Invariant Space Research Articles

Strong solutions of the Navier–Stokes equations based on the maximal Lorentz regularity theorem in Besov spaces

We show existence and uniqueness theorem of local strong solutions to the Navier–Stokes equations with arbitrary initial data and external forces in the homogeneous Besov space with both negative and positive differential orders which is an invariant space under the change of scaling. If the initial data and external forces are small, then the local solutions can be extended globally in time. Our solutions also belong to the Serrin class in the usual Lebesgue space. The method is based on the maximal Lorentz regularity theorem of the Stokes equations in the homogeneous Besov spaces. As an application, we may handle such singular data as the Dirac measure and the single layer potential supported on the sphere.

Weak-Type Boundedness of the Fourier Transform on Rearrangement Invariant Function Spaces

AbstractWe study several questions about the weak-type boundedness of the Fourier transform ℱ on rearrangement invariant spaces. In particular, we characterize the action of ℱ as a bounded operator from the minimal Lorentz space Λ(X) into the Marcinkiewicz maximal space M(X), both associated with a rearrangement invariant space X. Finally, we also prove some results establishing that the weak-type boundedness of ℱ, in certain weighted Lorentz spaces, is equivalent to the corresponding strong-type estimates.

Nonlocal vertices and analyticity: Landau equations and general Cutkosky rule

We study the analyticity properties of amplitudes in theories with nonlocal vertices of the type occurring in string field theory and a wide class of nonlocal field theory models. Such vertices are given in momentum space by entire functions of rapid decay in certain (including Euclidean) directions ensuring UV finiteness but are necessarily of rapid increase in others. A parametric representation is obtained by integrating out the loop (Euclidean) momenta after the introduction of generalized Schwinger parameters. Either in the original or parametric representation, the well-defined resulting amplitudesare then continued in the complex space of the external momenta invariants. We obtain the alternative forms of the Landau equations determining the singularity surfaces showing that the nonlocal vertices serve as UV regulators but do not affect the local singularity structure. As a result the full set of singularities known to occur in local field theory also occurs here: normal and anomalous thresholds as well as acnodes, crunodes, and cusps that may under certain circumstances appear even on the physical sheet. Singularities of the second type also appear as shown from the parametric representation. We obtain the general Cutkosky discontinuity rule for encircling a singularity by employing contour deformations only in the finite plane. The unitarity condition (optical theorem) is thendiscussed as a special application of the rule across normal thresholds and the hermitian analyticity property of amplitudes.

Effect of noncommutativity on the spectrum of free particle and harmonic oscillator in rotationally invariant noncommutative phase space

We consider rotationally invariant noncommutative algebra with tensors of noncommutativity constructed with the help of additional coordinates and momenta. The algebra is equivalent to the well-known noncommutative algebra of canonical type. In the noncommutative phase space, rotational symmetry influence of noncommutativity on the spectrum of free particle and the spectrum of harmonic oscillator is studied up to the second-order in the parameters of noncommutativity. We find that because of momentum noncommutativity, the spectrum of free particle is discrete and corresponds to the spectrum of harmonic oscillator in the ordinary space (space with commutative coordinates and commutative momenta). We obtain the spectrum of the harmonic oscillator in the rotationally invariant noncommutative phase space and conclude that noncommutativity of coordinates affects its mass. The frequency of the oscillator is affected by the coordinate noncommutativity and the momentum noncommutativity. On the basis of the results, the eigenvalues of squared length operator are found and restrictions on the value of length in noncommutative phase space with rotational symmetry are analyzed.

Surjectivity of differential operators and linear topological invariants for spaces of zero solutions

We provide a sufficient condition for a linear differential operator with constant coefficients $P(D)$ to be surjective on $C^\infty(X)$ and $\mathscr{D}'(X)$, respectively, where $X\subseteq\mathbb{R}^d$ is open. Moreover, for certain differential operators this sufficient condition is also necessary and thus a characterization of surjectivity for such differential operators on $C^\infty(X)$, resp. on $\mathscr{D}'(X)$, is derived. Additionally, we obtain for certain surjective differential operators $P(D)$ on $C^\infty(X)$, resp. $\mathscr{D}'(X)$, that the spaces of zero solutions $C_P^\infty(X)=\{u\in C^\infty(X);\, P(D)u=0\}$, resp. $\mathscr{D}_P'(X)=\{u\in\mathscr{D}'(X);\,P(D)u=0\}$ possess the linear topological invariant $(\Omega)$ introduced by Vogt and Wagner in [27], resp. its generalization $(P\Omega)$ introduced by Bonet and Doma\'nski in [1].

A Stealth Integrity Targeted Cyber‐Attack in Distributed Electric Power Networks with Local Model Information

AbstractElectric power networks are critical infrastructures, and their correct operation is of vital importance. Nowadays, these systems are prone to cyber‐attacks because of new vulnerabilities in the system and access to shared networks. In this paper, a novel Stealth Integrity Targeted Attack (SITA) is proposed in the context of distributed power systems. A distributed power system comprises several sub‐networks, or zones with dedicated control and monitoring centers. The overall system is represented by linear time invariant state space models with coupled dynamical and algebraic equations. In the proposed strategy, the attacker has access to only one of the sub networks; therefore, the attacker only requires local information about one of the power system zones. Primarily, the proposed attack policy is defined based on zero‐dynamics of the sub network. The intruder injects predesigned signals to both the local generation unit controller as well as local unsecured and controllable loads in the attacked zone. Moreover, the local measurement system, or the sensors of the targeted zone are tampered. Furthermore, it will be proved that although the neighbor zones have physical connections with the attacked zone, the injected adversary signals are designed as they do not impact other zones directly in order to conceal the local attack from neighbor control centers as much as possible. We provide some advice to system administrators to make the intrusion unfeasible or to reveal the attack. The simulations on IEEE‐118 bus test system illustrate the validity of the assertions.

Rearrangements and Leibniz-type rules of mean oscillations

We shall prove a rearrangement inequality in probability measure spaces in order to obtain sharp Leibniz-type rules of mean oscillations in Lp-spaces and rearrangement invariant Banach function spaces.

FFT-based homogenization on periodic anisotropic translation invariant spaces

In this paper we derive a discretisation of the equation of quasi-static elasticity in homogenization in form of a variational formulation and the so-called Lippmann-Schwinger equation, in anisotropic spaces of translates of periodic functions. We unify and extend the truncated Fourier series approach, the constant finite element ansatz and the anisotropic lattice derivation. The resulting formulation of the Lippmann-Schwinger equation in anisotropic translation invariant spaces unifies and analyses for the first time both the Fourier methods and finite element approaches in a common mathematical framework. We further define and characterize the resulting periodised Green operator. This operator coincides in case of a Dirichlet kernel corresponding to a diagonal matrix with the operator derived for the Galerkin projection stemming from the truncated Fourier series approach and to the anisotropic lattice derivation for all other Dirichlet kernels. Additionally, we proof the boundedness of the periodised Green operator. The operator further constitutes a projection if and only if the space of translates is generated by a Dirichlet kernel. Numerical examples for both the de la Vall\'ee Poussin means and Box splines illustrate the flexibility of this framework.

Translation–Modulation Invariant Banach Spaces of Ultradistributions

We introduce and study a new class of translation-modulation invariant Banach spaces of ultradistributions. These spaces show stability under Fourier transform and tensor products; furthermore, they have a natural Banach convolution module structure over a certain associated Beurling algebra, as well as a Banach multiplication module structure over an associated Wiener-Beurling algebra. We also investigate a new class of modulation spaces, the Banach spaces of ultradistributions $\mathcal{M}^F$ on $\mathbb{R}^{d}$, associated to translation-modulation invariant Banach spaces of ultradistributions $F$ on $\mathbb{R}^{2d}$.

Momentum space topology of QCD

We discuss the possibility to consider quark matter as the topological material. We consider hadronic phase (HP), the quark–gluon plasma phase (QGP), and the hypothetical color–flavor locking (CFL) phase. In those phases we identify the relevant topological invariants in momentum space. The formalism is developed, which relates those invariants and massless fermions that reside on vortices and at the interphases. This formalism is illustrated by the example of vortices in the CFL phase.

Domain Space Transfer Extreme Learning Machine for Domain Adaptation.

Extreme learning machine (ELM) has been applied in a wide range of classification and regression problems due to its high accuracy and efficiency. However, ELM can only deal with cases where training and testing data are from identical distribution, while in real world situations, this assumption is often violated. As a result, ELM performs poorly in domain adaptation problems, in which the training data (source domain) and testing data (target domain) are differently distributed but somehow related. In this paper, an ELM-based space learning algorithm, domain space transfer ELM (DST-ELM), is developed to deal with unsupervised domain adaptation problems. To be specific, through DST-ELM, the source and target data are reconstructed in a domain invariant space with target data labels unavailable. Two goals are achieved simultaneously. One is that, the target data are input into an ELM-based feature space learning network, and the output is supposed to approximate the input such that the target domain structural knowledge and the intrinsic discriminative information can be preserved as much as possible. The other one is that, the source data are projected into the same space as the target data and the distribution distance between the two domains is minimized in the space. This unsupervised feature transformation network is followed by an adaptive ELM classifier which is trained from the transferred labeled source samples, and is used for target data label prediction. Moreover, the ELMs in the proposed method, including both the space learning ELM and the classifier, require just a small number of hidden nodes, thus maintaining low computation complexity. Extensive experiments on real-world image and text datasets are conducted and verify that our approach outperforms several existing domain adaptation methods in terms of accuracy while maintaining high efficiency.

Rational Models of the Complement of a Subpolyhedron in a Manifold with Boundary

AbstractLet W be a compact simply connected triangulated manifold with boundary and let K ⊂ W be a subpolyhedron. We construct an algebraic model of the rational homotopy type of W\K out of a model of the map of pairs (K, K⋂∂W) ↪ (W, ∂W) under some high codimension hypothesis.We deduce the rational homotopy invariance of the configuration space of two points in a compact manifold with boundary under 2-connectedness hypotheses. Also, we exhibit nice explicit models of these configuration spaces for a large class of compact manifolds.

Extension of matrix pencil reduction to abelian categories

Matrix pencils, or pairs of matrices, are used in a variety of applications. By the Kronecker decomposition theorem, they admit a normal form. This normal form consists of four parts, one part based on the Jordan canonical form, one part made of nilpotent matrices, and two other dual parts, which we call the observation and control part. The goal of this paper is to show that large portions of that decomposition are still valid for pairs of morphisms of modules or abelian groups, and more generally in any abelian category. In the vector space case, we recover the full Kronecker decomposition theorem. The main technique is that of reduction, which extends readily to the abelian category case. Reductions naturally arise in two flavors, which are dual to each other. There are a number of properties of those reductions which extend remarkably from the vector space case to abelian categories. First, both types of reduction commute. Second, at each step of the reduction, one can compute three sequences of invariant spaces (objects in the category), which generalize the Kronecker decomposition into nilpotent, observation and control blocks. These sequences indicate whether the system is reduced in one direction or the other. In the category of modules, there is also a relation between these sequences and the resolvent set of the pair of morphisms, which generalizes the regular pencil theorem. We also indicate how this allows to define invariant subspaces in the vector space case, and study the notion of strangeness as an example.

Superposition operators, Hardy spaces, and Dirichlet type spaces

For 0<p<∞ and α>−1 the space of Dirichlet type Dαp consists of those functions f which are analytic in the unit disc D and satisfy ∫D(1−|z|)α|f′(z)|pdA(z)<∞. The space Dp−1p is the closest one to the Hardy space Hp among all the Dαp. Our main object in this paper is studying similarities and differences between the spaces Hp and Dp−1p (0<p<∞) regarding superposition operators. Namely, for 0<p<∞ and 0<s<∞, we characterize the entire functions φ such that the superposition operator Sφ with symbol φ maps the conformally invariant space Qs into the space Dp−1p, and, also, those which map Dp−1p into Qs and we compare these results with the corresponding ones with Hp in the place of Dp−1p. We also study the more general question of characterizing the superposition operators mapping Dαp into Qs and Qs into Dαp, for any admissible triplet of numbers (p,α,s).

On two notions of exponential dichotomy

ABSTRACTIn this paper, we propose a weaker form of hyperbolicity in which the conditional stability along the stable and unstable directions is allowed to be non-uniform depending on each particular vector, instead of on each invariant space. We also give explicit examples of the notion. Moreover, we show that the size of the set of vectors for which the spoiling of the uniform exponential behaviour has a prescribed bound, measured in terms of the Lebesgue measure on the unit sphere, may grow to infinity with any given speed.

Composite system in rotationally invariant noncommutative phase space

Composite system is studied in noncommutative phase space with preserved rotational symmetry. We find conditions on the parameters of noncommutativity on which commutation relations for coordinates and momenta of the center-of-mass of composite system reproduce noncommutative algebra for coordinates and momenta of individual particles. Also, on these conditions, the coordinates and the momenta of the center-of-mass satisfy noncommutative algebra with effective parameters of noncommutativity which depend on the total mass of the system and do not depend on its composition. Besides, it is shown that on these conditions the coordinates in noncommutative space do not depend on mass and can be considered as kinematic variables, the momenta are proportional to mass as it has to be. A two-particle system with Coulomb interaction is studied and the corrections to the energy levels of the system are found in rotationally invariant noncommutative phase space. On the basis of this result the effect of noncommutativity on the spectrum of exotic atoms is analyzed.

Special Metric Invariants

In the present paper we obtain new metric invariants of metric spaces. These invariants can be used for classification of the finite metric spaces, their recognition and in the research of Tammes’ problem for sphere.

Topology, edge states, and zero-energy states of ultracold atoms in one-dimensional optical superlattices with alternating on-site potentials or hopping coefficients

One-dimensional superlattices with periodic spatial modulations of onsite potentials or tunneling coefficients can exhibit a variety of properties associated with topology or symmetry. Recent developments of ring-shaped optical lattices allow a systematic study of those properties in superlattices with or without boundaries. While superlattices with additional modulating parameters are shown to have quantized topological invariants in the augmented parameter space, we also found localized or zero-energy states associated with symmetries of the Hamiltonians. Probing those states in ultracold-atoms is possible by utilizing recently proposed methods analyzing particle depletion or the local density of states. Moreover, we summarize feasible realizations of configurable optical superlattices using currently available techniques.

Beryllium Bonding in the Light of Modern Quantum Chemical Topology Tools.

We apply several modern quantum chemical topology (QCT) tools to explore the chemical bonding in well established beryllium bonds. By using the interacting quantum atoms (IQA) approach together with electron distribution functions (EDF) and the natural adaptive orbitals (NAdOs) picture, we show that, in agreement with orbital-based analyses, the interaction in simple σ and π complexes formed by BeX2 (X = H, F, Cl) with water, ammonia, ethylene, and acetylene is dominated by electrostatic terms, albeit covalent contributions cannot be ignored. Our detailed analysis proves that several σ back-donation channels are relevant in these dimers, actually controlling the conformational preference in the π adducts. A number of one-electron beryllium bonds are also studied. Orbital invariant real space arguments clearly show that the role of covalency and charge transfer cannot be ignored.

An E11 invariant gauge fixing

We consider the nonlinear realisation of the semi-direct product of [Formula: see text] and its vector representation which leads to a space-time with tangent group that is the Cartan involution invariant subalgebra of [Formula: see text]. We give an alternative derivation of the invariant tangent space metric that this space–time possesses and compute this metric at low levels in eleven, five and four dimensions. We show that one can gauge fix the nonlinear realisation in an [Formula: see text] invariant manner.