No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

Herein we describe new frontiers in mathematical modeling and statistical analysis of oscillatory biomedical signals, motivated by our recent studies of network formation in the human brain during the early stages of life and studies forty years ago on cardiorespiratory patterns during sleep in infants and animal models. The frontiers involve new nonlinear-type time-frequency analysis of signals with multiple oscillatory components, and efficient particle filters for joint state and parameter estimators together with uncertainty quantification in hidden Markov models and empirical Bayes inference.

Surface conformal parameterizations have been widely applied to various tasks in computer graphics. In this paper, we develop a convergent conformal energy minimization (CCEM) iterative algorithm via the line-search gradient descent method with a quadratic approximation for the computation of disk-shaped conformal parameterizations of simply connected open triangular meshes. In addition, we prove the global convergence of the proposed CCEM iterative algorithm. Moreover, under some mild assumptions, we prove the existence of the nontrivial solution, which is a local minimum of the conformal energy with a bijective boundary map. Numerical experiments indicate that the efficiency of the proposed CCEM algorithm is highly improved and the accuracy is competitive with that of state-of-the-art algorithms.

The interplay between excitatory and inhibitory neurons imparts rich functions of the brain. To understand the synaptic mechanisms underlying neuronal computations, a fundamental approach is to study the dynamics of excitatory and inhibitory synaptic inputs of each neuron. The traditional method of determining input conductance, which has been applied for decades, employs the synaptic current-voltage (I-V) relation obtained via voltage clamp. Due to the space clamp effect, the measured conductance is different from the local conductance on the dendrites. Therefore, the interpretation of the measured conductance remains to be clarified. Using theoretical analysis, electrophysiological experiments, and realistic neuron simulations, here we demonstrate that there does not exist a transform between the local conductance and the conductance measured by the traditional method, due to the neglect of a nonlinear interaction between the clamp current and the synaptic current in the traditional method. Consequently, the conductance determined by the traditional method may not correlate with the local conductance on the dendrites, and its value could be unphysically negative as observed in experiment. To circumvent the challenge of the space clamp effect and elucidate synaptic impact on neuronal information processing, we propose the concept of effective conductance which is proportional to the local conductance on the dendrite and reflects directly the functional influence of synaptic inputs on somatic membrane potential dynamics, and we further develop a framework to determine the effective conductance accurately. Our work suggests re-examination of previous studies involving conductance measurement and provides a reliable approach to assess synaptic influence on neuronal computation.

No abstract uploaded!

The purpose of this research monograph is to survey some recent developments in the analysis of shock reflection-diffraction, to present our original mathematical proofs of von Neumann’s conjectures for potential flow, to collect most of the related results and new techniques in the analysis of partial differential equations achieved in the last decades, and to discuss a set of fundamental open problems relevant to the directions of future research in this and related areas.

No abstract uploaded!

As an important issue in special relativity, Wigner rotation is notoriously difficult for beginners for two major reasons: this physical phenomenon is highly unintuitive, and the mathematics behind it can be extremely challenging. To remove the first obstacle, we introduce a clear and easy toy model under the guidance of group theory. To overcome the second, a concise mathematical method is developed by the integration of geometric algebra and active-frame formalism.

This paper presents entropy symmetrization and high-order accurate entropy stable schemes for the relativistic magnetohydrodynamic (RMHD) equations. It is shown that the conservative RMHD equations are not symmetrizable and do not admit a thermodynamic entropy pair. To address this issue, a symmetrizable RMHD system, equipped with a convex thermodynamic entropy pair, is first proposed by adding a source term into the equations, providing an analogue to the nonrelativistic Godunov--Powell system. Arbitrarily high-order accurate entropy stable finite difference schemes are developed on Cartesian meshes based on the symmetrizable RMHD system. The crucial ingredients of these schemes include (i) affordable explicit entropy conservative fluxes which are technically derived through carefully selected parameter variables, (ii) a special high-order discretization of the source term in the symmetrizable RMHD system, and (iii) suitable high-order dissipative operators based on essentially nonoscillatory reconstruction to ensure the entropy stability. Several numerical tests demonstrate the accuracy and robustness of the proposed entropy stable schemes.

Numerical schemes provably preserving the positivity of density and pressure are highly desirable for ideal magnetohydrodynamics (MHD), but the rigorous positivity-preserving (PP) analysis remains challenging. The difficulties mainly arise from the intrinsic complexity of the MHD equations as well as the indeterminate relation between the PP property and the divergence-free condition on the magnetic field. This paper presents the first rigorous PP analysis of conservative schemes with the Lax--Friedrichs (LF) flux for 1D and multidimensional ideal MHD. The significant innovation is the discovery of the theoretical connection between the PP property and a discrete divergence-free (DDF) condition. This connection is established through the generalized LF splitting properties, which are alternatives to the usually expected LF splitting property that does not hold for ideal MHD. The generalized LF splitting properties involve a number of admissible states strongly coupled by the DDF condition, making their derivation very difficult. We derive these properties via a novel equivalent form of the admissible state set and an important inequality, which is skillfully constructed by technical estimates. Rigorous PP analysis is then presented for finite volume and discontinuous Galerkin schemes with the LF flux on uniform Cartesian meshes. In the 1D case, the PP property is proved for the first-order scheme with proper numerical viscosity, and also for arbitrarily high-order schemes under conditions accessible by a PP limiter. In the 2D case, we show that the DDF condition is necessary and crucial for achieving the PP property. It is observed that even slightly violating the proposed DDF condition may cause failure to preserve the positivity of pressure. We prove that the 2D LF type scheme with proper numerical viscosity preserves both the positivity and the DDF condition. Sufficient conditions are derived for 2D PP high-order schemes, and extension to 3D is discussed. Numerical examples provided in the supplementary material further confirm the theoretical findings.

Suppose that $\eta$ is a Schramm-Loewner evolution ($\SLE_\kappa$) in a smoothly bounded simply connected domain $D \subset \C$ and that $\phi$ is a conformal map from $\D$ to a connected component of $D \setminus \eta([0,t])$ for some $t>0$. The multifractal spectrum of $\eta$ is the function $(-1,1) \to [0,\infty)$ which, for each $s \in (-1,1)$, gives the Hausdorff dimension of the set of points $x \in \partial \D$ such that $|\phi'( (1-\epsilon) x)| = \epsilon^{-s+o(1)}$ as $\epsilon \to 0$. We rigorously compute the a.s.\ multifractal spectrum of $\SLE$, confirming a prediction due to Duplantier. As corollaries, we confirm a conjecture made by Beliaev and Smirnov for the a.s.\ bulk integral means spectrum of $\SLE$, we obtain the optimal H\"older exponent for a conformal map which uniformizes the complement of an $\SLE$ curve, and we obtain a new derivation of the a.s.\ Hausdorff dimension of the $\SLE$ curve for $\kappa \leq 4$. Our results also hold for the $\SLE_\kappa(\ul\rho)$ processes with general vectors of weight $\ul\rho$.