Search

The sunflower equation describes the motion of the tip of a plant due to the auxin transportation under the influence of gravity. This work proposes the fractional-order generalization to this delay differential equation. The equation contains two fractional orders and infinitely many equilibrium points. The problem is important because the coefficients in the linearized equation near the equilibrium points are delay-dependent. We provide a detailed stability analysis of each equilibrium point using linearized stability. We find the boundary of the stable region by setting the purely imaginary value to the characteristic root. This gives the conditions for the existence of the critical values of the delay at which the stability properties change. We observed the following bifurcation phenomena: stable for all the delay values, a single stable region in the delayed interval, and a stability switch. We also observed a multi-scroll chaotic attractor for some values of the parameters., Comment: 15 pages, 26 figures

This manuscript deals with the stability and bifurcation analysis of the equation $D^{2\alpha}x(t)+c D^{\alpha}x(t)=a x(t)+b x(t-\tau)$, where $0<\alpha<1$ and $\tau>0$. We sketch the boundaries of various stability regions in the parameter plane under different conditions on $\alpha$ and $b$. First, we provide the stability analysis of this equation with $\tau=0$. Change in the stability of the delayed counterpart is possible only when the characteristic roots cross the imaginary axis. This leads to various delay-independent as well as delay-dependent stability results. The stability regions are bifurcated on the basis of the following behaviors with respect to the delay $\tau$ viz. stable region for all $\tau>0$, unstable region, single stable region, stability switch, and instability switch., Comment: 55 pages, 100 figures

We examine five machine learning-based architectures to estimate the droplet size distributions obtained using digital inline holography. The architectures, namely, U-Net, R2 U-Net, Attention U-Net, V-Net, and Residual U-Net are trained using synthetic holographic images. Our assessment focuses on evaluating the training, validation, and prediction performance of these architectures. We found that U-Net and R2 U-Net to be the most proficient, displaying consistent performance trends and achieving the highest Intersection Over Union (IOU) scores compared to the other three architectures. We employ additional training using experimental holographic images for the two top-performing architectures to validate their efficacy further. Subsequently, they are employed to segment an experimental dataset illustrating the bag breakup phenomenon, facilitating the extraction of size distribution. The extracted size distribution from U-Net and R2 U-Net segmentation is then compared with the analytical model proposed by \cite{jackiw2022prediction} by employing the gamma and log-normal distributions. Our findings indicate that the gamma distribution provides a more accurate prediction of the multi-modal size distribution than the log-normal distribution owing to its long exponential tail. The present study offers valuable insights into the effectiveness of machine learning architectures in estimating particle/droplet sizes, highlighting their practical application in real-world experimental scenarios., Comment: 36 pages, 11 figures, International Journal of Multiphase Flow

Recent studies have demonstrated promising performance of ChatGPT and GPT-4 on several medical domain tasks. However, none have assessed its performance using a large-scale real-world electronic health record database, nor have evaluated its utility in providing clinical diagnostic assistance for patients across a full range of disease presentation. We performed two analyses using ChatGPT and GPT-4, one to identify patients with specific medical diagnoses using a real-world large electronic health record database and the other, in providing diagnostic assistance to healthcare workers in the prospective evaluation of hypothetical patients. Our results show that GPT-4 across disease classification tasks with chain of thought and few-shot prompting can achieve performance as high as 96% F1 scores. For patient assessment, GPT-4 can accurately diagnose three out of four times. However, there were mentions of factually incorrect statements, overlooking crucial medical findings, recommendations for unnecessary investigations and overtreatment. These issues coupled with privacy concerns, make these models currently inadequate for real world clinical use. However, limited data and time needed for prompt engineering in comparison to configuration of conventional machine learning workflows highlight their potential for scalability across healthcare applications., Comment: This manuscript is pre-print and in peer review. Supplementary materials will be published later

For the fractional order systems \[D^\alpha x(t)=f(x),\quad 0<\alpha\leq 1,\] one can have a critical value of $\alpha$ viz $\alpha_*$ such that the system is stable for $0<\alpha<\alpha_*$ and unstable for $\alpha_*<\alpha\leq 1$. In general, if such system is stable for some $\alpha_0\in(0,1)$ then it remains stable for all $\alpha<\alpha_0.$ In this paper, we show that there are some delay differential equations \[D^\alpha x(t)=f(x(t),x(t-\tau))\] of the fractional order which behave in an exactly opposite way. These systems are unstable for higher values of fractional order and stable for the lower values. The striking observation is the example which is chaotic for $\alpha=0.27$ but stable for $\alpha=1$. This cannot be observed in the fractional differential equations (FDEs) without delay. We provide the complete bifurcation scenarios in the scalar FDEs., Comment: 20 pages, 18 figures

Fractional derivative and delay are important tools in modeling memory properties in the natural system. This work deals with the stability analysis of a fractional order delay differential equation \begin{equation*} D^\alpha x(t)=\delta x(t-\tau)-\epsilon x(t-\tau)^3-px(t)^2+q x(t). \end{equation*} We provide linearization of this system in a neighbourhood of equilibrium points and propose linearized stability conditions. To discuss the stability of equilibrium points, we propose various conditions on the parameters $\delta$, $\epsilon$, $p$, $q$ and $\tau$. Even though there are five parameters involved in the system, we are able to provide the stable region sketch in the $q\delta-$plane for any positive $\epsilon$ and $p$. This provides the complete analysis of stability of the system. Further, we investigate chaos in the proposed model. This system exhibits chaos for a wide range of delay parameter., Comment: 25 pages, 19 figures, 1 table