This paper studies a class of p-Laplacian equations with cubic polynomial nonlinearity Δ _p v + (v-a_1)(v-a_2)(v-a_3) = 0 on complete Riemannian manifolds M with lower Ricci curvature bounds, where a_1< a_2 < a_3 are real constants and Δ _p v = div(|∇ v|^p-2∇ v) denotes the p-Laplacian. Depending on whether the solution lies in the intervals (a_1,a_2), (a_2,a_3) or (a_1,a_3) , we employ, respectively, a logarithmic transformation or a hyperbolic tangent transformation to convert the original equation to another one for further analysis. Through a detailed analysis of the lower-bound estimate for the linearized operator of the new equation, and by combining Saloff-Coste’s Sobolev inequality with Moser iteration, we establish Cheng-Yau type gradient estimates under an additional assumption on p. As applications, the Liouville theorem and a Harnack inequality are further proved.

We consider the following coupled Ginzburg-Landau system in ℝ^3 -ϵ^2 Δ w^+ +[A_+(|w^+|^2-t^+^2)+B(|w^-|^2-t^-^2)]w^+=0, -ϵ^2 Δ w^- +[A_-(|w^-|^2-t^-^2)+B(|w^+|^2-t^+^2)]w^-=0, where w=(w^+, w^-)∈ℂ^2 and the constant coefficients satisfy A_+, A_->0, B^20, t^+^2+ t^-^2=1. If B<0, then for every ϵ small enough, we construct a family of entire solutions w_ϵ (z̃, t)∈ℂ^2 in the cylindrical coordinates (z̃, t)∈ℝ^2 ×ℝ for this system via the approach introduced by J. Dávila, M. del Pino, M. Medina and R. Rodiac in arXiv:1901.02807. These solutions are 2π-periodic in t and have multiple interacting vortex helices. The main results are the extensions of the phenomena of interacting helical vortex filaments for the classical (single) Ginzburg-Landau equation in ℝ^3 which has been studied in arXiv:1901.02807. Our results negatively answer the Gibbons conjecture for the Allen-Cahn equation in Ginzburg-Landau system version, which is an extension of the question originally proposed by H. Brezis.

We study the following coupled Ginzburg-Landau system in R-3 [-is an element of(2)Delta w+(A+|w+|(2)-t(+2))+B(|w-|(2)-t(-2))] w(+) = 0, -is an element of(2)Delta w-+ (|A-|W-(2)-t(-2)) + B(|w+|(2)-t(+2))] A-w-= 0, where w = (w(+), w-) is an element of C-2 and the constants satisfy A+, A->0, B-2 < A+A-, t(+/-) > 0, t(+2) + t(-2) = 1. Assuming B < 0, we construct, for each sufficiently small E, a family of entire solutions w is an element of(z, t) is an element of C-2 written in cylindrical coordinates (z, t) is an element of R-2 & times; R. These solutions are 2n-periodic in t and exhibit configurations of multiple interacting helical vortex filaments. Our main results extend the phenomena of interacting helical vortices, previously known for the classical (single-component) Ginzburg-Landau equation in R-3 by Davila, del Pino, Medina and Rodiac (JEMS, Vol 22, 2022), to the coupled system. Moreover, these results give a negative answer to a natural analogue of the well-known Gibbons conjecture for the Allen-Cahn equation, adapted to the Ginzburg-Landau system-an extension of the question originally raised by Brezis.

We concern standing wave solutions with frequency λ to a two dimensional Gross-Pitaevskii equation with a trap potential under the unit mass constraint, which is used to describe Bose-Einstein condensates with attractive interaction. First, we investigate the necessary conditions for existence of the solutions with concentration phenomena directed along closed smooth curves. Next, not only imposing stationary and non-degeneracy conditions on the curves with respect to an auxiliary weighted length involving the trap potential, but also adding some other technical assumptions, we select a sequence {λ_j} of the frequency λ with -λ_j→ +∞ and construct solutions with concentration directed along the curves. Our result partially answers the conjecture raised in [A. Ambrosetti, A. Malchiodi, W.-M. Ni, Comm. Math. Phys. 2003] about necessary condition for solution concentrating at submanifolds. The solutions constructed in this paper are concentrating on curves whose length are non-uniformly bounded, and hence the situation is quite different from that in [M. del Pino, M. Kowalczyk, J. Wei, Comm. Pure Appl. Math. 2007].

The accurate calculation of the angle of refraction of X-rays passing through an object is essential in X-ray phase-contrast imaging. While the wave optics-based method is commonly employed to calculate the angle of refraction, it presents several limitations. First, in cases where the object induces significant phase variations, the angle of refraction becomes divergent. Second, the method fails to adequately account for point-source illumination conditions, particularly the influence of the finite X-ray source size on the angle of refraction. In this study, we demonstrate that a geometric optics-based method can effectively simulate propagation-based X-ray phase-contrast imaging with a low-brilliance X-ray source and compute the angle of refraction more accurately than the wave optics-based method. Our studies reveal that the geometric optics-based method can robustly determine the angle of refraction, even under conditions of substantial phase variations within the object. Furthermore, we show that reducing both the X-ray source size and the detector pixel size increases the angle of refraction in both simulations and experiments. Additionally, our results highlight that the angle of refraction is not invariant. Instead, it increases with the system’s total length and as the object moves closer to the light source. For systems with a Fresnel number of N ≥ 1, our method exhibits full compatibility with wave optics methods and can be extended to grating-based X-ray interferometry. The approach offers a robust alternative for calculating the angle of refraction under diverse imaging conditions.

Calculating the angle of refraction when X-rays pass through an object is a fundamental concern in X-ray phase-contrast imaging. When using wave optics to calculate the angle of refraction, several challenges arise. First, when the phase of the object changes significantly, the angle of refraction is divergent. Second, the method is unable to address the case of point-source illumination, specifically regarding the influence of the X-ray source's focal spot size on the angle of refraction. Here, a geometric optics based method is established to simulate the propagation-based X-ray phase-contrast imaging (pbXPCI) and calculate the angle of refraction. The results indicate that the angle of refraction can be effectively determined by geometric optics even when the phase of the object changes significantly, and reducing the focal spot size of the X-ray source and the pixel size of the detector will result in a larger angle of refraction. The results also indicate that the angle of refraction would vary with the object’s position and the system’s length, rather than remaining constant regardless of its location.

In this paper, we use the Saloff-Coste Sobolev inequality and Nash-Moser iteration method to study the local and global behaviors of positive solutions to the nonlinear elliptic equation ∆pu+∆qu+h(u) = 0 defined on a complete Riemannian manifold (M, g) with Ricci lower bound, where q ≥ p > 1 are constants and , with z ∈ {p, q}, is the usual z-Laplace operator. Under some assumptions on h(u), we derive gradient estimates and Liouville type theorems for positive solutions to the above equation. In particular, we show that, if an entire positive solution u to ∆pu+∆qu = 0 (1 < p ≤ q) on a complete non-compact Riemannian manifold M with non-negative Ricci curvature and dimM = n ≥ 3 satisfies for some x0 ∈ M, then u is a constant.

X-ray phase-contrast imaging presents a significant advancement in the field of X-ray imaging, surpassing traditional X-ray absorption imaging in detecting hydrogen substances. It effectively addresses the limitations of the latter in providing contrast for imaging weakly absorbing objects, thereby opening up vast potential applications in biomedical research, materials science, and industrial inspection. This article initially explores the fundamental principles of X-ray phase-contrast imaging and several prevalent imaging techniques. Notably, imaging devices such as grating-based Talbot–Lau interferometers emerge as the most promising in phase-contrast imaging due to their exceptional compatibility and imaging quality. Furthermore, this article introduces key parameters for assessing the quality of grating phase-contrast imaging, specifically image noise and sensitivity, along with their calculation methods. These insights are valuable for optimizing grating-based phase-contrast imaging devices. Lastly, this article examines potential applications and advancements in the key components of X-ray phase-contrast imaging while addressing current challenges and future directions in its technological development. This article aims to provide insights and inspiration for scholars interested in this field.

In the X-ray single-grating imaging system, the acquisition of frequency information is the key step of phase-contrast and scattering information recovery. In the process of information extraction, it is easy to lead to the degradation of imaging quality due to the Moire Artifact, thus limiting the development and application of X-ray single-grating imaging system. In order to address the above problems, in this article, based on the theoretical analysis of the generation principle of Moire Artifact in imaging system, the advantages and disadvantages of grating rotation method are analyzed, and a method of suppressing Moire artifacts by adjusting grating projection frequency is proposed. The experimental results show that the method proposed here can suppress the Moire noise in the background noise, resulting in a reduction of more than 50% in the standard deviation of the background noise. High quality phase-contrast and scattering images are obtained experimentally, which is of great value to the development of X-ray single-grating imaging technology.

Background. Major psychiatric disorders (MPDs) are delineated by distinct clinical features. However, overlapping symptoms and transdiagnostic effectiveness of medications have challenged the traditional diagnostic categorisation. We investigate if there are shared and illness specific disruptions in the regional functional efficiency (RFE) of the brain across these disorders. Methods. We included 364 participants (118 schizophrenia [SCZ], 80 bipolar disorder [BD], 91 major depressive disorder [MDD], and 75 healthy controls [HCs]). Resting-state fMRI was used to caclulate the RFE based on the static amplitude of low-frequency fluctuation, regional homogeneity, and degree centrality and corresponding dynamic measures indicating variability over time. We used principal component analysis to obtain static and dynamic RFE values. We conducted functional and genetic annotation and enrichment analysis based on abnormal RFE profiles. Results. SCZ showed higher static RFE in the cortico-striatal regions and excessive variability in the cortico-limbic regions. SCZ and MDD shared lower static RFE with higher dynamic RFE in sensorimotor regions than BD and HCs. We observed association between static RFE abnormalities with reward and sensorimotor functions and dynamic RFE abnormalities with sensorimotor functions. Differential spatial expression of genes related to glutamatergic synapse and calcium/cAMP signaling was more likely in the regions with aberrant RFE. Conclusions. SCZ shares more regions with disrupted functional integrity, especially in sensorimotor regions, with MDD rather than BD. The neural patterns of these transdiagnostic changes appear to be potentially driven by gene expression variations relating to glutamatergic synapses and calcium/cAMP signaling. The aberrant sensorimotor, cortico-striatal, and cortico-limbic integrity may collectively underlie neurobiological mechanisms of MPDs

We consider the generalized parabolic Cahn-H equation u1 = -A[Au - W'(u)] + W"(u)[Au - W'(u)], V(I,x) Acirc;XRd, where d = 1, 3, the function W(-) is the standard d well potential and the time region R:= J (0,00), ((-00,0), if d = 1, if d = 3. For d = 1 and any given integer k > 2, we const solution u(t, x) with k interfaces, which has the fo u(t, x) approximate to Sigma(-1)2+1w(x - y;(1) - {11+(-1)*] as! -> +00, where w is the solution to the Allen-Cahn equation omega"-W'(omega) 0 and omega w(y) > 0 in R, omega(0) = 0 omega(+/-infinity+ +/- 1. The dynamics of the interfaces are determined by a Toda system and the functions y,(t) with j = 1,..., k have the forms Int+C For d = 3 and any integer m>1, we construct an ancient radial solution of the form u(t, x) approximate to Sigma(-1)2+1@(x -p,(1)) - }{11+(-1)''] 7-1 where all p(1)(1),..., Pm (1) satisfy another Toda system and have the forms Pi(1) +0(1), 1=1,..., m. In particular, these Toda systems are totally different from those driving the multiple interfaces of solutions to the parabolic Allen-Cahn equation established by M. del Pino and K. Gkikas [Proc. R. Soc. Edinb. Sect. A, 148 (2018), 6: 1165-1199; and Ann. Inst. H. Poincar & eacute; Anal. Non Lin & eacute;aire 35 (2018), 1: 187-215].

We consider the nonlinear problem of anisotropic Fife-Greenlee equation epsilon(2)div(del(a(y))u) + (u-P(y))(1-u(2)) = 0 in Omega, del(a(y))u center dot nu= 0 on partial derivative Omega, where Omega is a bounded domain in R-2 with smooth boundary, epsilon is a small positiveparameter,nu denotes the unit outward normal of partial derivative Omega, an dP(y) is a smoothfunction on (Omega) over bar. The operator del(a(y)) u is defined by del(a(y))u = (a(1)(y)u(y1),a(2)(y)u(y2)) with a(y) = (a(1)(y),a(2)(y)), where a(1)(y) and a(2)(y) are two positive smooth functions on <(Omega)over bar. Let Gamma ={y is an element of Omega :P(y) = 0}be a simple smooth curve in Omega that intersects partial derivative Omega orthogonally at exactly two points and divide the domain Omega into two parts Omega(-) ={y is an element of Omega : P(y) > 0} and Omega(+) ={y is an element of Omega :P(y) < 0}. In addition,Upsilon ={y is an element of Omega(+):P(y) = -3}is a simple smooth curve that intersects partial derivative Omega orthogonally at exactly two points and divide the domain Omega(+) into two parts Omega(+,1) = {y is an element of Omega(+): 0>P(y) > -3} and Omega(+,2) = {y is an element of Omega(+): -3 > P(y)>-4}. By assuming some additional constraints on the functions a(y), P(y) as wellas the curves Gamma, Upsilon and partial derivative Omega, we construct a solution with coexistence of two interfaces approaching Gamma and Upsilon such that: as epsilon -> 0, u(epsilon) -> -1 in Omega(-), u(epsilon)-> +1 in Omega(+,1), u(epsilon) -> -3 in Omega(+,2). Some other solutions with interfaces will also be constructed in a similar way

We consider the generalized parabolic Cahn–Hilliard equation u_t=-Δ[ Δ u -W'(u)] +W”(u)[ Δ u -W'(u)] , for (t, x)∈ℝ×ℝ^n, where n=2 or n≥ 4 , W(· ) is the typical double-well potential function and ℝ is given by ℝ={[ (0, ∞ ), if n=2,; (-∞ , 0), if n≥ 4. ]. We construct a radial solution u ( t , x ) possessing an interface. At main order this solution consists of a traveling copy of the steady state ω (|x|) , which satisfies ω”(y)-W'(ω (y))=0 . Its interface is resemble at main order copy of the sphere of the following form |x|= 4 -2(n-3)(n-1)^2t, for (t, x)∈ℝ×ℝ^n, which is a solution to the Willmore flow in Differential Geometry. When n=1 or 3, the result consists of trivial interface solutions.

We consider the nonlinear problem of anisotropic Allen-Cahn equa-tion epsilon 2div(Va(y)u) +P(y)u(1 - u2) = 0 in ohm, Va(y)u center dot nu = 0 on partial differential ohm, where ohm is a bounded domain in R2 with smooth boundary, epsilon is a small positive parameter, nu denotes the unit outward normal of partial differential ohm, and P(y) is a uniformly positive smooth potential on ohm over bar . The operator Va(y)u is defined by Va(y)u = (a1(y)uy1 , a2(y)uy2) with a(y) = (a1(y), a2(y)), where a1(y) and a2(y) are two positive smooth functions on ohm over bar . Let Gamma be an interior curve intersecting orthogonally with partial differential ohm at exactly two points or a closed simple curve in ohm, and dividing ohm into two parts. Moreover, Gamma is a non-degenerate geodesic embedded in the Riemannian manifold R2 associated with metric P(y)(a2(y)dy1 (R) dy1 + a1(y)dy2 (R) dy2). By assuming some additional constraints on the functions a(y), P(y) and the curves Gamma, partial differential ohm, we prove that there exists a solution u epsilon with an interface such that: as epsilon -> 0, u epsilon approaches +1 in one part of ohm, while tends to -1 in the other part, except a small neighborhood of Gamma.

We consider one dimensional generalized parabolic Cahn-Hilliard equation $$ u_t=-\partial_{xx}\big[\partial_{xx}u-W'(u)\big]+W''(u)\big[\partial_{xx} u -W'(u)\big], \qquad \forall\, (t,x)\in [0,+\infty)\times {\mathbb R}, $$ where the function $W(\cdot)$ is the standard double-well potential. For any given positive integer $k\geq2$, we construct a solution $u(t,x)$ with $k$ interfaces, which has the form $$ u(t,x)\approx\sum_{j=1}^k(-1)^{j+1}\omega\big(x-\gamma_j(t)\big)-\frac{1+(-1)^k}{2}\qquad \text{as}\ t\rightarrow +\infty, $$ where $\omega$ is the solution to the Allen-Cahn equation $$ \omega''-W'(\omega)=0,\quad\omega'>0\quad\mbox{in }{\mathbb R}, \quad \omega(0)=0, \quad \omega(\pm\infty)=\pm 1. $$ The interfaces are described by the functions $\gamma_j(t)$ with $j=1,\cdots,k$, which are determined by a Toda system and have the forms $$ \gamma_j(t)=\frac{1}{2\sqrt{2}}\left(j-\frac{k+1}{2}\right)\ln t +O(1). $$ The Toda system is different from the one that determine the dynamics of the multiple interfaces of solutions to one dimensional parabolic Allen-Cahn equation established by M. del Pino and K. Gkikas in {\em Proc. R. Soc. Edinb. Sect. A}, 148 (2018), 6: 1165-1199.

Applying different micro-bits with different sizes to carry out step-by-step drilling is a common technique to obtain micro-stepped holes. However, the possibility of repeated misalignments in the clamping of micro-bits will affect the machining accuracy of the micro-stepped holes. Furthermore, for difficult processing materials such as die steel and cemented carbide, it is difficult to obtain ideal machining results through drilling. Under this background, micro-stepped hole was processed on die steel based on the electrical discharge machining (EDM) wear of micro-bits. Firstly, a high-speed rotating micro-bit was used to perform micro-EDM on die steel to obtain a basic micro-hole. During the micro-EDM process, the micro-bit was worn and thus its length was reduced. Secondly, the worn micro-bit shook along a defined axis to carry out micro-EDM on the die steel workpiece, thus machining micro-stepped hole on the basic micro-hole. This method avoided the clamping error of micro-bit found in other techniques and improved machining accuracy of micro-stepped hole. Through the analysis of variance (ANOVA) of processing parameters and the establishment of surface response model, the appropriate processing parameters were selected. Finally, under the action of appropriate processing parameters, micro-stepped hole was obtained by the proposed method.

. We consider the following coupled Ginzburg-Landau system in R 3 where w = ( w + ,w − ) ∈ C 2 and the constant coefficients satisfy If B < 0, then for every ǫ small enough, we construct a family of entire solutions w ǫ (˜ z,t ) ∈ C 2 in the cylindrical coordinates (˜ z,t ) ∈ R 2 × R for this system via the approach introduced by J. D´avila, M. del Pino, M. Medina and R. Rodiac in arXiv:1901.02807 . These solutions are 2 π -periodic in t and have multiple interacting vortex helices. The main results are the extensions of the phenomena of interacting helical vortex filaments for the classical (single) Ginzburg-Landau equation in R 3 which has been studied in arXiv:1901.02807 . Our results negatively answer the Gibbons conjecture [10] for the Allen-Cahn equation in Ginzburg-Landau system version, which is an extension of the question originally proposed by H. Brezis.

When there is mechanical drift in X-ray phase contrast imaging system, the position of the grating will produce random error, and the intensity of the obtained image will have a certain deviation, and the information retrieved by using the phase step method may be accompanied with Moire artifacts. In order to overcome this limitation, we introduce the convolutional neural network (CNN) to address it. The training data is downloaded from Kaggle, and the fringe graph with random deviation is combined as the network input, while the label is defined as the first-order difference image along the horizontal direction of the image. Both simulation and experiment show that CNN can not only retrieve the phase signal of the sample, but also remove some Moire artifacts with regular shape to improve the image quality. As a result, the utilization rate of X-ray in imaging system can be improved.

BACKGROUND:Suicidal ideation (SI) is a common symptom of major depressive disorder (MDD). Accumulating studies demonstrated that MDD with SI was associated with static alterations in brain activity and functional connectivity. However, given that brain is a highly dynamic system, the changes of brain dynamic patterns in MDD with SI remain unknown.METHODS:We included 60 MDD patients with SI (MDD-SI), 58 MDD patients without SI (MDD-NSI), and 58 healthy controls (HCs) who underwent resting-state functional magnetic resonance imaging. The sliding-window approach was used to calculate the dynamic fractional amplitude of low-frequency fluctuation (dfALFF) and dynamic degree centrality (dDC) to characterize the temporal dynamic regional activity and distant functional connectivity. We compared dfALFF and dDC across groups and further conducted correlations between abnormal dynamic metrics and the severity of suicidality.RESULTS:In terms of the dynamic regional activity, MDD-SI showed decreased dfALFF in the left lingual gyrus and right middle occipital gyrus compared with MDD-NSI; in terms of the dynamic distant connectivity, MDD-SI showed decreased dDC in the right middle frontal gyrus compared with MDD-NSI. The decreased dDC in the right middle frontal gyrus was correlated with increased severity of suicidality.LIMITATIONS:The relatively small sample size.CONCLUSIONS:We demonstrate the specific brain dynamic patterns of MDD-SI in regional activity and distant functional connectivity compared to MDD-NSI. Especially the decreased temporal variability of the distant connectivity in the middle frontal gyrus was associated with SI. These altered dynamic patterns may represent a potential neurobiological diathesis of SI in MDD.

In this paper, we propose a new method to prepare X-ray absorption gratings. The thermal composite absorption gratings with periods of 80 μm and 140 μm are successfully fabricated by using the difference in X-ray absorption between aluminum and silver film. The production process and the use of equipment are simple and easy to implement. A number of absorption gratings can be produced in one production, which greatly reduces the cost of gratings. Finally, the X-ray single absorption grating imaging experiment is used to verify the effectiveness of the thermal composite absorption grating.