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Visiting Faculty
 Bae, Myoungjean
 Affiliate Professor
 Partial Differential equations, Calculus of Variations, Mathematical fluid dynamics
 Office
 Tel / Fax 3786



RADIAL TRANSONIC SHOCK SOLUTIONS OF EULERPOISSON SYSTEM IN CONVERGENT NOZZLES
NUMBER  null 
AUTHOR  Bae, Myoungjean 
TITLE  RADIAL TRANSONIC SHOCK SOLUTIONS OF EULERPOISSON SYSTEM IN CONVERGENT NOZZLES 
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JOURNAL  DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMSSERIES S, 2018 
ABSTRACT  Given constant data, of density rho(0), velocity u(0)e(r), pressure rho(0) and electric force E(0)e(r) for supersonic flow at the entrance, and constant pressure p(ex) for subsonic flow at the exit, we prove that EulerPoisson system admits a unique transonic shock solution in a two dimensional convergent nozzle, provided that u(0) > 0, E0 > 0, and that E0 is sufficiently large depending on (p(0), u(0),p(0)) and the length of the nozzle. 

3D axisymmetric subsonic flows with nonzero swirl for the compressible EulerPoisson system
NUMBER  null 
AUTHOR  Bae, Myoungjean 
TITLE  3D axisymmetric subsonic flows with nonzero swirl for the compressible EulerPoisson system 
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JOURNAL  ANNALES DE L INSTITUT HENRI POINCAREANALYSE NON LINEAIRE, 2018 
ABSTRACT  We address the structural stability of 3D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler Poisson system in a cylinder supplemented with nonsmall boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3D axisymmetric flow with a nonzero swirl (= angular momentum density) component. With the newly introduced decomposition, a quasilinear elliptic system of second order is derived from the elliptic modes in Euler Poisson system for subsonic flows. Due to the nonzero swirl, the main difficulties lie in the solvability of a singular elliptic equation which concerns the angular component of the voracity in its cylindrical representation, and in analysis of streamlines near the axis r = 0. (C) 2017 Elsevier Masson SAS. All rights reserved. 
 Cha, Jae Choon
 Affiliate Professor
 Geometric Topology
 Office 1410
 Tel 3848 / Fax 3786

 Ha, SeungYeal
 Affiliate Professor
 Analysis, Hyperbolic Conservation Laws, Kinetic Theory
 Office 1410
 Tel 3848 / Fax 3786



A local sensitivity analysis for the kinetic CuckerSmale equation with random inputs
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  A local sensitivity analysis for the kinetic CuckerSmale equation with random inputs 
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JOURNAL  JOURNAL OF DIFFERENTIAL EQUATIONS, 2018 
ABSTRACT  We present a local sensitivity analysis for the kinetic CuckerSmale (CS) equation with random inputs. This is a companion work to our previous local sensitivity analysis for the particle CS model. Random inputs in the coefficients of the kinetic CS equation can be caused by diverse sources such as the incomplete measurement and interactions with unknown environments, and will enter the problem through the communication function or initial data. For the proposed random kinetic CS equation, we present sufficient conditions for the pathwise wellposedness and flocking estimates. For the local sensitivity analysis, we study the propagation of regularity of the kinetic density function in random space. (C) 2018 Elsevier Inc. All rights reserved. 

UNIFORM STABILITY OF THE CUCKERSMALE MODEL AND ITS APPLICATION TO THE MEANFIELD LIMIT
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  UNIFORM STABILITY OF THE CUCKERSMALE MODEL AND ITS APPLICATION TO THE MEANFIELD LIMIT 
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JOURNAL  KINETIC AND RELATED MODELS, 2018 
ABSTRACT  

Uniformintime transition from discrete dynamics to continuous dynamics in the CuckerSmale flocking
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  Uniformintime transition from discrete dynamics to continuous dynamics in the CuckerSmale flocking 
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JOURNAL  MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES, 2018 
ABSTRACT  We study a uniformintime convergence from the discretetime (in short, discrete) CuckerSmale (CS) model to the continuoustime CS model, which is valid for the whole time interval, as timestep tends to zero. Classical theory yields the convergence results which are valid only in any finitetime interval. Our uniform convergence estimate relies on two quantitative estimates asymptotic flocking estimate and uniform l(2)stability estimate with respect to initial data. In the previous literature, most studies on the CS flocking have been devoted to the continuoustime model with general communication weights, whereas flocking estimates have been done for the discretetime model with special network topologies such as the complete network with algebraically decaying communication weights and rooted leaderships. For the discrete CS model with a regular and algebraically decaying communication weight, asymptotic flocking estimate has been extensively studied in the previous literature. In contrast, for a general decaying communication weight, corresponding flocking dynamics has not been addressed in the literature due to the difficulty of extending the Lyapunov functional approach to the discrete model. In this paper, we present asymptotic flocking estimate for the discrete model using the Lyapunov functional approach. Moreover, we present a uniform l(2)stability estimate of the solution for the discrete CS model with respect to initial data. We combine asymptotic flocking estimate and uniform stability to derive a uniformintime convergence from the discrete CS model to the continuous CS model, as timestep tends to zero. 

LOCAL SENSITIVITY ANALYSIS FOR THE CUCKERSMALE MODEL WITH RANDOM INPUTS
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  LOCAL SENSITIVITY ANALYSIS FOR THE CUCKERSMALE MODEL WITH RANDOM INPUTS 
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JOURNAL  KINETIC AND RELATED MODELS, 2018 
ABSTRACT  We present pathwise flocking dynamics and local sensitivity analysis for the CuckerSmale(CS) model with random communications and initial data. For the deterministic communications, it is well known that the CS model can model emergent local and global flocking dynamics depending on initial data and integrability of communication function. However, the communication mechanism between agents is not a priori clear and needs to be figured out from observed phenomena and data. Thus, uncertainty in communication is an intrinsic component in the flocking modeling of the CS model. In this paper, we provide a class of admissible random uncertainties which allows us to perform the local sensitivity analysis for flocking and establish stability to the random CS model with uncertain communication. 

Remarks on the stability properties of the KuramotoSakaguchiFokkerPlanck equation with frustration
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  Remarks on the stability properties of the KuramotoSakaguchiFokkerPlanck equation with frustration 
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JOURNAL  ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK, 2018 
ABSTRACT  We present stability properties of the KuramotoSakaguchiFokkerPlanck (in short, KSFP) equation with frustration arising from the synchronization modeling of a large ensemble of weakly coupled oscillators under the effect of uniform frustration (phaselag). Even in the absence of FokkerPlanck term (i.e., the diffusion term), the existence of frustration destroys a gradient flow structure of the original kinetic Kuramoto equation. Thus, useful machineries from the gradient flow theory cannot be used in the largetime behavior of the equation as it is. In this paper, we address two stability estimates for the KSFP equation. First, we present a sufficient framework leading to the asymptotic stability of the incoherent state. Our stability framework has been formulated in terms of the probability density function of natural frequencies, coupling strength, diffusion coefficient, size of frustration and initial data. Second, we show that the KSFP equation is structurally stable with respect to the size of frustration. More precisely, we show that the solution to the KSFP equation tends to the corresponding solution to the KSFP with zero frustration, as the size of frustration tends to zero under a suitable framework. 

Interplay of the unitspeed constraint and timedelay in CuckerSmale flocking
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  Interplay of the unitspeed constraint and timedelay in CuckerSmale flocking 
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JOURNAL  JOURNAL OF MATHEMATICAL PHYSICS, 2018 
ABSTRACT  We study the emergence of the monocluster flocking due to the interplay between the unitspeed constraint and timedelayed interactions in the evolution of the CuckerSmale ensemble. Several flocking models with unitspeed constraint have been extensively used in the flocking modeling of selfpropelled multiagent systems in the control theory community. Timedelayed interactions can be caused by the finite propagation speed constraint in communications. In the previous literature, these two physical mechanisms have been studied separately. In this paper, we investigate these combined physical effects in a common framework and study how the interplay between these mechanisms affects asymptotic flocking dynamics. For this, we provide a sufficient framework for a monocluster flocking in terms of system parameters (e.g., timedelay, coupling strength, particle numbers) and initial data. We also provide several numerical simulations and compare them with analytical results. Published by AIP Publishing. 

EMERGENT BEHAVIORS OF THERMODYNAMIC CUCKERSMALE PARTICLES
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  EMERGENT BEHAVIORS OF THERMODYNAMIC CUCKERSMALE PARTICLES 
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JOURNAL  SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 2018 
ABSTRACT  We study the flocking dynamics of a generalized CuckerSmale (CS) model for thermodynamic particles. The original CS model deals with flocking dynamics of mechanical particles, in which the position and momentum are only kinematic observables. Thus, the CS model cannot describe a phenomenon caused by different internal energy (temperatures) between particles. Recently, a new particle model for flocking was proposed to model a flocking dynamics in an ensemble of thermodynamic particles on timeindependent communication networks, where spatial dependence in the communication was neglected. In this paper, we extend a thermodynamically consistent particle to incorporate communications depending on the positions of the particle and study its asymptotic dynamics, such as formation and nonexistence of monocluster flockings in terms of an initial configuration, coupling strength, and ansatz of the communication weights. 

Remarks on the complete synchronization for the Kuramoto model with frustrations
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  Remarks on the complete synchronization for the Kuramoto model with frustrations 
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JOURNAL  ANALYSIS AND APPLICATIONS, 2018 
ABSTRACT  The synchronous dynamics of many limitcycle oscillators can be described by phase models. The Kuramoto model serves as a prototype model for phase synchronization and has been extensively studied in the last 40 years. In this paper, we deal with the complete synchronization problem of the Kuramoto model with frustrations on a complete graph. We study the robustness of complete synchronization with respect to the network structure and the interaction frustrations, and provide sufficient frameworks leading to the complete synchronization, in which all frequency differences of oscillators tend to zero asymptotically. For a uniform frustration and unit capacity, we extend the applicable range of initial configurations for the complete synchronization to be distributed on larger arcs than a half circle by analyzing the detailed dynamics of the order parameters. This improves the earlier results [S.Y. Ha. H. Kim and J. Park, Remarks on the complete frequency synchronization of Kuramoto oscillators. Nonlinearity 28 (2015) 14411462: Z. Li and S.Y. Ha, Uniqueness and wellordering of emergent phaselocked states for the Kuramoto model with frustration and inertia, Math. Models Methods Appl. Sci. 26 (2016) 357382.] which can be applicable only for initial configurations confined in a half circle. 

REMARKS ON THE CRITICAL COUPLING STRENGTH FOR THE CUCKERSMALE MODEL WITH UNIT SPEED
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  REMARKS ON THE CRITICAL COUPLING STRENGTH FOR THE CUCKERSMALE MODEL WITH UNIT SPEED 
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JOURNAL  DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 2018 
ABSTRACT  We present a nontrivial lower bound for the critical coupling strength to the CuckerSmale model with unit speed constraint and shortrange communication weight from the viewpoint of a monocluster(global) flocking. For a longrange communication weight, the critical coupling strength is zero in the sense that the monocluster flocking emerges from arty initial configurations for any positive coupling strengths, whereas for a shortrange communication weight, a monocluster flocking can emerge from an initial configuration only for a sufficiently large coupling strength. Our main interest lies on the condition of nonflocking. We provide a positive lower bound for the critical coupling strength. We also present numerical simulations for the upper and lower bounds for the critical coupling strength depending on initial configurations and compare them with analytical results. 

Emergence of PhaseLocking in the Kuramoto Model for Identical Oscillators with Frustration
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  Emergence of PhaseLocking in the Kuramoto Model for Identical Oscillators with Frustration 
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JOURNAL  SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS, 2018 
ABSTRACT  We present an exponential synchronization for the finitedimensional Kuramoto model for identical oscillators under the effect of frustration. In the presence of frustration, the total phase balance law and gradientflow structure are destroyed. Hence the gradient flow approach cannot be applied for the emergence of phaselocking, and the Lyapunov functional approach based on l(2)energy and phase diameter only show some results in a restricted set of initial configurations smaller than a half circle. In this paper, we show that an initial phase configuration whose order parameter is bounded below by some positive value, which is only depending on the size of frustration, leads to the complete phase synchronization or the bipolar state exponentially fast. This extends earlier results on the asymptotic phaselocking for the identical oscillators in which initial phase configurations are confined in a half circle. 

Remarks on the slow relaxation for the fractional Kuramoto model for synchronization
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  Remarks on the slow relaxation for the fractional Kuramoto model for synchronization 
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JOURNAL  JOURNAL OF MATHEMATICAL PHYSICS, 2018 
ABSTRACT  The collective behavior of an oscillatory system is ubiquitous in our nature, and one interesting issue in the dynamics of manybody oscillatory systems is the relaxation dynamics toward relative equilibria such as phaselocked states. For the Kuramoto model, relaxation dynamics occurs exponentially fast for generic initial data. However, some synchronization phenomena observed in our nature exhibit a slow subexponential relaxation. Thus, as one of the possible attempts for such slow relaxation, a secondorder inertia term was added to the Kuramoto model in the previous literature so that the resulting secondorder model can exhibit a slow relaxation dynamics for some range of inertia and coupling strength. In this paper, wepresent another Kuramoto type model exhibiting a slowalgebraic relaxation. More precisely, our proposed model replaces the classical derivative by the Caputo fractional derivative in the original Kuramoto model. For this new model, we present several sufficient frameworks for fractional complete synchronization and practical synchronization. Published by AIP Publishing. 

On the global solvability of the coupled kineticfluid system for flocking with large initial data
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  On the global solvability of the coupled kineticfluid system for flocking with large initial data 
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JOURNAL  MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES, 2018 
ABSTRACT  We study the dynamics of infinitely many CuckerSmale (CS) flocking particles under the interplay of random communication and compressible fluids in planar wave case. For the dynamics of an ensemble of flocking particles, we use the kinetic CuckerSmaleFokkerPlanck (CSFP) equation with a degenerate diffusion, whereasfor the fluid component, we use the compressible NavierStokes (NS) equations. These two subsystems are coupled via the drag force. For this coupled model, we present a global existence of classical solutions for arbitrarily large initial data which may contain vacuum. 

Emergent behaviors of the SchrodingerLohe model on cooperativecompetitive networks
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  Emergent behaviors of the SchrodingerLohe model on cooperativecompetitive networks 
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JOURNAL  JOURNAL OF DIFFERENTIAL EQUATIONS, 2017 
ABSTRACT  We present several sufficient frameworks leading to the emergent behaviors of the coupled Schrtidinger Lohe (S L) model under the same onebody external potential on cooperativecompetitive networks. The S L model was first introduced as a possible phenomenological model exhibiting quantum synchronization and its emergent dynamics on alltoall cooperative networks has been treated via two distinct approaches, Lyapunov functional approach and the finitedimensional reduction based on pairwise correlations. In this paper, we further generalize the finitedimensional dynamical systems approach for pairwise correlation functions on cooperativecompetitive networks and provide several sufficient frameworks leading to the collective exponential synchronization. For small systems consisting of three and four quantum subsystem, we also show that the system for pairwise correlations can be reduced to the Lotka Volterra model with cooperative and competitive interactions, in which lots of interesting dynamical patterns appear, e.g., existence of closed orbits and limitcycles. (C) 2017 Elsevier Inc. All rights reserved. 

THE WIGNERLOHE MODEL FOR QUANTUM SYNCHRONIZATION AND ITS EMERGENT DYNAMICS
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  THE WIGNERLOHE MODEL FOR QUANTUM SYNCHRONIZATION AND ITS EMERGENT DYNAMICS 
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JOURNAL  NETWORKS AND HETEROGENEOUS MEDIA, 2017 
ABSTRACT  We present the WignerLohe model for quantum synchronization which can be derived from the SchrodingerLohe model using the Wigner formalism. For identical onebody potentials, we provide a priori sufficient framework leading the complete synchronization, in which L2distances between all wave functions tend to zero asymptotically. 

DYNAMICAL SYSTEM APPROACH TO SYNCHRONIZATION OF THE COUPLED SCHRODINGERLOHE SYSTEM
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  DYNAMICAL SYSTEM APPROACH TO SYNCHRONIZATION OF THE COUPLED SCHRODINGERLOHE SYSTEM 
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JOURNAL  QUARTERLY OF APPLIED MATHEMATICS, 2017 
ABSTRACT  We study wave function synchronization of the SchrodingerLohe model, which describes the dynamics of the ensemble of coupled quantum Lohe oscillators with infinite states. To do this, we first derive a coupled system of ordinary differential equations for the Lx(2) inner products between distinct wave functions. For the same onebody potentials, we show that the inner products of two wave functions converge to unity for some restricted class of initial data, so complete wave function synchronization emerges asymptotically when the dynamical system approach is used. Moreover, for the family of onebody potentials consisting of realvalue translations of the same base potential, we show that the inner products for a twooscillator system follow the motion of harmonic oscillators in a small coupling regime, and then as the coupling strength increases, the inner products converge to constant values; this behavior yields convergence toward constant values for the Lx(2) differences between distinct wave functions. 

Emergent Dynamics of a Generalized Lohe Model on Some Class of Lie Groups
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  Emergent Dynamics of a Generalized Lohe Model on Some Class of Lie Groups 
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JOURNAL  JOURNAL OF STATISTICAL PHYSICS, 2017 
ABSTRACT  We introduce a Lohe group which is a new class of matrix Lie groups and present a continuous dynamical system for the synchronization of group elements in a Lohe group. The Lohe group includes classical Lie groups such as the orthogonal, unitary, and symplectic groups, and since Lohe groups need not be compact, global existence of ODEs may fail. The proposed dynamical system generalizes the Lohe model (Lohe in J Phys A 43:465301, 2010; Lohe in J Phys A 42:395101395126, 2009) itself a nonabelian generalization of the Kuramoto model, and alongside we also generalize the analytical framework (Ha and Ryoo in J Stat Phys 163:411439, 2016) of emergent and unique phaselocked states. For the construction of the phaselocked states, we introduce Lyapunov functions measuring the ensemble diameter and the dissimilarity between two Lohe flows, and derive Gronwalltype differential inequalities for them. The global existence of solutions then become a consequence of the boundedness of these Lyapunov functions. Our sufficient framework for the emergent dynamics is formulated in terms of coupling strength and initial states, and it leads to the global existence of solutions and the formation and uniqueness of a phaselocked asymptotic state. As a concrete example, we demonstrate how our theory can show emergent phenomenon on the Heisenberg group, where all initial configurations tend to a unique phaselocked state exponentially fast. 

A quest toward a mathematical theory of the dynamics of swarms
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  A quest toward a mathematical theory of the dynamics of swarms 
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JOURNAL  MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES, 2017 
ABSTRACT  This paper addresses some preliminary steps toward the modeling and qualitative analysis of swarms viewed as living complex systems. The approach is based on the methods of kinetic theory and statistical mechanics, where interactions at the microscopic scale are nonlocal, nonlinearly additive and modeled by theoretical tools of stochastic game theory. Collective learning theory can play an important role in the modeling approach. We present a kinetic equation incorporating the CuckerSmale flocking force and stochastic game theoretic interactions in collision operators. We also present a sufficient framework leading to the asymptotic velocity alignment and global existence of smooth solutions for the proposed kinetic model with a special kernel. Analytic results on the global existence and flocking dynamics are presented, while the last part of the paper looks ahead to research perspectives. 

Emergent Dynamics of a Thermodynamically Consistent Particle Model
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  Emergent Dynamics of a Thermodynamically Consistent Particle Model 
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JOURNAL  ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 2017 
ABSTRACT  We present a thermodynamically consistent particle (TCP) model motivated by the theory of multitemperature mixture of fluids in the case of spatially homogeneous processes. The proposed model incorporates the CuckerSmale (CS) type flocking model as its isothermal approximation. However, it is more complex than the CS model, because the mutual interactions are not only mechanical but are also affected by the temperature effect as individual particles may exhibit distinct internal energies. We develop a framework for asymptotic weak and strong flocking in the context of the proposed model. 

Emergent dynamics of CuckerSmale flocking particles in a random environment
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  Emergent dynamics of CuckerSmale flocking particles in a random environment 
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JOURNAL  JOURNAL OF DIFFERENTIAL EQUATIONS, 2017 
ABSTRACT  We present a new kinetic CuckerSmaleFokkerPlanck (CSFP) type equation with a degenerate diffusion, which describes the dynamics for an ensemble of infinitely many CuckerSmale particles in a random environment. The asymptotic dynamics of the CSFP equation exhibits a thresholdlike phenomenon depending on the relative strength between the coupling strength and the noise strength. In the small coupling regime, the noise effect becomes dominant, which induces the velocity variance to increase to infinity exponentially fast. In contrast, the velocity alignment effect is strong in the large coupling regime, and the velocity variance tends to zero exponentially fast. We present the global existence of classical solutions to the CSFP equation for a sufficiently smooth initial datum without smallness in its size. For the kinetic CSFP equation with a metric dependent communication weight, we provide a uniformintime meanfield limit from the stochastic CSmodel to the kinetic CSFP equation without convergence rate. (C) 2016 Elsevier Inc. All rights reserved. 

On the global wellposedness of BV weak solutions to the KuramotoSakaguchi equation
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  On the global wellposedness of BV weak solutions to the KuramotoSakaguchi equation 
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JOURNAL  JOURNAL OF DIFFERENTIAL EQUATIONS, 2017 
ABSTRACT  The Kuramoto model is a prototype phase model describing the synchronous behavior of weakly coupled limitcycle oscillators. When the number of oscillators is sufficiently large, the dynamics of Kuramoto ensemble can be effectively approximated by the corresponding meanfield equation, namely the KuramotoSakaguchi (KS) equation. This KS equation is a kind of scalar conservation law with a nonlocal flux function due to the meanfield interactions among oscillators. In this paper, we provide a unique global solvability of bounded variation (BV) weak solutions to the kinetic KS equation for identical oscillators using the method of fronttracking in hyperbolic conservation laws. Moreover, we also show that our BV weak solutions satisfy localintime L1stability with respect to BVinitial data. For the ensemble of identical Kuramoto oscillators, we explicitly construct an exponentially growing BV weak solution generated from BV perturbation of incoherent state for any positive coupling strength. This implies the nonlinear instability of incoherent state in a positive coupling strength regime. We provide several numerical examples and compare them with our analytical results. (C) 2016 Elsevier Inc. All rights reserved. 

On the Emergence and Orbital Stability of PhaseLocked States for the Lohe Model
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  On the Emergence and Orbital Stability of PhaseLocked States for the Lohe Model 
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JOURNAL  JOURNAL OF STATISTICAL PHYSICS, 2016 
ABSTRACT  We study the emergence and orbital stability of phaselocked states of the Lohe model, which was proposed as a nonabelian generalization of the Kuramoto phase model for synchronization. Lohe introduced a firstorder system of matrixvalued ordinary differential equations for quantum synchronization and numerically observed the asymptotic formation and orbital stability of phaselocked states of the Lohe model. In this paper, we provide an analytical framework to confirm Lohes observations of emergent phaselocked states. This extends earlier special results on lower dimensions to any finite dimension. For the construction and orbital stability of phaselocked states, we introduce Lyapunov functions to measure the ensemble diameter and dissimilarity between two Lohe flows, and using the timeevolution estimates of these Lyapunov functions, we present an admissible set of initial states, and show that an admissible initial state leads to a unique phaselocked asymptotic state. 

Practical quantum synchronization for the SchrodingerLohe system
NUMBER  null 
AUTHOR  Ha, SeungYeal 
TITLE  Practical quantum synchronization for the SchrodingerLohe system 
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JOURNAL  JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL, 2016 
ABSTRACT  We present a practical synchronization method for the SchrodingerLohe (SL) system distinct potentials. The SL model describes the spatialtemporal evolution of the wave functions of quantum Lohe oscillators on a network with Lohe couplings. When the potential effects are ignored, complete wave function synchronization (CWFS) can emerge in the sense that the L2distance between wave functions exponentially approaches zero for a class of initial wave functions. In contrast, when the Lohe oscillators are under the effect of potential forces, CWFS cannot occur. In this study, we employ a weaker concept of quantum synchronization for discussing the asymptotic collective behavior of the SL model. This concept leads to practical synchronization of the SL model. In practical synchronization, the L2distance between wave functions can be upper bounded by the inverse power of the square root of the coupling strength; as the coupling strength increases. Thus, the L2discrepancy between wave functions arbitrarily decreases as the coupling strength increases. We present a sufficient analytical framework for this practical synchronization, which is a generalization of the earlier result in (Choi SH and Ha SY 2014 J. Phys. A: Math. Theor. 47 355104) 
 Hong, SeokCheol
 Affiliate Professor
 Theoretical and Computational Biophysics
 Office 8214
 Tel 2660 / Fax 3820



Decoding Single Molecule Time Traces with Dynamic Disorder
NUMBER  C17009 
AUTHOR  Hwang, Wonseok,Hong, SeokCheol,Hyeon, Changbong 
TITLE  Decoding Single Molecule Time Traces with Dynamic Disorder 
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JOURNAL  PLOS COMPUTATIONAL BIOLOGY, 2016 
ABSTRACT  Single molecule time trajectories of biomolecules provide glimpses into complex folding landscapes that are difficult to visualize using conventional ensemble measurements. Recent experiments and theoretical analyses have highlighted dynamic disorder in certain classes of biomolecules, whose dynamic pattern of conformational transitions is affected by slower transition dynamics of internal state hidden in a low dimensional projection. A systematic means to analyze such data is, however, currently not well developed. Here we report a new algorithm D Variational Bayesdouble chain Markov model (VBDCMM) D to analyze single molecule time trajectories that display dynamic disorder. The proposed analysis employing VBDCMM allows us to detect the presence of dynamic disorder, if any, in each trajectory, identify the number of internal states, and estimate transition rates between the internal states as well as the rates of conformational transition within each internal state. Applying VBDCMM algorithm to single molecule FRET data of HDNA in 100 mMNa+ solution, followed by data clustering, we show that at least 6 kinetic paths linking 4 distinct internal states are required to correctly interpret the duplextriplex transitions of HDNA. 

Deciphering Kinetic Information from SingleMolecule FRET Data That Show Slow Transitions
NUMBER  C17037 
AUTHOR  Hyeon, Changbong,Hong, SeokCheol 
TITLE  Deciphering Kinetic Information from SingleMolecule FRET Data That Show Slow Transitions 
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JOURNAL  JOURNAL OF PHYSICAL CHEMISTRY B, 2015 
ABSTRACT  Singlemolecule: FRET is One Of the most powerful and widely used biophysical techniques in biological sciences. It, however often suffers from limitations such as weak signal and limited measurement time intrinsic to single Molecule fluorescence measurements. Despite several ameliorative measures taken to increase measurement time, it is nearly impossible to acquire meaningful kinetic information on a molecule if conformational transitions of the molecule are ultraslow such that transition times ((orig)) are. comparable to or longer than measurement times (delta t) limited by the finite lifetime of fluorescent dye. Here, to extract a reliable and accurate mean transition time from, a series of short time traces with ultraslow kinetics, we suggest seheme called sHaRPer (serialized Handshaking Repeated Permutation with end removal) that concatenates multiple time traces because data acquisition frequency f and measurement time (delta t) affect the estimation of mean transition time (), we provide Mathematical criteria that f, delta t, and should satisfy to make close enough to (orig) Although application of the sHaRPer: methods a potential risk of distorting the time,constants of individual kinetic phases if the data are described with kinetic partitioning, We, also provide criteria to avoid such distortion. Our sHaRPer method is a useful way to handle singlemolecule data:With,slow transition kinetics This,study:provides a practical vide to use sHaRPer. 

Destabilization of iMotif by Submolar Concentrations of a Monovalent Cation
NUMBER  C15010 
AUTHOR  Hyeon, Changbong,Hong, SeokCheol 
TITLE  Destabilization of iMotif by Submolar Concentrations of a Monovalent Cation 
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JOURNAL  JOURNAL OF PHYSICAL CHEMISTRY B, 2014 
ABSTRACT  Counterions are crucial for selfassembly of nucleic acids. Submolar monovalent cations are generally deemed to stabilize various types of base pairs in nucleic acids such as WatsonCrick and Hoogsteen base pairs via screening of electrostatic repulsion. Besides monovalent cations, acidic pH is required for imotif formation because protons facilitate pairing between cytosines. Here we report that Li+ ions destabilize imotif, whereas other monovalent cations, Na+ and K+, have the usual stabilizing effect. The thermodynamics data alone, however, cannot reveal which mechanism, enhanced unfolding or suppressed folding or both, is responsible for the Li+induced destabilization. To gain further insight, we examined the kinetics of imotif. To deal with slow kinetics of imotif, we developed a method dubbed HaRP to construct a long FRET time trace to observe a sufficient number of transitions. Our kinetics analysis shows clearly that Li+ ions promote unfolding of imotif but do not hinder its folding, lending strong support for our hypothesis on the origin of this unusual effect of Li+. Although the subangstrom size of Li+ ions allows them to infiltrate the space between cytosines in competition with protons, they cannot adequately fulfill the role of protons in mediating the hydrogen bonding of cytosine pairs. 

Direct observation of the formation of DNA triplexes by singlemolecule FRET measurements
NUMBER  null 
AUTHOR  Hong, SeokCheol 
TITLE  Direct observation of the formation of DNA triplexes by singlemolecule FRET measurements 
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JOURNAL  CURRENT APPLIED PHYSICS, 2012 
ABSTRACT  In this report we investigated the effects of various biological and chemical factors (DNA sequence, pH, ions, and molecularity) on the formation of DNA triplexes through singlemolecule FRET technique. Using this method, we determined how the third strand bound to a DNA duplex and how stable the triplex structure was under various conditions. From this study, we not only verified a variety of wellknown features of DNA triplex but also discovered or experimentally supported several interesting behaviors: at neutral pH, a pyrimidinemotif triplex can be formed; the parallel arrangement was not only possible but also dominant over the antiparallel arrangement for a purinemotif triplex. We demonstrated that our method is a versatile analytical tool in studying structural aspects of nucleic acids, particularly nonclassical DNA structures, and provides insights into physical mechanism of such structures. (C) 2012 Elsevier B. V. All rights reserved. 
 Ki, Haseo
 Affiliate Professor
 Number Theory
 Office 1404B
 Tel 3715 / Fax 3786



Pair correlation of zeros of the real and imaginary parts of the Riemann zetafunction
NUMBER  null 
AUTHOR  Ki, Haseo 
TITLE  Pair correlation of zeros of the real and imaginary parts of the Riemann zetafunction 
ARCHIVE  
FILE  
JOURNAL  JOURNAL OF NUMBER THEORY, 2018 
ABSTRACT  We show that if the Riemann Hypothesis is true for the Riemann zetafunction, zeta(s), and 0 < alpha < 1/2, then all but a finite number of the zeros of R zeta(a, it), I zeta(a + it), and similar functions are simple. We also study the pair correlation of the zeros of these functions assuming the Riemann Hypothesis is true and 0 < a <= 1/2. (C) 2017 Elsevier Inc. All rights reserved. 

Additive problems with smooth integers
NUMBER  null 
AUTHOR  Ki, H. 
TITLE  Additive problems with smooth integers 
ARCHIVE  
FILE  
JOURNAL  ACTA ARITHMETICA, 2016 
ABSTRACT  

On the zeros of Weng zeta functions for Chevalley groups
NUMBER  null 
AUTHOR  Ki, Haseo 
TITLE  On the zeros of Weng zeta functions for Chevalley groups 
ARCHIVE  
FILE  
JOURNAL  MANUSCRIPTA MATHEMATICA, 2015 
ABSTRACT  We prove that all but finitely many zeros of Wengs zeta function for a Chevalley group defined over Q are simple and on the critical line. 

A uniqueness theorem for functions in the extended Selberg class
NUMBER  null 
AUTHOR  Ki, Haseo 
TITLE  A uniqueness theorem for functions in the extended Selberg class 
ARCHIVE  
FILE  
JOURNAL  MATHEMATISCHE ZEITSCHRIFT, 2014 
ABSTRACT  We study the uniqueness of functions in the extended Selberg class. It was shown in Ki (Adv Math 231, 24842490, 2012) that if for a nonzero complex number c the inverse images L1(1) (c) and L2(1) (c) of two functions satisfying the same functional equation in the extended Selberg class are the same, then L1(s) and L2(s) are identical. Here we prove that this holds even without the assumption that they satisfy the same functional equation. 

A remark on the uniqueness of the Dirichlet series with a Riemanntype function equation
NUMBER  null 
AUTHOR  Ki, Haseo 
TITLE  A remark on the uniqueness of the Dirichlet series with a Riemanntype function equation 
ARCHIVE  
FILE  
JOURNAL  ADVANCES IN MATHEMATICS, 2012 
ABSTRACT  We show that if for a nonzero complex number c the inverse images L1(1) (c) and L2(1) (c) of two functions in the extended Selberg class are the same, then L1(s) and L2(s) must be identical. (c) 2012 Elsevier Inc. All rights reserved. 

On the zeros of degree one Lfunctions from the extended Selberg class
NUMBER  M11010 
AUTHOR  Ki, Haseo 
TITLE  On the zeros of degree one Lfunctions from the extended Selberg class 
ARCHIVE  
FILE  
JOURNAL  ACTA ARITHMETICA, 2011 
ABSTRACT  
 Kwon, Chulan
 Affiliate Professor
 Office 8410
 Tel 2594 / Fax

 Lee, KiAhm
 Affiliate Professor
 Analysis, Partial Differential Equations
 Office 1410
 Tel 3848 / Fax 3786



Higher order convergence rates in theory of homogenization III: Viscous HamiltonJacobi equations
NUMBER  null 
AUTHOR  Lee, KiAhm 
TITLE  Higher order convergence rates in theory of homogenization III: Viscous HamiltonJacobi equations 
ARCHIVE  
FILE  
JOURNAL  JOURNAL OF DIFFERENTIAL EQUATIONS, 2018 
ABSTRACT  In this paper, we establish the higher order convergence rates in periodic homogenization of viscous HamiltonJacobi equations, which is convex and grows quadratically in the gradient variable. We observe that although the nonlinear structure governs the first order approximation, the nonlinear effect is absorbed as an external source term of a linear equation in the second and higher order approximation. Moreover, we find that the geometric shape of the initial data has to be chosen carefully according to the effective Hamiltonian, in order to achieve the higher order convergence rates. (C) 2018 Elsevier Inc. All rights reserved. 

Harnack inequality and pinching estimates for anisotropic curvature flow of hypersurfaces
NUMBER  null 
AUTHOR  Lee, KiAhm 
TITLE  Harnack inequality and pinching estimates for anisotropic curvature flow of hypersurfaces 
ARCHIVE  
FILE  
JOURNAL  JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2018 
ABSTRACT  We obtain a differential Harnack inequality for anisotropic curvature flow of convex hypersurfaces in Euclidean space with its speed given by a curvature function of homogeneity degree one in a certain class, and restrictions depending only on the initial data and the anisotropic factor which reflects the influence of the ambient space. Moreover, the pinching estimate for such flows is derived from the maximum principle for tensors. (C) 2018 Elsevier Inc. All rights reserved. 

A regularity condition and temporal asymptotics for chemotaxisfluid equations
NUMBER  null 
AUTHOR  Lee, KiAhm 
TITLE  A regularity condition and temporal asymptotics for chemotaxisfluid equations 
ARCHIVE  
FILE  
JOURNAL  NONLINEARITY, 2018 
ABSTRACT  We consider two dimensional chemotaxis equations coupled to the NavierStokes equations. We present a new localized regularity criterion that is localized in a neighborhood at each point. Secondly, we establish temporal decays of the regular solutions under the assumption that the initial mass of biological cell density is sufficiently small. Both results are improvements of previously known results given in Chae et al (2013 Discrete Continuous Dyn. Syst. A 33 227197) and Chae et al (2014 Commun. PDE 39 120535) 

Dualsemiparametric regression using weighted Dirichlet process mixture
NUMBER  null 
AUTHOR  Lee, KiAhm 
TITLE  Dualsemiparametric regression using weighted Dirichlet process mixture 
ARCHIVE  
FILE  
JOURNAL  COMPUTATIONAL STATISTICS & DATA ANALYSIS, 2018 
ABSTRACT  An efficient and flexible Bayesian approach is proposed for a dualsemiparametric regression model that models mean function semiparametrically and estimates the distribution of the error term nonparametrically. Using a weighted Dirichlet process mixture (WDPM), a Bayesian approach has been developed on the assumption that the distributions of the response variables are unknown. The WDPM approach is especially useful for real applications that have heterogeneous error distributions or come from a mixture of distributions. In the mean function, the unknown functions are estimated using natural cubic smoothing splines. For the error terms, several different WDPMs are proposed using different weights that depend on the distances between the covariates. Their marginal likelihoods are derived, and the computation of marginal likelihood for WDPM is provided. Efficient Markov chain Monte Carlo (MCMC) algorithms are also provided. The Bayesian approaches based on different WDPMs are compared with the parametric error model and the Dirichlet process mixture (DPM) error model in terms of the Bayes factor using a simulation study, suggesting better performance of the Bayesian approach based on WDPM. The advantage of the proposed Bayesian approach is also demonstrated using the credit rating data. (C) 2017 Elsevier B.V. All rights reserved. 

An elliptic free boundary arising from the jump of conductivity
NUMBER  null 
AUTHOR  Lee, KiAhm 
TITLE  An elliptic free boundary arising from the jump of conductivity 
ARCHIVE  
FILE  
JOURNAL  NONLINEAR ANALYSISTHEORY METHODS & APPLICATIONS, 2017 
ABSTRACT  In this paper we consider a quasilinear elliptic PDE, div(A(x, u)del u) = 0, where the underlying physical problem gives rise to a jump for the conductivity A(x, u), across a level surface for u. Our analysis concerns Lipschitz regularity for the solution u, and the regularity of the level surfaces, where A(x,u) has a jump and the solution u does not degenerate. In proving Lipschitz regularity of solutions, we introduce a new and unexpected type of ACFmonotonicity formula with two different operators, that might be of independent interest, and surely can be applied in other related situations. The proof of the monotonicity formula is done through careful computations, and (as a byproduct) a slight generalization to a specific type of variable matrixvalued conductivity is presented. (C) 2017 Elsevier Ltd. All rights reserved. 

The EvansKrylov theorem for nonlocal parabolic fully nonlinear equations
NUMBER  null 
AUTHOR  Lee, KiAhm 
TITLE  The EvansKrylov theorem for nonlocal parabolic fully nonlinear equations 
ARCHIVE  
FILE  
JOURNAL  NONLINEAR ANALYSISTHEORY METHODS & APPLICATIONS, 2017 
ABSTRACT  In this paper, we prove the EvansKrylov theorem for nonlocal parabolic fully nonlinear equations. (C) 2017 Elsevier Ltd. All rights reserved. 

Qualitative properties of multibubble solutions for nonlinear elliptic equations involving critical exponents
NUMBER  null 
AUTHOR  Choi, Woocheol,Lee, KiAhm 
TITLE  Qualitative properties of multibubble solutions for nonlinear elliptic equations involving critical exponents 
ARCHIVE  
FILE  
JOURNAL  ADVANCES IN MATHEMATICS, 2016 
ABSTRACT  The objective of this paper is to obtain qualitative characteristics of multibubble solutions to the LaneEmdenFowler equations with slightly subcritical exponents given any dimension n >= 3. By examining the linearized problem at each mbubble solution, we provide a number of estimates on the first (n + 2)meigenvalues and their corresponding eigenfunctions. Specifically, we present a new and unified proof of the classical theorems due to BahriLiRey (1995) [2] and Rey (1999) [24] which state that if n >= 4 or n = 3, respectively, then the Morse index of a multibubble solution is governed by a certain symmetric matrix whose component consists of a combination of Greens function, the Robin function, and their first and second derivatives. (C) 2016 Elsevier Inc. Ali rights reserved. 

Holder regularity and uniqueness theorem on weak solutions to the degenerate KellerSegel system
NUMBER  null 
AUTHOR  Lee, KiAhm 
TITLE  Holder regularity and uniqueness theorem on weak solutions to the degenerate KellerSegel system 
ARCHIVE  
FILE  
JOURNAL  NONLINEAR ANALYSISTHEORY METHODS & APPLICATIONS, 2016 
ABSTRACT  In this paper, we present local Holder estimates for the degenerate KellerSegel system (KSm) below in the range of m > 1 and q > 2 before a blowup of solutions. To deal with difficulties caused by the degeneracy of the operator, we find uniform estimates depending on supnorm of the density function and modified the energy estimates and intrinsic scales considered in Porous Medium Equation. As its application, the uniqueness of weak solution to (KSm) is also showed for the case q > max (1 + m/2, 2) in the class of Holder continuous functions by proving L1contraction in this class. (C) 2015 Elsevier Ltd. All rights reserved. 

CordesNirenberg type estimates for nonlocal parabolic equations
NUMBER  null 
AUTHOR  Kim, YongCheol,Lee, KiAhm 
TITLE  CordesNirenberg type estimates for nonlocal parabolic equations 
ARCHIVE  
FILE  
JOURNAL  NONLINEAR ANALYSISTHEORY METHODS & APPLICATIONS, 2016 
ABSTRACT  In this paper, we obtain Cordes Nirenberg type estimates for nonlocal parabolic equations on the more flexible solution space LT(infinity) (Lomega(1)) than the classical solution space B(RT(n)) consisting of all bounded functions on RT(n). (C) 2015 Elsevier Ltd. All rights reserved. 

Regularity for Fully Nonlinear Integrodifferential Operators with Regularly Varying Kernels
NUMBER  null 
AUTHOR  Lee, KiAhm 
TITLE  Regularity for Fully Nonlinear Integrodifferential Operators with Regularly Varying Kernels 
ARCHIVE  
FILE  
JOURNAL  POTENTIAL ANALYSIS, 2016 
ABSTRACT  In this paper, the regularity results for the integrodifferential operators of the fractional Laplacian type by Caffarelli and Silvestre (Comm. Pure Appl. Math. 62, 597638, 2009) are extended to those for the integrodifferential operators associated with symmetric, regularly varying kernels at zero. In particular, we obtain the uniform Harnack inequality and Holder estimate of viscosity solutions to the nonlinear integrodifferential equations associated with the kernels Ksigma,Kbeta satisfying Ksigma,Kbeta (y) asymptotic to 2  sigma/y(n+sigma) (log 2/y(2))(beta(2sigma)) with respect to sigma is an element of(0, 2) close to 2 (for a given beta is an element of R), where the regularity estimates do not blow up as the order sigma is an element of (0, 2) tends to 2. 

Parabolic Harnack inequality of viscosity solutions on Riemannian manifolds
NUMBER  null 
AUTHOR  Lee, KiAhm 
TITLE  Parabolic Harnack inequality of viscosity solutions on Riemannian manifolds 
ARCHIVE  
FILE  
JOURNAL  JOURNAL OF FUNCTIONAL ANALYSIS, 2014 
ABSTRACT  We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold M with the sectional curvature bounded from below by kappa for kappa >= 0. In the elliptic case, Wang and Zhang [24] recently extended the results of [5] to nonlinear elliptic equations in nondivergence form on such M, where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on M. The Harnack inequality of nonnegative viscosity solutions to the elliptic equations is also proved. (C) 2014 Elsevier Inc. All rights reserved. 

Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian
NUMBER  null 
AUTHOR  Lee, KiAhm 
TITLE  Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian 
ARCHIVE  
FILE  
JOURNAL  JOURNAL OF FUNCTIONAL ANALYSIS, 2014 
ABSTRACT  In this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional Laplacians and the critical exponents. This work can be seen as a nonlocal analog of the results of Han (1991) [24] and Rey (1990) [35]. (C) 2014 Elsevier Inc. All rights reserved. 

Geometric properties of the Gelfand problem through a parabolic approach
NUMBER  null 
AUTHOR  Lee, KiAhm 
TITLE  Geometric properties of the Gelfand problem through a parabolic approach 
ARCHIVE  
FILE  
JOURNAL  JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2014 
ABSTRACT  We consider the asymptotic profiles of the nonlinear parabolic flows (e(u))(t) = Delta u+ lambda e(u) to show the geometric properties of minimal solutions of the following elliptic nonlinear eigenvalue problems known as the Gelfand problem: Delta posed in a strictly convex domain Omega subset of Rn. In this work, we show that there isa strictly increasing function f(s) such that f (1)((x)) is convex for 0 < lambda <= lambda*, i.e., we prove that level set of is convex. Moreover, we also present the boundary condition of co which guarantees the fconvexity of solution . (C) 2014 Elsevier Inc. All rights reserved. 

Regularity results for fully nonlinear parabolic integrodifferential operators
NUMBER  null 
AUTHOR  Lee, KiAhm 
TITLE  Regularity results for fully nonlinear parabolic integrodifferential operators 
ARCHIVE  
FILE  
JOURNAL  MATHEMATISCHE ANNALEN, 2013 
ABSTRACT  In this paper, we consider the regularity theory for fully nonlinear parabolic integrodifferential equations with symmetric kernels. We are able to find parabolic versions of AlexandrovBackelmanPucci estimate with . And we show a Harnack inequality, Holder regularity, and regularity of the solutions by obtaining decay estimates of their level sets. 

alphaGauss Curvature flows with flat sides
NUMBER  null 
AUTHOR  Lee, Kiahm 
TITLE  alphaGauss Curvature flows with flat sides 
ARCHIVE  
FILE  
JOURNAL  JOURNAL OF DIFFERENTIAL EQUATIONS, 2013 
ABSTRACT  In this paper, we study the deformation of the 2dimensional convex surfaces in R3 whose speed at a point on the surface is proportional to alphapower of positive part of Gauss Curvature. First, for 1/2 < alpha <= 1, we show that there is smooth solution if the initial data is smooth and strictly convex and that there is a viscosity solution with C1,C1estimate before the collapsing time if the initial surface is only convex. Moreover, we show that there is a waiting time effect which means the flat spot of the convex surface will persist for a while. We also show the interface between the flat side and the strictly convex side of the surface remains smooth on 0 < t < T0 under certain necessary regularity and nondegeneracy initial conditions, where T0 is the vanishing time of the flat side. (C) 2012 Elsevier Inc. All rights reserved. 

Asymptotic behavior in degenerate parabolic fully nonlinear equations and its application to elliptic eigenvalue problems
NUMBER  null 
AUTHOR  Lee, KiAhm 
TITLE  Asymptotic behavior in degenerate parabolic fully nonlinear equations and its application to elliptic eigenvalue problems 
ARCHIVE  
FILE  
JOURNAL  JOURNAL OF DIFFERENTIAL EQUATIONS, 2013 
ABSTRACT  We study the fully nonlinear parabolic equation F(D(2)u(m))  u(t) = 0 in Omega x (0, +infinity), m >= 1, with the Dirichlet boundary condition and positive initial data in a smooth bounded domain Omega subset of Rn, provided that the operator F is uniformly elliptic and positively homogeneous of order one. We prove that the renormalized limit of parabolic flow u(x, t) as t > +infinity is the corresponding positive eigenfunction which solves F(D2 phi)+mu phi(P) = 0(.) in Omega, where 0 < p := 1/m <= 1 and mu > 0 is the corresponding eigenvalue. We also show that some geometric property of the positive initial data is preserved by the parabolic flow, under the additional assumptions that Omega is convex and F is concave. As a consequence, the positive eigenfunction has such geometric property, that is, log(phi) is concave in the case p = 1, and phi(1p/2) is concave for 0< p <1. (C) 2013 Elsevier Inc. All rights reserved. 

Highly oscillating thin obstacles
NUMBER  null 
AUTHOR  Lee, Kiahm 
TITLE  Highly oscillating thin obstacles 
ARCHIVE  
FILE  
JOURNAL  ADVANCES IN MATHEMATICS, 2013 
ABSTRACT  The focus of this paper is on a thin obstacle problem where the obstacle is defined on the intersection between a hyperplane Gamma in Rn and a periodic perforation Tepsilon of Rn, depending on a small parameters epsilon > 0. As epsilon > 0, it is crucial to estimate the frequency of intersections and to determine this number locally. This is done using strong tools from uniform distribution. By employing classical estimates for the discrepancy of sequences of type {k alpha}(k=1)(infinity), alpha is an element of R, we are able to extract rather precise information about the set Gamma boolean AND Tepsilon. As epsilon > 0, we determine the limit u of the solution u(epsilon) to the obstacle problem in the perforated domain, in terms of a limit equation it solves. We obtain the typical strange term behavior for the limit problem, but with a different constant taking into account the contribution of all different intersections, that we call the averaged capacity. Our result depends on the normal direction of the plane, but holds for a.e. normal on the unit sphere in Rn. 

Regularity Results for Fully Nonlinear IntegroDifferential Operators with Nonsymmetric Positive Kernels: Subcritical Case
NUMBER  null 
AUTHOR  Lee, KiAhm 
TITLE  Regularity Results for Fully Nonlinear IntegroDifferential Operators with Nonsymmetric Positive Kernels: Subcritical Case 
ARCHIVE  
FILE  
JOURNAL  POTENTIAL ANALYSIS, 2013 
ABSTRACT  We introduce a class of fully nonlinear integrodifferential operators with possible nonsymmetric kernels. For the index sigma of the operator in (1, 2) (subcritical case), we introduce a very general class of fully nonlinear integrodifferential operators and obtain a comparison principle, a nonlocal version of the AlexandroffBackelmanPucci estimate, a Harnack inequality, a Holder regularity, and an interior C1,Calpharegularity for equations associated with such a class. 

The Viscosity Method for the Homogenization of Soft Inclusions
NUMBER  null 
AUTHOR  Lee, Kiahm 
TITLE  The Viscosity Method for the Homogenization of Soft Inclusions 
ARCHIVE  
FILE  
JOURNAL  ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 2012 
ABSTRACT  In this paper, we consider periodic soft inclusions T (epsilon) with periodicity epsilon, where the solution, u (epsilon) , satisfies semilinear elliptic equations of nondivergence in with Neumann data on . The difficulty lies in the nondivergence structure of the operator where the standard energy method, which is based on the divergence theorem, cannot be applied. The main object is to develop a viscosity method to find the homogenized equation satisfied by the limit of u (epsilon) , referred to as u, as epsilon approaches to zero. We introduce the concept of a compatibility condition between the equation and the Neumann condition on the boundary for the existence of uniformly bounded periodic first correctors. The concept of a second corrector is then developed to show that the limit, u, is the viscosity solution of a homogenized equation. 

Fully degenerate MongeAmpere equations
NUMBER  null 
AUTHOR  Lee, Kiahm 
TITLE  Fully degenerate MongeAmpere equations 
ARCHIVE  
FILE  
JOURNAL  JOURNAL OF DIFFERENTIAL EQUATIONS, 2012 
ABSTRACT  In this paper, we consider the following nonlinear eigenvalue problem for the MongeAmpere equation: find a nonnegative weakly convex classical solution f satisfying {det D2 f = f(p) in Omega, f = phi on partial derivative Omega for a strictly convex smooth domain Omega subset of R2 and 0 < p < 2. When {f = 0} contains a convex domain, we find a classical solution which is smooth on {f > 0} and whose freeboundary partial derivative{f = 0} is also smooth. (C) 2012 Published by Elsevier Inc. 
 Lee, Yongnam
 Affiliate Professor
 Algebraic Geometry
 Office 1404B
 Tel 3715 / Fax 3786



A Birational Contraction of Genus 2 Tails in the Moduli Space of Genus 4 Curves I
NUMBER  null 
AUTHOR  Lee, Yongnam 
TITLE  A Birational Contraction of Genus 2 Tails in the Moduli Space of Genus 4 Curves I 
ARCHIVE  
FILE  
JOURNAL  INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2014 
ABSTRACT  We show that for a. (2 3, 7 10), the log canonical model M4(a) of the pair (M4, ad) is isomorphic to the moduli space M hs 4 of hsemistable curves constructed in [13] and that there is a birational morphism. : M hs 4. M4(2 3) which contracts the locus of curves C1. p C2 consisting of genus 2 curves meeting in a node p such that p is a Weierstrass point of C1 or C2. To obtain this morphism, we construct a compact moduli space M hs 2,1 of pointed genus 2 curves that have nodes, ordinary cusps, and tacnodes as singularities, and prove that it is isomorphic to Rullas flip constructed in [23]. 

Construction of surfaces of general type from elliptic surfaces via Gorenstein smoothing
NUMBER  M11008 
AUTHOR  Keum, JongHae,Park, Heesang,Lee, Yongnam 
TITLE  Construction of surfaces of general type from elliptic surfaces via Gorenstein smoothing 
ARCHIVE  arXiv:1008.1222 
FILE  
JOURNAL  MATHEMATISCHE ZEITSCHRIFT, 2012 
ABSTRACT  We present methods to construct interesting surfaces of general type via Gorenstein smoothing of a singular surface obtained from an elliptic surface. By applying our methods to special Enriques surfaces, we construct new examples of a minimal surface of general type with , and K (2) a parts per thousand currency sign 4. 
 Oum, Sangil
 Affiliate Professor
 Graph Theory, Combinatorics, Combinatorial Optimization
 Office
 Tel / Fax 3786



CHARACTERIZATION OF CYCLE OBSTRUCTION SETS FOR IMPROPER COLORING PLANAR GRAPHS
NUMBER  null 
AUTHOR  Oum, SangIl 
TITLE  CHARACTERIZATION OF CYCLE OBSTRUCTION SETS FOR IMPROPER COLORING PLANAR GRAPHS 
ARCHIVE  
FILE  
JOURNAL  SIAM JOURNAL ON DISCRETE MATHEMATICS, 2018 
ABSTRACT  For nonnegative integers k, d(1),...,d(k), a graph is (d(1),...,d(k))colorable if its vertex set can be partitioned into k parts so that the ith part induces a graph with maximum degree at most d(i) for all i is an element of{1,...,k}. A class C of graphs is balanced kpartitionable and unbalanced kpartitionable if there exists a nonnegative integer D such that all graphs in C are (D,...,D)colorable and (0,...,0, D)colorable, respectively, where the tuple has length k. A set X of cycles is a cycle obstruction set of a class C of planar graphs if every planar graph containing none of the cycles in X as a subgraph belongs to C. This paper characterizes all cycle obstruction sets of planar graphs to be balanced kpartitionable and unbalanced kpartitionable for all k; namely, we identify all inclusionwise minimal cycle obstruction sets for all k. 

Evencycle decompositions of graphs with no oddK4minor
NUMBER  null 
AUTHOR  Oum, Sangil 
TITLE  Evencycle decompositions of graphs with no oddK4minor 
ARCHIVE  
FILE  
JOURNAL  EUROPEAN JOURNAL OF COMBINATORICS, 2017 
ABSTRACT  An evencycle decomposition of a graph G is a partition of E(G) into cycles of even length. Evidently, every Eulerian bipartite graph has an evencycle decomposition. Seymour (1981) proved that every 2connected loopless Eulerian planar graph with an even number of edges also admits an evencycle decomposition. Later, Zhang (1994) generalized this to graphs with no K5minor. Our main theorem gives sufficient conditions for the existence of evencycle decompositions of graphs in the absence of odd minors. Namely, we prove that every 2connected loopless Eulerian oddK4minorfree graph with an even number of edges has an evencycle decomposition. This is best possible in the sense that oddK4minorfree cannot be replaced with oddK5minorfree. The main technical ingredient is a structural characterization of the class of oddK4minorfree graphs, which is due to Lovasz, Seymour, Schrijver, and Truemper. (C) 2017 Elsevier Ltd. All rights reserved. 

Dynamic coloring of graphs having no K5 minor
NUMBER  null 
AUTHOR  Oum, Sangil 
TITLE  Dynamic coloring of graphs having no K5 minor 
ARCHIVE  
FILE  
JOURNAL  DISCRETE APPLIED MATHEMATICS, 2016 
ABSTRACT  We prove that every simple connected graph with no K5 minor admits a proper 4coloring such that the neighborhood of each vertex v having more than one neighbor is not monochromatic, unless the graph is isomorphic to the cycle of length 5. This generalizes the result on planar graphs by 5.J. Kim, W.J. Park and the second author [Discrete Appl. Math. 161 (2013) 220722121. (C) 2016 Elsevier B.V. All rights reserved. 
 Park, Jongil
 Affiliate Professor
 Topology
 Office 1521
 Tel 2521 / Fax 3786



Milnor fibers and symplectic fillings of quotient surface singularities
NUMBER  null 
AUTHOR  Park, Jongil 
TITLE  Milnor fibers and symplectic fillings of quotient surface singularities 
ARCHIVE  
FILE  
JOURNAL  ADVANCES IN MATHEMATICS, 2018 
ABSTRACT  We determine a onetoone correspondence between Milnor fibers and minimal symplectic fillings of a quotient surface singularity (up to diffeomorphism type) by giving an explicit algorithm to compare them mainly via techniques from the minimal model program for 3folds and Pinkhams negative weight smoothing. As byproducts, we show that:  Milnor fibers associated to irreducible components of the reduced versal deformation space of a quotient surface singularity are not diffeomorphic to each other with a few obvious exceptions. For this, we classify minimal symplectic fillings of a quotient surface singularity up to diffeomorphism.  Any symplectic filling of a quotient surface singularity is obtained by a sequence of rational blowdowns from a special resolution (socalled the maximal resolution) of the singularity, which is an analogue of the onetoone correspondence between the irreducible components of the reduced versal deformation space and the socalled Presolutions of a quotient surface singularity. (C) 2018 Elsevier Inc. All rights reserved. 

Lefschetz Fibrations on Knot Surgery 4Manifolds Via Stallings Twist
NUMBER  null 
AUTHOR  Park, Jongil 
TITLE  Lefschetz Fibrations on Knot Surgery 4Manifolds Via Stallings Twist 
ARCHIVE  
FILE  
JOURNAL  MICHIGAN MATHEMATICAL JOURNAL, 2017 
ABSTRACT  In this paper, we construct a family of simply connected minimal symplectic 4manifolds that admit arbitrarily many nonisomorphic Lefschetz fibration structures with the same genus fiber. We obtain these families by performing knot surgery on an elliptic surface E(2) using connected sums of n copies of fibered knots, which in turn are obtained by Stallings twist from the square knot. Thus, all of these 4 manifolds are homotopy E (2) surfaces. We show that they admit 2(n) mutually nonisomorphic Lefschetz fibration structures of fiber genus (4n + 1) by comparing their monodromy groups that are induced from the corresponding monodromy factorizations. 

SMOOTHLY EMBEDDED RATIONAL HOMOLOGY BALLS
NUMBER  null 
AUTHOR  Park, Jongil 
TITLE  SMOOTHLY EMBEDDED RATIONAL HOMOLOGY BALLS 
ARCHIVE  
FILE  
JOURNAL  JOURNAL OF THE KOREAN MATHEMATICAL SOCIETY, 2016 
ABSTRACT  In this paper we prove the existence of rational homology balls smoothly embedded in regular neighborhoods of certain linear chains of smooth 2spheres by using techniques from minimal model program for 3dimensional complex algebraic variety. 

Families of nondiffeomorphic 4manifolds with the same SeibergWitten invariants
NUMBER  null 
AUTHOR  Park, Jongil 
TITLE  Families of nondiffeomorphic 4manifolds with the same SeibergWitten invariants 
ARCHIVE  
FILE  
JOURNAL  JOURNAL OF SYMPLECTIC GEOMETRY, 2015 
ABSTRACT  In this article, we show that, at least for nonsimply connected case, there exist an infinite family of nondiffeomorphic symplectic 4manifolds with the same SeibergWitten invariants. The main techniques are knot surgery and a covering method developed in Fintushel and Sterns paper [6]. 

A COMPLEX SURFACE OF GENERAL TYPE WITH p(g)=0, K2=2 AND H1=Z/4Z
NUMBER  M11009 
AUTHOR  Park, Jongil,Park, Heesang 
TITLE  A COMPLEX SURFACE OF GENERAL TYPE WITH p(g)=0, K2=2 AND H1=Z/4Z 
ARCHIVE  arXiv:1012.5871 
FILE  
JOURNAL  TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 2013 
ABSTRACT  We construct a new minimal complex surface of general type with p(g) = 0, K2 = 2 and H1 = Z/4Z (in fact, pi(alg)(1) = Z/4Z), which settles the existence question for numerical Campedelli surfaces with all possible algebraic fundamental groups. The main techniques involved in the construction are a rational blowdown surgery and a QGorenstein smoothing theory. 

SURFACES OF GENERAL TYPE WITH p(g)=1 AND q=0
NUMBER  M11007 
AUTHOR  Park, Jongil,Park, Heesang 
TITLE  SURFACES OF GENERAL TYPE WITH p(g)=1 AND q=0 
ARCHIVE  arXiv:0906.5195 
FILE  
JOURNAL  JOURNAL OF THE KOREAN MATHEMATICAL SOCIETY, 2013 
ABSTRACT  We construct a new family of simply connected minimal complex surfaces of general type with p(g) = 1, q = 0, and K2 = 3, 4, 5, 6, 8 using a QGorenstein smoothing theory. 

Lefschetz Fibration Structures on Knot Surgery 4Manifolds
NUMBER  null 
AUTHOR  Park, Jongil 
TITLE  Lefschetz Fibration Structures on Knot Surgery 4Manifolds 
ARCHIVE  
FILE  
JOURNAL  MICHIGAN MATHEMATICAL JOURNAL, 2011 
ABSTRACT  
 Jeong, HyeongChai
 KIAS Visiting Professor
 Office
 Tel 2593 / Fax
