arXiv subjects

Yi-Hsuan Lin

Publications and source records attributed to Yi-Hsuan Lin.

At least 73 records · Page 4Linked to original sources

arXiv2020

A unified analytical theory of heteropolymers for sequence-specific phase behaviors of polyelectrolytes and polyampholytes

The physical chemistry of liquid-liquid phase separation (LLPS) of polymer solutions bears directly on the assembly of biologically functional droplet-like bodies from proteins and nucleic acids. These biomolecular condensates include certain extracellular materials, and intracellular compartments that are characterized as "membraneless organelles". Analytical theories are a valuable, computationally efficient tool for addressing general principles. LLPS of neutral homopolymers are quite well described by theory; but it has been a challenge to develop general theories for the LLPS of heteropolymers involving charge-charge interactions. Here we present a novel theory that combines a random-phase-approximation treatment of polymer density fluctuations and an account of intrachain conformational heterogeneity based upon renormalized Kuhn lengths to provide predictions of LLPS properties as a function of pH, salt, and charge patterning along the chain sequence. Advancing beyond more limited analytical approaches, our LLPS theory is applicable to a wide variety of charged sequences ranging from highly charged polyelectrolytes to neutral or nearly neutral polyampholytes. The new theory should be useful in high-throughput screening of protein and other sequences for their LLPS propensities and can serve as a basis for more comprehensive theories that incorporate non-electrostatic interactions. Experimental ramifications of our theory are discussed.

q-bio.BM↗

arXiv2019

Monotonicity-based inversion of the fractional Schrödinger equation I. Positive potentials

We consider the inverse problems of for the fractional Schrödinger equation by using monotonicity formulas. We provide if-and-only-if monotonicity relations between positive bounded potentials and their associated nonlocal Dirichlet-to-Neumann maps. Based on the monotonicity relation, we can prove uniqueness for the nonlocal Calderón problem in a constructive manner. Secondly, we offer a reconstruction method for an unknown obstacles in a given domain. Our method is independent of the dimension $n\geq 2$ and only requires the background solution of the fractional Schrödinger equation.

math.AP↗

arXiv2019

The Calderón problem for a space-time fractional parabolic equation

In this article we study an inverse problem for the space-time fractional parabolic operator $(\partial_t-Δ)^s+Q$ with $0<s<1$ in any space dimension. We uniquely determine the unknown bounded potential $Q$ from infinitely many exterior Dirichlet-to-Neumann type measurements. This relies on Runge approximation and the dual global weak unique continuation properties of the equation under consideration. In discussing weak unique continuation of our operator, a main feature of our argument relies on a Carleman estimate for the associated fractional parabolic Caffarelli-Silvestre extension. Furthermore, we also discuss constructive single measurement results based on the approximation and unique continuation properties of the equation.

math.AP↗

arXiv2019

Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations

We study various partial data inverse boundary value problems for the semilinear elliptic equation $Δu+ a(x,u)=0$ in a domain in $\mathbb R^n$ by using the higher order linearization technique introduced in [LLS 19, FO19]. We show that the Dirichlet-to-Neumann map of the above equation determines the Taylor series of $a(x,z)$ at $z=0$ under general assumptions on $a(x,z)$. The determination of the Taylor series can be done in parallel with the detection of an unknown cavity inside the domain or an unknown part of the boundary of the domain. The method relies on the solution of the linearized partial data Calderón problem [FKSU09], and implies the solution of partial data problems for certain semilinear equations $Δu+ a(x,u) = 0$ also proved in [KU19]. The results that we prove are in contrast to the analogous inverse problems for the linear Schrödinger equation. There recovering an unknown cavity (or part of the boundary) and the potential simultaneously are long-standing open problems, and the solution to the Calderón problem with partial data is known only in special cases when $n \geq 3$.

math.AP↗

arXiv2019

Reconstruction of unknown cavity by single measurement

In this paper we propose a domain sampling type reconstruction scheme for an inverse boundary value problem to identify an unknown cavity by single measurement on the accessible boundary of a known electric or heat conductive medium. Here the single measurement is to give single current or heat flux which can have a small support over the boundary, and we measure the corresponding voltage or temperature over the whole boundary. For this inverse boundary value problem, we adapted the single NRT (no response test) introduced by Luke and Potthast for inverse scattering problem and show that it can provide such a domain sampling type reconstruction scheme.

math.AP↗

arXiv2019

Inverse problems for elliptic equations with power type nonlinearities

We introduce a method for solving Calderón type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. Assuming the knowledge of a nonlinear Dirichlet-to-Neumann map, we determine both a potential and a conformal manifold simultaneously in dimension $2$, and a potential on transversally anisotropic manifolds in dimensions $n \geq 3$. In the Euclidean case, we show that one can solve the Calderón problem for certain semilinear equations in a surprisingly simple way without using complex geometrical optics solutions.

math.AP↗

arXiv2019

Boundary determination of electromagnetic and Lamé parameters with corrupted data

We study boundary determination for an inverse problem associated to the time-harmonic Maxwell equations and another associated to the isotropic elasticity system. We identify the electromagnetic parameters and the Lamé moduli for these two systems from the corresponding boundary measurements. In a first step we reconstruct Lipschitz magnetic permeability, electric permittivity and conductivity on the surface from the ideal boundary measurements. Then, we study inverse problems for Maxwell equations and the isotropic elasticity system assuming that the data contains measurement errors. For both systems, we provide explicit formulas to reconstruct the parameters on the boundary as well as its rate of convergence formula.

math.AP↗

arXiv2018

The Calderón problem for the fractional Schrödinger equation with drift

We investigate the Calderón problem for the fractional Schrödinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior measurements. In particular, in contrast to its local analogue, this nonlocal problem does \emph{not} enjoy a gauge invariance. The uniqueness result is complemented by an associated logarithmic stability estimate under suitable apriori assumptions. Also uniqueness under finitely many \emph{generic} measurements is discussed. Here the genericity is obtained through \emph{singularity theory} which might also be interesting in the context of hybrid inverse problems. Combined with the results from \cite{GRSU18}, this yields a finite measurements constructive reconstruction algorithm for the fractional Calderón problem with drift. The inverse problem is formulated as a partial data type nonlocal problem and it is considered in any dimension $n\geq 1$.

math.AP↗

arXiv2018

Coarse-Grained Residue-Based Models of Disordered Protein Condensates: Utility and Limitations of Simple Charge Pattern Parameters

Biomolecular condensates undergirded by phase separations of proteins and nucleic acids serve crucial biological functions. To gain physical insights into their genetic basis, we study how liquid-liquid phase separation (LLPS) of intrinsically disordered proteins (IDPs) depends on their sequence charge patterns using a continuum Langevin chain model wherein each amino acid residue is represented by a single bead. Charge patterns are characterized by the `blockiness' measure $κ$ and the `sequence charge decoration' (SCD) parameter. Consistent with random phase approximation (RPA) theory and lattice simulations, LLPS propensity as characterized by critical temperature $T^*_{\rm cr}$ increases with increasingly negative SCD for a set of sequences showing a positive correlation between $κ$ and $-$SCD. Relative to RPA, the simulated sequence-dependent variation in $T^*_{\rm cr}$ is often---though not always---smaller, whereas the simulated critical volume fractions are higher. However, for a set of sequences exhibiting an anti-correlation between $κ$ and $-$SCD, the simulated $T^*_{\rm cr}$'s are quite insensitive to either parameters. Additionally, we find that blocky sequences that allow for strong electrostatic repulsion can lead to coexistence curves with upward concavity as stipulated by RPA, but the LLPS propensity of a strictly alternating charge sequence was likely overestimated by RPA and lattice models because interchain stabilization of this sequence requires spatial alignments that are difficult to achieve in real space. These results help delineate the utility and limitations of the charge pattern parameters and of RPA, pointing to further efforts necessary for rationalizing the newly observed subtleties.

q-bio.BM↗

arXiv2018

Radiating and non-radiating sources in elasticity

In this work, we study the inverse source problem of a fixed frequency for the Navier's equation. We investigate that nonradiating external forces. If the support of such a force has a convex or non-convex corner or edge on their boundary, the force must be vanishing there. The vanishing property at corners and edges holds also for sufficiently smooth transmission eigenfunctions in elasticity. The idea originates from the enclosure method: The energy identity and new type exponential solutions for the Navier's equation.

math.AP↗

arXiv2018

Quaternionic loci in Siegel's modular threefold

Let $\mathcal Q_D$ be the set of moduli points on Siegel's modular threefold whose corresponding principally polarized abelian surfaces have quaternionic multiplication by a maximal order $\mathcal O$ in an indefinite quaternion algebra of discriminant $D$ over $\mathbb Q$ such that the Rosati involution coincides with a positive involution of the form $α\mapstoμ^{-1}\overlineαμ$ on $\mathcal O$ for some $μ\in\mathcal O$ with $μ^2+D=0$. In this paper, we first give a formula for the number of irreducible components in $\mathcal Q_D$, strengthening an earlier result of Rotger. Then for each irreducible component of genus $0$, we determine its rational parameterization in terms of a Hauptmodul of the associated Shimura curve.

math.NT↗

arXiv2018

On localizing and concentrating electromagnetic fields

We consider field localizing and concentration of electromagnetic waves governed by the time-harmonic anisotropic Maxwell system in a bounded domain. It is shown that there always exist certain boundary inputs which can generate electromagnetic fields with energy localized/concentrated in a given subdomain while nearly vanishing in another given subdomain. The theoretical results may have potential applications in telecommunication, inductive charging and medical therapy. We also derive a related Runge approximation result for the time-harmonic anisotropic Maxwell system with partial boundary data.

math.AP↗

arXiv2018

A Lattice Model of Charge-Pattern-Dependent Polyampholyte Phase Separation

In view of recent intense experimental and theoretical interests in the biophysics of liquid-liquid phase separation (LLPS) of intrinsically disordered proteins (IDPs), heteropolymer models with chain molecules configured as self-avoiding walks on the simple cubic lattice are constructed to study how phase behaviors depend on the sequence of monomers along the chains. To address pertinent general principles, we focus primarily on two fully charged 50-monomer sequences with significantly different charge patterns. Each monomer in our models occupies a single lattice site and all monomers interact via a screened pairwise Coulomb potential. Phase diagrams are obtained by extensive Monte Carlo sampling performed at multiple temperatures on ensembles of 300 chains in boxes of sizes ranging from $52\times 52\times 52$ to $246\times 246\times 246$ to simulate a large number of different systems with the overall polymer volume fraction $ϕ$ in each system varying from $0.001$ to $0.1$. Phase separation in the model systems is characterized by the emergence of a large cluster connected by inter-monomer nearest-neighbor lattice contacts and by large fluctuations in local polymer density. The simulated critical temperatures, $T_{\rm cr}$, of phase separation for the two sequences differ significantly, whereby the sequence with a more "blocky" charge pattern exhibits a substantially higher propensity to phase separate. The trend is consistent with our sequence-specific random-phase-approximation (RPA) polymer theory, but the variation of the simulated $T_{\rm cr}$ with a previously proposed "sequence charge decoration" pattern parameter is milder than that predicted by RPA. Ramifications of our findings for the development of analytical theory and simulation protocols of IDP LLPS are discussed.

q-bio.BM↗

arXiv2018

Leading and second order homogenization of an elastic scattering problem for highly oscillating anisotropic medium

We consider the scattering of elastic waves by highly oscillating anisotropic periodic media with bounded support. Applying the two-scale homogenization, we first obtain a constant coefficient second-order partial differential elliptic equation that describes the wave propagation of the effective or overall wave field. We study the rate of convergence by introducing complimentary boundary correctors. To account for dispersion induced by the periodic structure, we further pursue a higher-order homogenization. We then investigate the rate of convergence and formally obtain a fourth-order differential equation that demonstrates the anisotropic dispersion

math.AP↗

arXiv2017

Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators

Let $A\in\mathrm{Sym}(n\times n)$ be an elliptic 2-tensor. Consider the anisotropic fractional Schrödinger operator $\mathscr{L}_A^s+q$, where $\mathscr{L}_A^s:=(-\nabla\cdot(A(x)\nabla))^s$, $s\in (0, 1)$ and $q\in L^\infty$. We are concerned with the simultaneous recovery of $q$ and possibly embedded soft or hard obstacles inside $q$ by the exterior Dirichlet-to-Neumann (DtN) map outside a bounded domain $Ω$ associated with $\mathscr{L}_A^s+q$. It is shown that a single measurement can uniquely determine the embedded obstacle, independent of the surrounding potential $q$. If multiple measurements are allowed, then the surrounding potential $q$ can also be uniquely recovered. These are surprising findings since in the local case, namely $s=1$, both the obstacle recovery by a single measurement and the simultaneous recovery of the surrounding potential by multiple measurements are longstanding problems and still remain open in the literature. Our argument for the nonlocal inverse problem is mainly based on the strong uniqueness property and Runge approximation property for anisotropic fractional Schrödinger operators.

math.AP↗

arXiv2017

Global uniqueness for the semilinear fractional Schrödinger equation

We study global uniqueness in an inverse problem for the fractional semilinear Schrödinger equation $(-Δ)^{s}u+q(x,u)=0$ with $s\in (0,1)$. We show that an unknown function $q(x,u)$ can be uniquely determined by the Cauchy data set. In particular, this result holds for any space dimension greater than or equal to $2$. Moreover, we demonstrate the comparison principle and provide a $L^\infty$ estimate for this nonlocal equation under appropriate regularity assumptions.

math.AP↗

arXiv2017

Charge Pattern Matching as a "Fuzzy" Mode of Molecular Recognition for the Functional Phase Separations of Intrinsically Disordered Proteins

Biologically functional liquid-liquid phase separation of intrinsically disordered proteins (IDPs) is driven by interactions encoded by their amino acid sequences. Little is currently known about the molecular recognition mechanisms for distributing different IDP sequences into various cellular membraneless compartments. Pertinent physics was addressed recently by applying random-phase-approximation (RPA) polymer theory to electrostatics, which is a major energetic component governing IDP phase properties. RPA accounts for charge patterns and thus has advantages over Flory-Huggins and Overbeek-Voorn mean-field theories. To make progress toward deciphering the phase behaviors of multiple IDP sequences, the RPA formulation for one IDP species plus solvent is hereby extended to treat polyampholyte solutions containing two IDP species. The new formulation generally allows for binary coexistence of two phases, each containing a different set of volume fractions $(ϕ_1,ϕ_2)$ for the two different IDP sequences. The asymmetry between the two predicted coexisting phases with regard to their $ϕ_1/ϕ_2$ ratios for the two sequences increases with increasing mismatch between their charge patterns. This finding points to a multivalent, stochastic, "fuzzy" mode of molecular recognition that helps populate various IDP sequences differentially into separate phase compartments. An intuitive illustration of this trend is provided by Flory-Huggins models, whereby a hypothetical case of ternary coexistence is also explored. Augmentations of the present RPA theory with a relative permittivity $ε_{\rm r}(ϕ)$ that depends on IDP volume fraction $ϕ=ϕ_1+ϕ_2$ lead to higher propensities to phase separate, in line with the case with one IDP species we studied previously. ...

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arXiv2017

The Calderón problem for variable coefficients nonlocal elliptic operators

In this paper, we introduce an inverse problem of a Schrödinger type variable nonlocal elliptic operator $(-\nabla\cdot(A(x)\nabla))^{s}+q)$, for $0<s<1$. We determine the unknown bounded potential $q$ from the exterior partial measurements associated with the nonlocal Dirichlet-to-Neumann map for any dimension $n\geq2$. Our results generalize the recent initiative [16] of introducing and solving inverse problem for fractional Schrödinger operator $((-Δ)^{s}+q)$ for $0<s<1$. We also prove some regularity results of the direct problem corresponding to the variable coefficients fractional differential operator and the associated degenerate elliptic operator.

math.AP↗