Nontrivial Components Research Articles

Maxwell–Dirac system in cosmology

Within the scope of a Bianchi type-I (BI) cosmological model, we study the interacting system of spinor and electromagnetic fields and its role in the evolution of the Universe. In some earlier studies, it was found that in the case of a pure spinor field, the presence of nontrivial nondiagonal components of energy–momentum tensor (EMT) leads to some severe restrictions both on the space–time geometry and/or spinor field itself, whereas in the case of electromagnetic field with induced nonlinearity, such components impose severe restrictions on metric functions and the components of the vector potential. It is shown that in the case of interacting spinor and electromagnetic fields, the restrictions are not as severe as in the other cases and in this case, a nonlinear and massive spinor field with different components of vector potential can survive in a general Bianchi type-I space–time.

Clear-Plenoxels: Floaters free radiance fields without neural networks

Directly optimizing radiance fields with explicit 3D primitives, such as voxels and 3D Gaussians, demonstrates impressive effectiveness in both training and rendering. However, optimizing explicit 3D structures from limited training views using differentiable volumetric rendering often results in ghosting artifacts (floaters), i.e., it suffers from early-stage errors in unseen space and remains as evident ghosting artifacts as gradient becomes very sparse spatially with optimization procedure processes. Thus, to address this problem, this paper proposes Clear-Plenoxels, which involves three simple yet non-trivial components: 1) a Visual Hull-based initialization strategy for maximizing the utilization of object resolution and effectively rejecting false updates during early-stage optimization by learning rate assignment per voxel grid; 2) an effective penalty function on local grid discrepancy for ensuring consistent rendering at different view directions, which can remove unnecessary voxels; 3) a mask guided transmittance supervision for each training ray, significantly guaranteeing depth prediction precision when the object's surface color is close to the background. Experiments on a public dataset demonstrate that our method successfully overcomes ghosting artifacts by directly optimizing explicit primitive strategy. The proposed method achieves approximately 1 dB improvement in voxel grid-based methods and matches, if not surpasses, state-of-the-art quality regarding novel view rendering. The code and visualization videos are publicly available at https://github.com/nortonBryan/Clear-plenoxels.

Yamabe systems, optimal partitions and nodal solutions to the Yamabe equation

We give conditions for the existence of regular optimal partitions, with an arbitrary number \ell\geq 2 of components, for the Yamabe equation on a closed Riemannian manifold (M,g) . To this aim, we study a weakly coupled competitive elliptic system of \ell equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if \dim M\geq 10 , (M,g) is not locally conformally flat, and satisfies an additional geometric assumption whenever \dim M=10 . Moreover, we show that the limit profiles of the components of the solution separate spatially as the competition parameter goes to -\infty , giving rise to an optimal partition. We show that this partition exhausts the whole manifold, and we prove the regularity of both the interfaces and the limit profiles, together with a free boundary condition. For \ell=2 the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.

Partially concentrating standing waves for weakly coupled Schrödinger systems

AbstractWe study the existence of standing waves for the following weakly coupled system of two Schrödinger equations "Equation missing"where $$V_1$$ V 1 and $$V_2$$ V 2 are radial potentials bounded from below. We address the case "Equation missing", $$m_2$$ m 2 constant, and prove the existence of a standing wave solution with both nontrivial components satisfying a prescribed asymptotic profile. In particular, the second component of such solution exhibits a concentrating behavior, while the first one keeps a quantum nature.

Extremal spectral radius and essential edge-connectivity

An edge-cut X of G is essential if G−X has at least two non-trivial components, the essential edge-connectivity of G is the minimum cardinality over all essential edge-cuts of G. In this paper, we determine the graphs attaining the maximum spectral radius among all graphs with minimum degree and essential edge-connectivity. Moreover, the extremal graphs are characterized. This generalizes some previous results on edge-connectivity.

Fluid Cohomology

The vorticity-streamfunction formulation for incompressible inviscid fluids is the basis for many fluid simulation methods in computer graphics, including vortex methods, streamfunction solvers, spectral methods, and Monte Carlo methods. We point out that current setups in the vorticity-streamfunction formulation are insufficient at simulating fluids on general non-simply-connected domains. This issue is critical in practice, as obstacles, periodic boundaries, and nonzero genus can all make the fluid domain multiply connected. These scenarios introduce nontrivial cohomology components to the flow in the form of harmonic fields. The dynamics of these harmonic fields have been previously overlooked. In this paper, we derive the missing equations of motion for the fluid cohomology components. We elucidate the physical laws associated with the new equations, and show their importance in reproducing physically correct behaviors of fluid flows on domains with general topology.

Spinor Field in FLRW Cosmology

Within the scope of a Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological model we study the role of a nonlinear spinor field in the evolution of the universe. In doing so, we exploit the FLRW models given in both Cartesian and spherical coordinates. It is found that if the FLRW model is given in the spherical coordinates the energy-momentum tensor (EMT) of the spinor field possesses nontrivial non-diagonal components, which is not the case for Cartesian coordinates. These non-diagonal components do not depend on either the spinor field nonlinearity or the parameter k that defines the type of curvature of the FLRW model. The presence of such components imposes some restrictions on the spinor field. The problem is studied for open, flat and close geometries and the spinor field is used to simulate different types of sources including dark energies. Some qualitative numerical solutions are given.

A Generalization of the Graph Packing Theorems of Sauer-Spencer and Brandt

We prove a common generalization of the celebrated Sauer-Spencer packing theorem and a theorem of Brandt concerning finding a copy of a tree inside a graph. This proof leads to the characterization of the extremal graphs in the case of Brandt’s theorem: If G is a graph and F is a forest, both on n vertices, and 3Δ(G) + ℓ* (F) ≤ n, then G and F pack unless n is even, \(G = {n \over 2}{K_2}\) and F = K1,n−1 where ℓ*(F) is the difference between the number of leaves and twice the number of nontrivial components of F.

Obstacles from interstellar matters and distortion in warp drive superluminal travel scenario

We investigate obstacles of superluminal “warp drive” travels from interactions with interstellar matter and from curvature effects. The effect of collision of interstellar dust particles and photons with the spacecraft will all lead to a pressure proportional to the apparent velocity of the spaceship v_s. The force exerted on the spacecraft from the curvature effect has two non-trivial components. The radial and longitudinal components scales as v_s^2 and v_s^4 respectively. The above obstacles become increasingly important when the spaceship travels at high superluminal speeds.

Time-dependent spinor field in a static cylindrically symmetric space-time

Within the scope of a static cylindrically symmetric space-time we study the behavior of a nonlinear spinor field that depends on time and radial coordinates. It is found that the presence of nontrivial non-diagonal components of the energy-momentum tensor (EMT) imposes some restrictions on both the metric functions and the spinor field. While for the time independent spinor field there occur three way restrictions resembling the Bianchi type-I Universe, in this case things become more complicated.

Weakly nonlinear surface waves on the plasma–vacuum interface

We consider the free boundary problem for a plasma–vacuum interface in ideal incompressible magnetohydrodynamics. Unlike the classical statement, where the vacuum magnetic field obeys the div-curl system of pre-Maxwell dynamics, we do not neglect the displacement current in the vacuum region and consider the Maxwell equations for electric and magnetic fields. Our aim is to construct highly oscillating surface wave solutions in weakly nonlinear regime to this plasma–vacuum interface problem.Under a necessary and sufficient stability condition for a piecewise constant background state, we construct approximate solutions at any arbitrarily large order of accuracy to the free boundary problem in three space dimensions when the initial discontinuity displays high frequency oscillations. Moreover, such approximate surface waves have nontrivial residual non-oscillatory components.

Least energy positive solutions of critical Schrödinger systems with mixed competition and cooperation terms: The higher dimensional case

Let Ω⊂RN be a smooth bounded domain. In this paper we investigate the existence of least energy positive solutions to the following Schrödinger system with d≥2 equations−Δui+λiui=|ui|p−2ui∑j=1dβij|uj|p in Ω,ui=0 on ∂Ω,i=1,...,d, in the case of a critical exponent 2p=2⁎=2NN−2 in high dimensions N≥5. We treat the focusing case (βii>0 for every i) in the variational setting βij=βji for every i≠j, dealing with a Brézis-Nirenberg type problem: −λ1(Ω)<λi<0, where λ1(Ω) is the first eigenvalue of (−Δ,H01(Ω)). We provide several sufficient conditions on the coefficients βij that ensure the existence of least energy positive solutions; these include the situations of pure cooperation (βij>0 for every i≠j), pure competition (βij≤0 for every i≠j) and coexistence of both cooperation and competition coefficients.In order to provide these results, we firstly establish the existence of nonnegative solutions with 1≤m≤d nontrivial components by comparing energy levels of the system with those of appropriate sub-systems. Moreover, based on new energy estimates, we obtain the precise asymptotic behaviors of the nonnegative solutions in some situations which include an analysis of a phase separation phenomena. Some proofs depend heavily on the fact that 1<p<2, revealing some different phenomena comparing to the special case N=4. Our results provide a rather complete picture in the particular situation where the components are divided in two groups. Besides, based on the results about the phase separation, we prove the existence of least energy sign-changing solution to the Brézis-Nirenberg problem−Δu+λu=μ|u|2⁎−2u,u∈H01(Ω), for μ>0, −λ1(Ω)<λ<0 for all N≥4, a result which is new in dimensions N=4,5.

Deformation and dewetting of liquid films under gas jets

Abstract

Spinors in Cylindrically Symmetric Space–Time

We studied the behavior of nonlinear spinor field within the scope of a static cylindrically symmetric space–time. It is found that the energy-momentum tensor (EMT) of the spinor field in this case possesses nontrivial non-diagonal components. The presence of non-diagonal components of the EMT imposes three-way restrictions either on the space–time geometry or on the components of the spinor field or on both. It should be noted that the analogical situation occurs in cosmology when the nonlinear spinor field is exploited as a source of gravitational field given by the Bianchi type-I cosmological model.

MHD instability of two fluids between parallel plates

The work focuses on the exact solution of the perturbative model for a plane-parallel flow of two immiscible fluids under constant transversal magnetic field (Hartmann flow). The special case of the perturbation’s symmetry is considered: it is supposed that perturbations intensity depends just on specified two parameters and time. Nevertheless, the considered case seems to be common enough and can be considered as one of the flow stability criteria. To receive exact solution, the governing equation system is analyzed. It is shown that in the considered case, it can be reduced to heat equations for the nontrivial components. The perturbations considered in this work are not necessary of harmonic type and the corresponding equations on the perturbations are not Orr-Sommerfeld equations.

Ground states for a coupled Schr\xf6dinger system with general nonlinearities

We study a coupled Schrödinger system with general nonlinearities. By using variational methods, we prove the existence and asymptotic behaviour of ground state solution for the system with periodic couplings. Moreover, we prove the existence and nonexistence of ground state solution for the system with non-periodic couplings via Nehari manifold method. Especially, the ground state solution with both nontrivial components is obtained, and the sign of nontrivial components is considered.

Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case

In this paper we investigate the existence of solutions to the following Schrodinger system in the critical case $$\begin{aligned} -\Delta u_{i}+\lambda _{i}u_{i}=u_{i}\sum _{j = 1}^{d}\beta _{ij}u_{j}^{2} \text { in } \Omega , \quad u_i=0 \text { on } \partial \Omega , \qquad i=1,\ldots ,d. \end{aligned}$$Here, $$\Omega \subset {\mathbb {R}}^{4}$$ is a smooth bounded domain, $$d\ge 2$$, $$-\lambda _{1}(\Omega ) 0$$ for every i, $$\beta _{ij}=\beta _{ji}$$ for $$i\ne j$$, where $$\lambda _{1}(\Omega )$$ is the first eigenvalue of $$-\Delta $$ with Dirichlet boundary conditions. Under the assumption that the components are divided into m groups, and that $$\beta _{ij}\ge 0$$ (cooperation) whenever components i and j belong to the same group, while $$\beta _{ij}<0$$ or $$\beta _{ij}$$ is positive and small (competition or weak cooperation) for components i and j belonging to different groups, we establish the existence of nonnegative solutions with m nontrivial components, as well as classification results. Moreover, under additional assumptions on $$\beta _{ij}$$, we establish existence of least energy positive solutions in the case of mixed cooperation and competition. The proof is done by induction on the number of groups, and requires new estimates comparing energy levels of the system with those of appropriate sub-systems. In the case $$\Omega ={\mathbb {R}}^4$$ and $$\lambda _1=\cdots =\lambda _m=0$$, we present new nonexistence results. This paper can be seen as the counterpart of Soave and Tavares (J Differ Equ 261:505–537, 2016) in the critical case, while extending and improving some results from Chen and Zou (Arch Ration Mech Anal 205:515–551, 2012) and Guo et al. (Nonlinearity 31:314–339, 2018).

Cluster Analysis of Polysemous Word Semantic Structure in the Light of Invariant Theory

The aim of the article is to disclose specific features of the cluster method of material organization to the semantic structure analysis of polysemous words. The object of study includes the most frequent polysemous words of Russian and English languages. The words for study should have a developed semantic structure thanks to which intra-word connections and a picture of the semantic component selection in the process of decoding figurative meaning can be visually presented. The main method of analysis is component analysis and consideration of non-trivial semantic components. In the process of word analysis, we intend to discover what underlies the retention of meanings within a single structure. Initially, the idea bases on the existence of some invariant content, which is stored in the folded form in the mind of a native speaker. In the present study, the application of cluster theory to the organization of the polysemous word semantic structure not only confirms the invariant theory, but also brings it to a new level of understanding what the meaning of the word is and how we understand the figurative meaning. Representation of the semantic structure in the form of interconnected semantic components, united in clusters on the basis of a criterion, represents a new approach to understanding the phenomenon of polysemy and explaining the diffusivity of figurative meanings. The configuration of invariant and differential components in the decoding process provides an understanding of figurative meanings. We find confirmation of this theory in the works on neurolinguistics and cognitive psychology.

Further developments on Brouwer's conjecture for the sum of Laplacian eigenvalues of graphs

Let G be a simple graph with order n and size m and having Laplacian eigenvalues μ1,μ2,…,μn−1,μn=0 and let Sk(G)=∑i=1kμi be the sum of k largest Laplacian eigenvalues of G. Brouwer conjectured that Sk(G)≤m+(k+12), for all k=1,2,…,n. We obtain upper bounds for Sk(G), in terms of the clique number ω, the order n and integers p≥0,r≥1,s1≥s2≥2 associated to the structure of the graph G. We discuss Brouwer's conjecture for two large families of graphs; the first family of graphs is obtained from a clique of size ω by identifying each of its vertices to a vertex of an arbitrary c-cyclic graph, and the second family is composed of the graphs in which the removal of the edges of the maximal complete bipartite subgraph gives a graph each of whose non-trivial components is a c-cyclic graph. We show among these two large families of graphs, the Brouwer's conjecture holds for various subfamilies of graphs depending upon the value of c, the order of the c-cyclic graphs, the clique number of the graph, the order of the maximal complete bipartite subgraph and the number of the c-cyclic components of the graph.

Finite-amplitude steady-state wave groups with multiple near-resonances in finite water depth

Finite-amplitude wave groups with multiple near-resonances are investigated to extend the existing results due to Liu et al. (J. Fluid Mech., vol. 835, 2018, pp. 624–653) from steady-state wave groups in deep water to steady-state wave groups in finite water depth. The slow convergence rate of the series solution in the homotopy analysis method and extra unpredictable high-frequency components in finite water depth make it hard to obtain finite-amplitude wave groups accurately. To overcome these difficulties, a solution procedure that combines the homotopy analysis method-based analytical approach and Galerkin method-based numerical approaches has been used. For weakly nonlinear wave groups, the continuum of steady-state resonance from deep water to finite water depth is established. As nonlinearity increases, the frequency bands broaden and more steady-state wave groups are obtained. Finite-amplitude wave groups with steepness no less than $0.20$ are obtained and the resonant sets configuration of steady-state wave groups are analysed in different water depths. For waves in deep water, the majority of non-trivial components appear around the primary ones due to four-wave, six-wave, eight-wave or even ten-wave resonant interactions. The dominant role of four-wave resonant interactions for steady-state wave groups in deep water is demonstrated. For waves in finite water depth, additional non-trivial high-frequency components appear in the spectra due to three-wave, four-wave, five-wave or even six-wave resonant interactions with the components around the primary ones. The amplitude of these high-frequency components increases further as the water depth decreases. Resonances composed by components only around the primary ones are suppressed while resonances composed by components around the primary ones and from the high-frequency domain are enhanced. The spectrum of steady-state resonant wave groups changes with the water depth and the significant role of three-wave resonant interactions in finite water depth is demonstrated.