Potential Kernel Research Articles

On stationary solutions of the 2D Doi–Onsager model

We study the 2D Doi–Onsager models with general potential kernel, with special emphasis on the classical Onsager kernel. Through application of topological methods from nonlinear functional analysis, in particular the Leray–Schauder degree theory, we obtain the uniqueness of the trivial solution for low temperatures as well as the local bifurcation structure of the solutions.

Spectral gap properties for linear random walks and Pareto’s asymptotics for affine stochastic recursions

Let $V=\mathbb R^d$ be the Euclidean $d$-dimensional space, $\mu$ (resp $\lambda$) a probability measure on the linear (resp affine) group $G=G L (V)$ (resp $H= \Aff (V))$ and assume that $\mu$ is the projection of $\lambda$ on $G$. We study asymptotic properties of the iterated convolutions $\mu^n *\delta_{v}$ (resp $\lambda^n*\delta_{v})$ if $v\in V$, i.e asymptotics of the random walk on $V$ defined by $\mu$ (resp $\lambda$), if the subsemigroup $T\subset G$ (resp. $\Sigma \subset H$) generated by the support of $\mu$ (resp $\lambda$) is ``large''. We show spectral gap properties for the convolution operator defined by $\mu$ on spaces of functions of degree $s\geq 0$ on $V$, which satisfy Holder type conditions. As a consequence of our analysis we get precise asymptotics for the potential kernel $\Sigma_{0}^{\infty} \mu^k * \delta_{v}$, which imply its asymptotic homogeneity. Under natural conditions the $H$-space $V$ is a $\lambda$-boundary; then we use the above results and radial Fourier Analysis on $V\setminus \{0\}$ to show that the unique $\lambda$-stationary measure $\rho$ on $V$ is homogeneous at infinity with respect to dilations $v\rightarrow t v$ (for $t>0$), with a tail measure depending essentially of $\mu$ and $\Sigma$. Our proofs are based on the simplicity of the dominant Lyapunov exponent for certain products of Markov-dependent random matrices, on the use of renewal theorems for ``tame'' Markov walks, and on the dynamical properties of a conditional $\lambda$-boundary dual to $V$.

Patient-specific scatter correction for flat-panel detector-based cone-beam CT imaging

A patient-specific scatter correction algorithm is proposed to mitigate scatter artefacts in cone-beam CT (CBCT). The approach belongs to the category of convolution-based methods in which a scatter potential function is convolved with a convolution kernel to estimate the scatter profile. A key step in this method is to determine the free parameters introduced in both scatter potential and convolution kernel using a so-called calibration process, which is to seek for the optimal parameters such that the models for both scatter potential and convolution kernel is able to optimally fit the previously known coarse estimates of scatter profiles of the image object. Both direct measurements and Monte Carlo (MC) simulations have been proposed by other investigators to achieve the aforementioned rough estimates. In the present paper, a novel method has been proposed and validated to generate the needed coarse scatter profile for parameter calibration in the convolution method. The method is based upon an image segmentation of the scatter contaminated CBCT image volume, followed by a reprojection of the segmented image volume using a given x-ray spectrum. The reprojected data is subtracted from the scatter contaminated projection data to generate a coarse estimate of the needed scatter profile used in parameter calibration. The method was qualitatively and quantitatively evaluated using numerical simulations and experimental CBCT data acquired on a clinical CBCT imaging system. Results show that the proposed algorithm can significantly reduce scatter artefacts and recover the correct CT number. Numerical simulation results show the method is patient specific, can accurately estimate the scatter, and is robust with respect to segmentation procedure. For experimental and in vivo human data, the results show the CT number can be successfully recovered and anatomical structure visibility can be significantly improved.

On potential kernels associated with random dynamical systems

Let \((\theta,\varphi)\) be a continuous random dynamical system defined on a probability space \((\Omega,\mathcal{F},\mathbb{P})\) and taking values on a locally compact Hausdorff space \(E\). The associated potential kernel \(V\) is given by \[ Vf(\omega ,x)= \int\limits_{0}^{\infty} f(\theta_{t}\omega,\varphi(t,\omega)x)dt, \quad \omega \in \Omega, x\in E.\] In this paper, we prove the equivalence of the following statements: 1. The potential kernel of \((\theta,\varphi)\) is proper, i.e. \(Vf\) is \(x\)-continuous for each bounded, \(x\)-continuous function \(f\) with uniformly random compact support. 2. \((\theta ,\varphi)\) has a global Lyapunov function, i.e. a function \(L:\Omega\times E \rightarrow (0,\infty)\) which is \(x\)-continuous and \(L(\theta_t\omega, \varphi(t,\omega)x)\downarrow 0\) as \(t\uparrow \infty\). In particular, we provide a constructive method for global Lyapunov functions for gradient-like random dynamical systems. This result generalizes an analogous theorem known for deterministic dynamical systems.

Dynamical Regulation Analysis Identifies Molecular Mechanisms of Fuzheng-Huayu Formula against Hepatitis B-Caused Liver Cirrhosis.

Fuzheng-Huayu (FZHY) tablet was formulated based on Chinese medicine theory in treating liver fibrosis. A clinical trial has indicated that FZHY can against hepatitis B-caused liver cirrhosis (HBC), but the underlying mechanism of FZHY efficacy is unclear. Here, we report that miRNA expression levels are remarkably changed when FZHY formula was used in HBC patient's treatment as a paradigm of trials. Then, we functionally characterize the significant impact of potential kernel miRNAs by miRNA-target network analysis. Enrichment analysis show that the FZHY formula dramatically effecting the molecular regulated module in HBC. Thus, we infer that FZHY plays a critical function in HBC treatment process and directly regulated many important pathways, including but not limited to cell cycle, p53 signaling pathway, and TGF-β signaling pathway, suggesting a new strategy for investigating the molecular mechanism of FZHY treatment.

Transcriptional profiling and dynamical regulation analysis identify potential kernel target genes of SCYL1-BP1 in HEK293T cells.

SCYL1-BP1 is thought to function in the p53 pathway through Mdm2 and hPirh2, and mutations in SCYL1-BP1 are associated with premature aging syndromes such as Geroderma Osteodysplasticum; however, these mechanisms are unclear. Here, we report significant alterations in miRNA expression levels when SCYL1-BP1 expression was inhibited by RNA interference in HEK293T cells. We functionally characterized the effects of potential kernel miRNA-target genes by miRNA-target network and protein-protein interaction network analysis. Importantly, we showed the diminished SCYL1-BP1 dramatically reduced the expression levels of EEA1, BMPR2 and BRCA2 in HEK293T cells. Thus, we infer that SCYL1-BP1 plays a critical function in HEK293T cell development and directly regulates miRNA-target genes, including, but not limited to, EEA1, BMPR2, and BRCA2, suggesting a new strategy for investigating the molecular mechanism of SCYL1-BP1.

Potential operators associated with Jacobi and Fourier–Bessel expansions

We study potential operators (Riesz and Bessel potentials) associated with classical Jacobi and Fourier–Bessel expansions. We prove sharp estimates for the corresponding potential kernels. Then we characterize those 1≤p,q≤∞, for which the potential operators are of strong type (p,q), of weak type (p,q) and of restricted weak type (p,q). These results may be thought of as analogues of the celebrated Hardy–Littlewood–Sobolev fractional integration theorem in the Jacobi and Fourier–Bessel settings. As an ingredient of our line of reasoning, we also obtain sharp estimates of the Poisson kernel related to Fourier–Bessel expansions.

Development and analysis of composite flour bread.

The study elucidates the effect of utilizing cereal-pulse-fruit seed composite flour in the development and quality analysis of leavened bread. The composite flour was prepared using refined wheat flour (WF), high protein soy flour (SF), sprouted mung bean flour (MF) and mango kernel flour (MKF). Three variations were formulated such as V-I (WF: SF: MF: MKF = 85:5:5:5), V-II (WF: SF: MF: MKF = 70:10:10:10), and V-III (WF: SF: MF: MKF = 60:14:13:13). Pertinent functional, physico-chemical and organoleptic attributes were studied in composite flour variations and their bread preparations. Physical characteristics of the bread variations revealed a percentage decrease in loaf height (14%) and volume (25%) and 20% increase in loaf weight with increased substitution of composite flour. The sensory evaluation of experimental breads on a nine-point hedonic scale revealed that V-I score was 5% higher than the standard bread. Hence, the present study highlighted the nutrient enrichment of bread on incorporation of a potential waste material mango kernel, soy and sprouted legume. Relevant statistical tests were done to analyze the significance of means for all tested parameters.

General Parameterized Time-Frequency Transform

Interest in parameterized time-frequency analysis for non-stationary signal processing is increasing steadily. An important advantage of such analysis is to provide highly concentrated time-frequency representation with signal-dependent resolution. In this paper, a general scheme, named as general parameterized time-frequency transform (GPTF transform), is proposed for carrying out parameterized time-frequency analysis. The GPTF transform is defined by applying generalized kernel based rotation operator and shift operator. It provides the availability of a single generalized time-frequency transform for applications on signals of different natures. Furthermore, by replacing kernel function, it facilitates the implementation of various parameterized time - frequency transforms from the same standpoint. The desirable properties and the dual definition in the frequency domain of GPTF transform are also described in this paper. One of the advantages of the GPTF transform is that the generalized kernel can be customized to characterize the time - frequency signature of non-stationary signal. As different kernel formulation has bias toward the signal to be analyzed, a proper kernel is vital to the GPTF. Thus, several potential kernels are provided and discussed in this paper to develop the desired parameterized time - frequency transforms. In real applications, it is desired to identify proper kernel with respect to the considered signal. This motivates us to propose an effective method to identify the kernel for the GPTF.

Performance evaluation of kernel fusion BLAS routines on the GPU: iterative solvers as case study

Programmers usually implement iterative methods that solve partial differential equations by expressing them using a sequence of basic kernels from libraries optimized for the graphics processing unit (GPU). The global runtime of the resulting combination is often penalized by the smallest and most inefficient vector operations. To improve the GPU exploitation, we identify and analyze the potential kernels to be fused according to the data dependence, data type and size, and GPU resources. This paper provides an extensive analysis of the impact of fusing vector operations [level 1 of Basic Linear Algebra Subprograms (BLAS)] on the performance of the GPU. The experimental evaluation shows that this optimization provides noticeable improvement especially for kernels with lower memory requirements and on more modern GPUs. It is worth noting that the fused BLAS operations can be very useful to help programmers efficiently code iterative methods to solve large linear systems of equations for the GPU. Iterative methods such as biconjugate gradient method (BCG) are one of the examples that can benefit from this optimization strategy. Indeed, kernel fusion of vector routines makes the most efficient GPU implementation of BCG run between $$1.09\times $$ 1.09 × and $$1.27\times $$ 1.27 × faster on three GPUs of different characteristics.

Sharp estimates of the potential kernel for the harmonic oscillator with applications

AbstractWe prove qualitatively sharp estimates of the potential kernel for the harmonic oscillator. These bounds are then used to show that theLp–Lqestimates of the associated potential operator obtained recently by Bongioanni and Torrea are in fact sharp.

Sharp estimates of the potential kernel for the harmonic oscillator with applications

AbstractWe prove qualitatively sharp estimates of the potential kernel for the harmonic oscillator. These bounds are then used to show that the Lp–Lq estimates of the associated potential operator obtained recently by Bongioanni and Torrea are in fact sharp.

Solitary waves in nematic liquid crystals

We study soliton solutions of a two-dimensional nonlocal NLS equation of Hartree-type with a Bessel potential kernel. The equation models laser propagation in nematic liquid crystals. Motivated by the experimental observation of spatially localized beams, see Conti et al. (2003), we show existence, stability, regularity, and radial symmetry of energy minimizing soliton solutions in R2. We also give theoretical lower bounds for the L2-norm (power) of these solitons, and show that small L2-norm initial conditions lead to decaying solutions. We also present numerical computations of radial soliton solutions. These solutions exhibit the properties expected by the infinite plane theory, although we also see some finite (computational) domain effects, especially solutions with arbitrarily small power.

Physiological Dynamics of Maize Nitrogen Uptake and Partitioning in Response to Plant Density and Nitrogen Stress Factors: II. Reproductive Phase

ABSTRACTImproved plant N utilization and partitioning is critical for future improvements in maize (Zea mays L.) grain yield (GY). The overall research objective was to gain understanding of the physiological mechanisms underpinning biomass (BM), N uptake partitioning, and GY processes during the reproductive period for two maize hybrids grown at varying plant density (PD) (low is 54,000 plant ha−1, medium is 79,000 plants ha−1, and high is 104,000 plants ha−1) and N inputs (low is 0 kg N ha−1, medium is 112 kg N ha−1, and high is 224 kg N ha−1) over four site–years. At the community level, maize GY was maximized in both genotypes at the medium PD and highest N rate. At maturity, grain harvest index improved as the whole‐plant N uptake increased following a linear‐plateau model and, for N allocation, both grain and shoot N concentrations increased similarly as BM increased. Around flowering (±15 d), dry mass and N partitioning rates were unmodified by treatment factors. Treatment factors only marginally influenced potential kernel number near flowering. Allometric analyses confirmed a lack of treatment impact on whole‐plant N uptake and N remobilization coefficients. Greater reproductive‐stage N uptake was associated with superior ear strength (kernel number and weight) and late shoot N remobilization, but GY was also positively related to vegetative‐stage N uptake. Future research should identify genotypic variation for overcoming the documented N uptake trade‐off mechanisms (vegetative‐stage N uptake vs. shoot N remobilization) as related to the maize GY improvement process.

On Harnack Inequality and Hölder Regularity for Isotropic Unimodal Lévy Processes

We prove the scale invariant Harnack inequality and regularity properties for harmonic functions with respect to an isotropic unimodal Lévy process with the characteristic exponent ψ satisfying some scaling condition. We derive sharp estimates of the potential measure and capacity of balls, and further, under the assumption that ψ satisfies the lower scaling condition, sharp estimates of the potential kernel of the underlying process. This allows us to establish the Krylov–Safonov type estimate, which is the key ingredient in the approach of Bass and Levin, that we follow.

Physiological Evaluations of Recent Drought‐Tolerant Maize Hybrids at Varying Stress Levels

Maize (Zea mays L.) improvement in drought‐stress tolerance poses a great challenge as the global need for food, feed, fiber, and fuel increases. Seed companies are developing and promoting drought‐tolerant hybrids, but their physiological drought‐tolerance mechanisms are not well understood. The research objective was to investigate the plant traits related to yield improvement for similar maturity hybrids classified as either drought‐tolerant (non‐transgenic) or conventional at varying plant density (PD) (two levels) and N rates (four levels) over 2 site‐years in northwestern Indiana. Physiological measurements included photosynthesis (A), transpiration (E), and leaf area index at multiple growth stages, as well as anthesis‐silking interval, potential kernel number, grain yield (GY) and its components. Intensive heat and drought stress occurred in the 30‐d period before and during flowering in 2012, but not in 2011. Overall, similar maturity drought‐ and non‐drought‐tolerant hybrids did not markedly differ in GY or most other traits, and hybrid responses to varying PD and N rates were similar. In both seasons, GY was impacted most by N rates. A complex N rate effect on A and E was tightly related to water supply (i.e., higher N had positive impact under non‐drought conditions). Hybrid differences in A and E were not significant at the leaf‐scale, but one drought‐tolerant hybrid had lower estimated cumulative A and E at the season‐long canopy scale. Under the non‐drought and specific‐drought conditions in these single‐location trials there was no indication that designated drought‐tolerant hybrids were more tolerant to high crowding intensity and/or low N stresses.

Time-dependent effective potential and exchange kernel of homogeneous electron gas

By minimizing the difference between the left- and the right-hand sides of the many-body time-dependent Schr\"odinger equation with the Slater-determinant wave function, we derive a nonadiabatic and self-interaction-free time-dependent single-particle effective potential, which is the generalization to the time-dependent case of the so-called localized Hartree-Fock potential. The new potential can be efficiently used within the framework of the time-dependent density-functional theory as we demonstrate by the evaluation of the wave vector and frequency-dependent exchange kernel ${f}_{x}^{h}(q,\ensuremath{\omega})$ of the homogeneous electron gas. This is found to be nonsingular and causal, and it satisfies the positiveness of the dissipation, in contrast to the earlier known kernel from the first-order perturbation theory, which makes our ${f}_{x}^{h}$ promising for applications.

Support vector machines for spike pattern classification with a leaky integrate-and-fire neuron

Spike pattern classification is a key topic in machine learning, computational neuroscience, and electronic device design. Here, we offer a new supervised learning rule based on Support Vector Machines (SVM) to determine the synaptic weights of a leaky integrate-and-fire (LIF) neuron model for spike pattern classification. We compare classification performance between this algorithm and other methods sharing the same conceptual framework. We consider the effect of postsynaptic potential (PSP) kernel dynamics on patterns separability, and we propose an extension of the method to decrease computational load. The algorithm performs well in generalization tasks. We show that the peak value of spike patterns separability depends on a relation between PSP dynamics and spike pattern duration, and we propose a particular kernel that is well-suited for fast computations and electronic implementations.

Hitting Times for Random Walks with Restarts

The time it takes a random walker in a lattice to reach the origin from another vertex x has infinite mean. If the walker can restart the walk at x at will, then the minimum expected hitting time $\gamma(x,0)$ (minimized over restarting strategies) is finite; it was called the “grade” of x by Dumitriu, Tetali, and Winkler. They showed that in a more general setting, the grade (a variant of the “Gittins index”) plays a crucial role in control problems involving several Markov chains. Here we establish several conjectures of Dumitriu, Tetali, and Winkler on the asymptotics of the grade in Euclidean lattices. In particular, we show that in the planar square lattice, $\gamma(x,0)$ is asymptotic to $2|x|^2\log|x|$ as $|x| \to \infty$. The proof hinges on the local variance of the potential kernel h being almost constant on the level sets of h. We also show how the same method yields precise second order asymptotics for hitting times of a random walk (without restarts) in a lattice disk.

Weighted estimates of truncated potential kernels in the variable exponent setting

In connection with the applications to the study of classical operators of harmonic analysis in variable exponents Morrey spaces, we prove estimates of weighted norms of potential kernels truncated to balls B(x, r). We admit a class of radial type weights. The conditions on the validity of estimates are given in terms of Zygmund type inequalities on weight or in terms of their Matuszewska–Orlich indices