds in the high temperature regime.

A graph regularized dimension reduction method for out-of-sample data

Among various dimension reduction techniques, Principal Component Analysis (PCA) is specialized in treating vector data, whereas Laplacian embedding is often employed for embedding graph data. Moreover, graph regularized PCA, a combination of both techniques, has also been developed to assist the learning of a low dimensional representation of vector data by incorporating graph data. However, these approaches are confronted by the out-of-sample problem: each time when new data is added, it has to be combined with the old data before being fed into the algorithm to re-compute the eigenvectors, leading to enormous computational cost. In order to address this problem, we extend the graph regularized PCA to the graph regularized linear regression PCA (grlrPCA). grlrPCA eliminates the redundant calculation on the old data by first learning a linear function and then directly applying it to the new data for its dimension reduction. Furthermore, we derive an efficient iterative algorithm to solve grlrPCA optimization problem and show the close relatedness of grlrPCA and unsupervised Linear Discriminant Analysis at infinite regularization parameter limit. The evaluations of multiple metrics on seven realistic datasets demonstrate that grlrPCA outperforms established unsupervised dimension reduction algorithms.

Before and Above: Spinoza and Symbolic Necessity

In Spinoza, God is without a name and without a shape. His essence is the very form of the necessity of nature, the infinite regularity, actuality and rationality of what there is. Nothing good, nothing bad in this. All representations of God as a legislator, a creator or a father, endowed with intentions, are only human projections produced by an inadequate understanding of what a cause is. A true cause is never separated from its effect, but is immanent to it, which means that it remains within it. As cause of himself, that is of nature, God is nothing but his own effectuation, and in that sense, he cannot be said transcendent, that is external, to what he produces. Are the readings of Spinoza that characterize him only as a thinker of “immanence” and/or “self-regulation” totally fair though ? Do they do justice to the major issue of the origin of the sacred as developped in the Theological-Political Treatise ? Where does sacredness come from for Spinoza ? Can it be reduced to a sheer error or illusion, a temporary hole in the tissue of immanence, or does it open a specific space in immanence that remains to be explored ? What exactly are the relationships between necessity and faith, between truth and its irreducible symbolic dimension? And what does “symbolic” means there where God is impersonal ? Following Spinoza scriptural hermeneutics, and discussing it also with thinkers like Levinas, we will develop a new approach to Spinoza’s concept of revelation, pertaining to his vision of the sacred as an economy of signs without a referent. In doing so, we would like to show that Spinoza’s critique of religious dogmatism or fanatism is not to be confused with the dismissal of the sacred, it is on the contrary propedeutical to the philosophical delineation of sacredness.

Superadditivity of Private Information for Any Number of Uses of the Channel.

The quantum capacity of a quantum channel is always smaller than the capacity of the channel for private communication. Both quantities are given by the infinite regularization of the coherent and the private information, respectively, which makes their evaluation very difficult. Here, we construct a family of channels for which the private and coherent information can remain strictly superadditive for unbounded number of uses, thus demonstrating that the regularization is necessary. We prove this by showing that the coherent information is strictly larger than the private information of a smaller number of uses of the channel. This implies that even though the quantum capacity is upper bounded by the private capacity, the nonregularized quantities can be interleaved.

Insights into Tikhonov regularization: application to trace gas column retrieval and the efficient calculation of total column averaging kernels

Abstract. Insights are given into Tikhonov regularization and its application to the retrieval of vertical column densities of atmospheric trace gases from remote sensing measurements. The study builds upon the equivalence of the least-squares profile-scaling approach and Tikhonov regularization method of the first kind with an infinite regularization strength. Here, the vertical profile is expressed relative to a reference profile. On the basis of this, we propose a new algorithm as an extension of the least-squares profile scaling which permits the calculation of total column averaging kernels on arbitrary vertical grids using an analytic expression. Moreover, we discuss the effective null space of the retrieval, which comprises those parts of a vertical trace gas distribution which cannot be inferred from the measurements. Numerically the algorithm can be implemented in a robust and efficient manner. In particular for operational data processing with challenging demands on processing time, the proposed inversion method in combination with highly efficient forward models is an asset. For demonstration purposes, we apply the algorithm to CO column retrieval from simulated measurements in the 2.3 μm spectral region and to O3 column retrieval from the UV. These represent ideal measurements of a series of spaceborne spectrometers such as SCIAMACHY, TROPOMI, GOME, and GOME-2. For both spectral ranges, we consider clear-sky and cloudy scenes where clouds are modelled as an elevated Lambertian surface. Here, the smoothing error for the clear-sky and cloudy atmosphere is significant and reaches several percent, depending on the reference profile which is used for scaling. This underlines the importance of the column averaging kernel for a proper interpretation of retrieved column densities. Furthermore, we show that the smoothing due to regularization can be underestimated by calculating the column averaging kernel on a too coarse vertical grid. For both retrievals, this effect becomes negligible for a vertical grid with 20–40 equally thick layers between 0 and 50 km.

Mass Transport at Infinite Regular Arrays of Microband Electrodes Submitted to Natural Convection: Theory and Experiments

Mass transport at infinite regular arrays of microband electrodes was investigated theoretically and experimentally in unstirred solutions. Even in the absence of forced hydrodynamics, natural convection limits the convection-free domain up to which diffusion layers may expand. Hence, several regimes of mass transport may take place according to the electrode size, gap between electrodes, time scale of the experiment, and amplitude of natural convection. They were identified through simulation by establishing zone diagrams that allowed all relative contributions to mass transport to be delineated. Dynamic and steady-state regimes were compared to those achieved at single microband electrodes. These results were validated experimentally by monitoring the chronoamperometric responses of arrays with different ratios of electrode width to gap distance and by mapping steady-state concentration profiles above their surface through scanning electrochemical microscopy.

Identifying Codes with Small Radius in Some Infinite Regular Graphs

Let $G=(V,E)$ be a connected undirected graph and $S$ a subset of vertices. If for all vertices $v \in V$, the sets $B_r(v) \cap S$ are all nonempty and different, where $B_r(v)$ denotes the set of all points within distance $r$ from $v$, then we call $S$ an $r$-identifying code. We give constructive upper bounds on the best possible density of $r$-identifying codes in four infinite regular graphs, for small values of $r$.

General Bounds for Identifying Codes in Some Infinite Regular Graphs

Consider a connected undirected graph $G=(V,E)$ and a subset of vertices $C$. If for all vertices $v \in V$, the sets $B_r(v) \cap C$ are all nonempty and pairwise distinct, where $B_r(v)$ denotes the set of all points within distance $r$ from $v$, then we call $C$ an $r$-identifying code. We give general lower and upper bounds on the best possible density of $r$-identifying codes in three infinite regular graphs.

Infinite Regular Hexagon Sequences on a Triangle

The well-known dual pair of Napoleon equilateral triangles intrinsic to each triangle is extended to infinite sequences of them, shown to be special casesof infinite regular hexagon sequences on each triangle. A set of hexagon-to-hexagon transformations, the hex operators, is defined for this purpose, a set forming an abelian monoid under function composition. The sequences result from arbitrary strings of hex operators applied to a particular truncation of a given triangle to a hexagon. The deep structure of the sequence constructions reveals surprising infinite sequences of nonconcentric, symmetric equilateral triangle pairs parallel to one of the sequences of hexagons and provides the most visually striking contribution. Extensive experimentation with a plane geometry educational program inspired all theorems, proofs of which utilize eigenvector analysis of polygons in the complex plane.