Tubular Domains Research Articles

The Gel’fand Problem on Expanding Tubular Domains in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^2$$\\end{document}: Existence and the Morse Index of Solutions

We discuss the existence and the Morse index of solutions to the problem ΔU+λeU=0inΩR,U=0on∂ΩR,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} \\Delta U+\\lambda e^U=0 & \ ext {in}\\ \\Omega _{R},\\\\ U=0 & \ ext {on}\\ \\partial \\Omega _{R}, \\end{array}\\right. } \\end{aligned}$$\\end{document}where ΩR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega _R$$\\end{document} is a tubular domain in the plane with fixed width. We obtain the existence of an increasing sequence Rk→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_k\\rightarrow \\infty $$\\end{document} as k→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\rightarrow \\infty $$\\end{document} and a corresponding sequence of solutions {URk}k≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{U_{R_k}\\}_{k\\ge 1}$$\\end{document} to the above problem. We investigate the energy of such solutions and show that their Morse index m(URk)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ extsf{m}}}(U_{R_k})$$\\end{document} satisfies m(URk)/Rk→2-η1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ extsf{m}}}(U_{R_k})/R_k\\rightarrow 2\\sqrt{-\\eta _1}$$\\end{document} as k→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\rightarrow \\infty $$\\end{document}, where η1<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta _1<0$$\\end{document} is the first eigenvalue of the linearized 1D Gel’fand problem.

Physics-informed neural network for solving Young–Laplace equation and identifying parameters

Capillarity is prevalent in nature, daily life, and industrial processes, governed by the fundamental Young–Laplace equation. Solving this equation not only deepens our understanding of natural phenomena but also yields insight into industrial advancements. To tackle the challenges posed by traditional numerical methods in parameter identification and complex boundary condition handling, the Young–Laplace physics-informed neural network (Y–L PINN) is established to solve the Young–Laplace equation within tubular domain. The computations on the classical capillary rise scenario confirm the accuracy of the proposed method on the basis of the comparison with Jurin's law, experimental data, and numerical results. Furthermore, the Y–L PINN method excels in parameter identification, e.g., contact angle, Bond number, and so on. These numerical examples even demonstrate its excellent predictive ability from the noisy data. For the complex boundary, it is rather convenient to obtain the liquid meniscus shapes in vessels, which is in good agreement with the experimental results. We further examine the variation of meniscus profile with wetting condition or discontinuous boundary. Importantly, the Y–L PINN method could directly solve the Young–Laplace equation with discontinuous wetting boundary without additional techniques. This work provides valuable insight for material wettability assessments, microstructure preparation, and microfluidics research.

Tracing chirality from molecular organization to triply-periodic network assemblies: Threading biaxial twist through block copolymer gyroids

Chirality transfer from the level of molecular structure up to mesoscopic lengthscales of supramolecular morphologies is a broad and persistent theme in self-assembled soft materials, from biological to synthetic matter. Here, we analyze the mechanism of chirality transfer in a prototypical self-assembly system, block copolymers (BCPs), in particular, its impact on one of the most complex and functionally vital phases: the cubic, triply-periodic, gyroid network. Motivated by recent experimental studies, we consider a self-consistent field model of ABC* triblock copolymers possessing an end-block of chiral chain chemistry and examine the interplay between chirality at the scale of networks in alternating double network phases and the patterns of segmental order within tubular network domains. We show that while segments in gyroids exhibit twist in both polar and nematic segmental order parameters, the magnitude of net nematic twist is generically much larger than polar twist, and more surprising, reverses handedness relative to the sense of polar order as well as the sense of dihedral twist of the network. Careful analysis of the intra-domain nematic order reveals that this unique chirality transfer mechanism relies on the strongly biaxial nature of segmental order in BCP networks and relates the biaxial twist to complex patterns of frame rotation of the principal directors in the intra-domain texture. Finally, we show that this mechanism of twist reversal leads to chirality selection of alternating gyroid networks in ABC* triblocks, in the limit of very weak chirality.

ERnet: a tool for the semantic segmentation and quantitative analysis of endoplasmic reticulum topology.

The ability to quantify structural changes of the endoplasmic reticulum (ER) is crucial for understanding the structure and function of this organelle. However, the rapid movement and complex topology of ER networks make this challenging. Here, we construct a state-of-the-art semantic segmentation method that we call ERnet for the automatic classification of sheet and tubular ER domains inside individual cells. Data are skeletonized and represented by connectivity graphs, enabling precise and efficient quantification of network connectivity. ERnet generates metrics on topology and integrity of ER structures and quantifies structural change in response to genetic or metabolic manipulation. We validate ERnet using data obtained by various ER-imaging methods from different cell types as well as ground truth images of synthetic ER structures. ERnet can be deployed in an automatic high-throughput and unbiased fashion and identifies subtle changes in ER phenotypes that may inform on disease progression and response to therapy.

Boundary quasi-analyticity and a Phragmén–Lindelöf type theorem in classes of functions of bounded type in tubular domains

A complete description is obtained of the Carleman classes on R n \mathbb {R}^n such that every function of bounded type in C + n \mathbb {C}^n_+ whose boundary values belong to the class under study is in fact a member of the corresponding Carleman class in C + n ∪ R n \mathbb {C}^n_+\cup \mathbb {R}^n . Also a refinement of the classical Salinas theorem is obtained, namely: under the conditions of the Salinas theorem on quasi-analyticity, instead of the assumption that a function belongs to the Carleman class in C + n ∪ R n \mathbb {C}^n_+\cup \mathbb {R}^n it suffices that its boundary values on R n \mathbb {R}^n belong to the Carleman class, and the function is of bounded type in C + n \mathbb {C}^n_+ .

Medial packing and elastic asymmetry stabilize the double-gyroid in block copolymers

Triply-periodic networks are among the most complex and functionally valuable self-assembled morphologies, yet they form in nearly every class of biological and synthetic soft matter building blocks. In contrast to simpler assembly motifs – spheres, cylinders, layers – networks require molecules to occupy variable local environments, confounding attempts to understand their formation. Here, we examine the double-gyroid network phase by using a geometric formulation of the strong stretching theory of block copolymer melts, a prototypical soft self-assembly system. The theory establishes the direct link between molecular packing, assembly thermodynamics and the medial map, a generic measure of the geometric center of complex shapes. We show that “medial packing” is essential for stability of double-gyroid in strongly-segregated melts, reconciling a long-standing contradiction between infinite- and finite-segregation theories. Additionally, we find a previously unrecognized non-monotonic dependence of network stability on the relative entropic elastic stiffness of matrix-forming to tubular-network forming blocks. The composition window of stable double-gyroid widens for both large and small elastic asymmetry, contradicting intuitive notions that packing frustration is localized to the tubular domains. This study demonstrates the utility of optimized medial tessellations for understanding soft-molecular assembly and packing frustration via an approach that is readily generalizable far beyond gyroids in neat block copolymers.

FER regulates endosomal recycling and is a predictor for adjuvant taxane benefit in breast cancer.

Elevated expression of non-receptor tyrosine kinase FER is an independent prognosticator that correlates with poor survival of high-grade and basal/triple-negative breast cancer (TNBC) patients. Here, we show that high FER levels are also associated with improved outcomes after adjuvant taxane-based combination chemotherapy in high-risk, HER2-negative patients. In TNBC cells, we observe a causal relation between high FER levels and sensitivity to taxanes. Proteomics and mechanistic studies demonstrate that FER regulates endosomal recycling, a microtubule-dependent process that underpins breast cancer cell invasion. Using chemical genetics, we identify DCTN2 as a FER substrate. Our work indicates that the DCTN2 tyrosine 6 is essential for the development of tubular recycling domains in early endosomes and subsequent propagation of TNBC cell invasion in 3D. In conclusion, we show that high FER expression promotes endosomal recycling and represents a candidate predictive marker for the benefit of adjuvant taxane-containing chemotherapy in high-risk patients, including TNBC patients.

Topological magnetic hysteresis in single crystals of CeAgSb2 ferromagnet

Closed-topology magnetic domains are usually observed in thin films and in an applied magnetic field. Here we report the observation of rectangular cross-section tubular ferromagnetic domains in thick single crystals of CeAgSb2 in zero applied field. Relatively low exchange energy, small net magnetic moment, and anisotropic in-plane crystal electric fields lower the domain wall energy and allow for the formation of the closed-topology patterns. The tubular domain structure irreversibly transforms into a dendritic pattern upon cycling the magnetic field. This transition between closed and open topologies results in a ‘topological magnetic hysteresis’— the actual hysteresis in magnetization, not due to the imperfections and pinning, but due to the difference in the pattern morphology. Similar physics was suggested before in pure type-I superconductors and is believed to be a generic feature of other nonlinear single (present case), or two-phase (type-I superconductor) systems where the effects similar to demagnetization (shape-dependent macroscopic variation of properties) lead to pattern formation.

An optimal control problem in a tubular thin domain with rough boundary

In this paper we analyze the asymptotic behavior of a control problem set by a convection-reaction-diffusion equation with mixed boundary conditions and defined in a tubular thin domain with rough boundary. The control term acts on a subset of the rough boundary where a Robin-type boundary condition and a catalyzed reaction mechanism are set. The reaction mechanism depends on a parameter α∈R. Such parameter establishes different regimes which also depend on the profile and geometry of the tube defined by a periodic function g:R2↦R. We see that, if ∂2g is not null (that is, when g really depends on the second variable), then three regimes with respect to α are established: α<2, α=2 (the critical value) and α>2. On the other hand, if ∂2g≡0, similar regimes are obtained but now with a different critical value. Indeed, if ∂2g≡0, then the critical value is achieved at α=1. For each one of these six regimes, we obtain the asymptotic behavior of the control problem when the cylindrical thin domain degenerates to the unit interval. We show that the problem is asymptotically controllable just when α assumes the critical values. Our analysis is mainly performed using the periodic unfolding method adapted to cylindrical coordinates in R3.

On traces of new BMOA type spaces in tubular domains over symmetric cones

We provide new sharp trace theorems in new BMOA type analytic function spaces in tubular domains over symmetric cones under certain condition on the Bergman kernel. We obtain based on known properties of r-lattices in tubular domains complete analogues of our previously known results obtained previously by the authors in the unit ball and then in bounded strongly pseudoconvex domains with smooth boundary. Proofs of all theorems in all domains are based mainly on same ideas.

Laplace Transforms for Analytic Functions in Tubular Domains

Laplace Transforms for Analytic Functions in Tubular Domains

Bifurcation structure of coexistence states for a prey–predator model with large population flux by attractive transition

This paper is concerned with a prey–predator model with population flux by attractive transition. Our previous paper (Oeda and Kuto, 2018, Nonlinear Anal. RWA, 44, 589–615) obtained a bifurcation branch (connected set) of coexistence steady states which connects two semitrivial solutions. In Oeda and Kuto (2018, Nonlinear Anal. RWA, 44, 589–615), we also showed that any positive steady-state approaches a positive solution of either of two limiting systems, and moreover, one of the limiting systems is an equal diffusive competition model. This paper obtains the bifurcation structure of positive solutions to the other limiting system. Moreover, this paper implies that the global bifurcation branch of coexistence states consists of two parts, one of which is a simple curve running in a tubular domain near the set of positive solutions to the equal diffusive competition model, the other of which is a connected set characterized by positive solutions to the other limiting system.

Abstract LB203: FER regulates endosomal recycling and is a candidate predictor for taxane benefit in breast cancer

Abstract To this day, targeted treatment options for patients with triple negative metastatic breast cancer (TNBC) are still virtually absent. Previously, we showed that elevated expression of the tyrosine kinase FER is an independent prognosticator that correlates with poor survival of high-grade and basal/TNBC patients. Here, we show that high FER levels are also associated with improved outcome after adjuvant taxane-based combination chemotherapy, in high-risk, HER2-negative patients. In TNBC cells, we observed a causal relation between high FER levels and sensitivity to taxanes. Our proteomics and mechanistic studies demonstrated that FER regulates endosomal recycling, a microtubule-dependent process that underpins breast cancer cell invasion. Using chemical genetics, we identified DCTN2 as a novel FER substrate. Our work indicates that the DCTN2 tyrosine 6 is essential for tubular recycling domains development in early endosomes and subsequent propagation of TNBC cell invasion in 3D. In conclusion, we show that high FER expression promotes endosomal recycling and represents a candidate predictive marker for benefit of adjuvant taxane-containing chemotherapy in high-risk patients, including TNBC patients. Citation Format: Sandra Tavares, Nalan Liv, Milena Pasolli, Mark Opdam, Max Ratze, Manuel Saornil, Lilian Sluimer, Rutger Hengeveld, Robert van Es, Erik van Werkhoven, Harmjan Vos, Holger Rehmann, Boudewijn Burgering, Hendrika Oosterkamp, Susanne Lens, Judith Klumperman, Sabine Linn, Patrick Derksen. FER regulates endosomal recycling and is a candidate predictor for taxane benefit in breast cancer [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2021; 2021 Apr 10-15 and May 17-21. Philadelphia (PA): AACR; Cancer Res 2021;81(13_Suppl):Abstract nr LB203.

A benchmark problem to evaluate implementational issues for three-dimensional flows of incompressible fluids subject to slip boundary conditions

We consider flows of an incompressible Navier–Stokes fluid in a tubular domain with Navier’s slip boundary condition imposed on the impermeable wall. We focus on several implementational issues associated with this type of boundary conditions within the framework of the standard Taylor-Hood mixed finite element method and present the computational results for flows in a tubular domain of finite length with one inlet and one outlet. In particular, we present the details regarding variants of the Nitsche method concerning the incorporation of the impermeability condition on the wall. We also find that the manner in which the normal to the boundary is numerically implemented influences the nature of the computational results. As a benchmark, we set up steady flows in a tube of finite length and compare the computational results with the analytical solutions. Finally, we identify various quantities of interest, such as the dissipation, wall shear stress, vorticity, pressure drop, and provide their precise mathematical definitions. We document how well these quantities are computationally approximated in the case of the benchmark.Although the geometry of the benchmark is simple, the correct computational results require careful selection of numerical methods and surprisingly non-trivial computational resources. Our goal is to test, using the setting with a known analytical solution, a robust computational tool that would be suitable for computations on real complex geometries that have relevance to problems in engineering and medicine. The model parameters in our computations are chosen based on flows in large arteries.

Nonexistence of Solutions for Dirichlet Problems with Supercritical Growth in Tubular Domains

Abstract We deal with Dirichlet problems of the form { Δ ⁢ u + f ⁢ ( u ) = 0 in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}\Delta u+f(u)&amp;\displaystyle=0&amp;&amp;% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&amp;\displaystyle=0&amp;&amp;\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a bounded domain of ℝ n {\mathbb{R}^{n}} , n ≥ 3 {n\geq 3} , and f has supercritical growth from the viewpoint of Sobolev embedding. In particular, we consider the case where Ω is a tubular domain T ε ⁢ ( Γ k ) {T_{\varepsilon}(\Gamma_{k})} with thickness ε &gt; 0 {{\varepsilon}&gt;0} and center Γ k {\Gamma_{k}} , a k-dimensional, smooth, compact submanifold of ℝ n {\mathbb{R}^{n}} . Our main result concerns the case where k = 1 {k=1} and Γ k {\Gamma_{k}} is contractible in itself. In this case we prove that the problem does not have nontrivial solutions for ε &gt; 0 {{\varepsilon}&gt;0} small enough. When k ≥ 2 {k\geq 2} or Γ k {\Gamma_{k}} is noncontractible in itself we obtain weaker nonexistence results. Some examples show that all these results are sharp for what concerns the assumptions on k and f.

On some new sharp embedding theorems for multifunctional Herz-type and Bergman-type spaces in tubular domains over symmetric cones

We introduce new multifunctional mixed norm analytic Herz-type spaces in tubular domains over symmetric cones and provide new sharp embedding theorems for them. Some results are new even in case of onefunctional holomorphic spaces. Some new related sharp results for new multifunctional Bergman-type spaces will be also provided under one condition on Bergman kernel.

Effective transmission conditions for second-order elliptic equations on networks in the limit of thin domains

We consider star-shaped tubular domains consisting of a number of non intersecting semi-infinite strips of small thickness that are connected by a central region of diameter proportional to the thickness of the strips. At the thin-domain limit, the region reduces to a network of half-lines with the same end point (junction). We show that the solutions of uniformly elliptic partial differential equations set on the domain with Neumann boundary conditions converge, in the thin-domain limit, to the unique solution of a second-order partial differential equation on the network satisfying an effective Kirchhoff-type transmission condition at the junction. The latter is found by solving an “ergodic”-type problem at infinity obtained after a first-order blow up at the junction.

On some new estimates for integrals of the Lusin’s square function in the unit polydisk

The purpose of the note is to obtain new estimates for the quasinorm of Hardy’s analytic classes of in the polydisk. We extend some classical onedimensional assertions to the case of several complex variables. Our results more precisely provide direct new extention of some known one variable theorems concerning area integral to the case of simplest product domains namely the unit polydisk in $\mathbb{C}^n$. Let further $D$ be a bounded or unbounded domain in $\mathbb{C}^n$. For example, tubular domain over symmetic cone or bounded pseudoconvex domain with smooth boundary. Our results can be probably extended to the case of products of such type complicated domains, namely even to $D\times\dots\times D$. This can be probably done based on some approaches we suggested and used in this paper. On the other hand our results in simpler case namely in the unit polydisk may also have various interesting applications in complex function theory in the unit polydisk.

Thin domain limit and counterexamples to strong diamagnetism

We study the magnetic Laplacian and the Ginzburg–Landau functional in a thin planar, smooth, tubular domain and with a uniform applied magnetic field. We provide counterexamples to strong diamagnetism, and as a consequence, we prove that the transition from the superconducting to the normal state is non-monotone. In some nonlinear regime, we determine the structure of the order parameter and compute the super-current along the boundary of the sample. Our results are in agreement with what was observed in the Little–Parks experiment, for a thin cylindrical sample.

Travelling wave behaviour arising in nonlinear diffusion problems posed in tubular domains

For a fixed bounded domain D⊂RN we investigate the asymptotic behaviour for large times of solutions to the p-Laplacian diffusion equation posed in a tubular domain∂tu=Δpu in D×R,t>0 with p>2, i.e., the slow diffusion case, and homogeneous Dirichlet boundary conditions on the tube boundary. Passing to suitable re-scaled variables, we show the existence of a travelling wave solution in logarithmic time that connects the level u=0 and the unique nonnegative steady state associated to the re-scaled problem posed in a lower dimension, i.e. in D⊂RN.We then employ this special wave to show that a wide class of solutions converge to the universal stationary profile in the middle of the tube and at the same time they spread in both axial tube directions, miming the behaviour of the travelling wave (and its reflection) for large times.The first main feature of our analysis is that wave fronts are constructed through a (nonstandard) combination of diffusion and absorbing boundary conditions, which gives rise to a sort of Fisher-KPP long-time behaviour. The second one is that the nonlinear diffusion term plays a crucial role in our analysis. Actually, in the linear diffusion framework p=2 solutions behave quite differently.