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The lack of an automated online platform for reporting citizens' complaints, coupled with the city corporations' struggles in managing them, presents significant challenges. Furthermore, the availability of resources is very limited to higher authorities for monitoring progress. The primary objective of this paper is to develop two Android applications and to categorize complaints automatically using a deep learning model created on the Teachable Machine. With the citizen-oriented application, individuals can easily report complaints by capturing pictures of their municipal issues. The authority version of the application provides categorized complaints, along with location and status details. Higher authorities can monitor the municipal progress, thereby enhancing transparency, and efficiency and promoting smart city development on a nationwide scale.

We exhibit various restrictions about the wellposedness of the Schr\"odinger product $$\mathcal{L}: z \longmapsto -\imath \int_0^t e^{\imath \Delta s}\big( z_s\cdot \Psi_s\big) ds $$ where $\Psi$ refers to the so-called linear solution of the stochastic Schr\"dinger problem. We focus more specifically on the case where $\Psi$ satisfies $$(\imath \partial_t-\Delta)\Psi=\dot{B}, \quad \Psi_0=0,\quad \quad t\in \mathbb{R}, \ x\in \mathbb{T},$$where $\dot{B}$ is a white noise in space with fractional time covariance of index $H>\frac12$. As an important consequence of our analysis, we obtain that if $H$ is close to $\frac12$ (that is $\dot{B}$ is close to a space-time white noise), then it is essentially impossible to treat the stochastic NLS problem $$(\imath \partial_t-\Delta)u=\lambda\, u^{p} \overline{u}^q+\dot{B}, \quad u_0=0,\quad \quad \lambda \in \{-1,1\}, \ p,q\geq 1, \ t\in \mathbb{R}, \ x\in \mathbb{T},$$using only a first-order expansion of the solution ("$u=\Psi+z$").

The study is devoted to the interpretation and wellposedness of a stochastic NLS model with a quadratic nonlinearity and a space-time fractional noise. We focus on a rough regime corresponding to the case where the Hurst indexes of the fractional noise are small. In this specific case, we exhibit an explicit renormalization procedure allowing to restore the (local) convergence of the approximated solutions. We follow a pathwise-type approach emphasizing the distinction between the stochastic objects at the core of the dynamics and the general deterministic machinery., Comment: 56 pages

This proposes a novel ensemble deep learning-based model to accurately classify, detect and localize different defect categories for aggressive pitches and thin resists (High NA applications).In particular, we train RetinaNet models using different ResNet, VGGNet architectures as backbone and present the comparison between the accuracies of these models and their performance analysis on SEM images with different types of defect patterns such as bridge, break and line collapses. Finally, we propose a preference-based ensemble strategy to combine the output predictions from different models in order to achieve better performance on classification and detection of defects. As CDSEM images inherently contain a significant level of noise, detailed feature information is often shadowed by noise. For certain resist profiles, the challenge is also to differentiate between a microbridge, footing, break, and zones of probable breaks. Therefore, we have applied an unsupervised machine learning model to denoise the SEM images to remove the False-Positive defects and optimize the effect of stochastic noise on structured pixels for better metrology and enhanced defect inspection. We repeated the defect inspection step with the same trained model and performed a comparative analysis for "robustness" and "accuracy" metric with conventional approach for both noisy/denoised image pair. The proposed ensemble method demonstrates improvement of the average precision metric (mAP) of the most difficult defect classes. In this work we have developed a novel robust supervised deep learning training scheme to accurately classify as well as localize different defect types in SEM images with high degree of accuracy. Our proposed approach demonstrates its effectiveness both quantitatively and qualitatively.

We study a wave equation in dimension $d\in \{1,2\}$ with a multiplicative space-time Gaussian noise. The existence and uniqueness of the Stratonovich solution is obtained under some conditions imposed on the Gaussian noise. The strategy is to develop some Strichartz type estimates for the wave kernel in weighted Besov spaces, by which we can prove the wellposedness of an associated Young-type equation. Those Strichartz bounds are of independent interest., Comment: 34 pages

We highlight a fundamental ill-posedness issue for nonlinear stochastic wave equations driven by a fractional noise. Namely, if the noise becomes too rough (i.e., the sum of its Hurst indexes becomes too small), then there is essentially no hope to provide an interpretation of the model, whether directly or through a Wick-type renormalization procedure. This phenomenon can be compared with the situation of a general SDE driven by a two-dimensional fractional noise of index $H\leq \frac{1}{4}$. Our results clarify and extend previous similar properties exhibited in [3] or in [14].

This paper is concerned with a wave equation in dimension $d\in \{1,2, 3\}$, with a multiplicative space-time Gaussian noise which is fractional in time and homogeneous in space. We provide necessary and sufficient conditions on the space-time covariance of the Gaussian noise, allowing the existence and uniqueness of a mild Skorohod solution.

We study a full discretization scheme for the stochastic linear heat equation \begin{equation*}\begin{cases}\partial_t \langle\Psi\rangle = \Delta \langle\Psi\rangle +\dot{B}\, , \quad t\in [0,1], \ x\in \mathbb{R},\\ \langle\Psi\rangle_0=0\, ,\end{cases}\end{equation*} when $\dot{B}$ is a very \emph{rough space-time fractional noise}. The discretization procedure is divised into three steps: $(i)$ regularization of the noise through a mollifying-type approach; $(ii)$ discretization of the (smoothened) noise as a finite sum of Gaussian variables over rectangles in $[0,1]\times \mathbb{R}$; $(iii)$ discretization of the heat operator on the (non-compact) domain $[0,1]\times \mathbb{R}$, along the principles of Galerkin finite elements method. We establish the convergence of the resulting approximation to $\langle\Psi\rangle$, which, in such a specific rough framework, can only hold in a space of distributions. We also provide some partial simulations of the algorithm.

We construct a $K$-rough path above either a space-time or a spatial fractional Brownian motion, in any space dimension $d$. This allows us to provide an interpretation and a unique solution for the corresponding parabolic Anderson model, understood in the renormalized sense. We also consider the case of a spatial fractional noise., Comment: This preprint is the result of splitting our original submission arXiv:2003.14367, which was slightly too long. The present preprint contains the construction of the Stratonovich solution

We study a stochastic Schr{\"o}dinger equation with a quadratic nonlinearity and a space-time fractional perturbation, in space dimension less than 3. When the Hurst index is large enough, we prove local well-posedness of the problem using classical arguments. However, for a small Hurst index, even the interpretation of the equation needs some care. In this case, a renormalization procedure must come into the picture, leading to a Wick-type interpretation of the model. Our fixed-point argument then involves some specific regularization properties of the Schr{\"o}dinger group, which allows us to cope with the strong irregularity of the solution., Comment: 43 pages

The theory of regularity structures enables the definition of the following parabolic Anderson model in a very rough environment: $\partial_{t} u_{t}(x) = \frac12 \Delta u_{t}(x) + u_{t}(x) \, \dot W_{t}(x)$, for $t\in\mathbb{R}_{+}$ and $x\in \mathbb{R}^{d}$, where $\dot W_{t}(x)$ is a Gaussian noise whose space time covariance function is singular. In this rough context, we shall give some information about the moments of $u_{t}(x)$ when the stochastic heat equation is interpreted in the Skorohod as well as the Stratonovich sense. Of special interest is the critical case, for which one observes a blowup of moments for large times., Comment: The paper has been split. The construction of Stratonovich solution has been removed from this version

We pursue our investigations, initiated in [8], about stochastic integration with respect to the non-commutative fractional Brownian motion (NC-fBm). Our main objective in this paper is to compare the pathwise constructions of [8] with a Skorohod-type interpretation of the integral. As a first step, we provide details on the basic tools and properties associated with non-commutative Malliavin calculus, by mimicking the presentation of Nualart's celebrated treatise [14]. Then we check that, just as in the classical (commutative) situation, Skorohod integration can indeed be considered in the presence of the NC-fBm, at least for a Hurst index H > 1 4.This finally puts us in a position to state and prove the desired comparison result, which can be regarded as an It{\^o}-Stratonovich correction formula for the NC-fBm., Comment: ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matem\'atica Pura e Aplicada, In press

For every $d\geq 1$, we consider the $d$-dimensional Hermitian fractional Brownian motion (HfBm), that is the process with values in the space of $(d\times d)$-Hermitian matrices and with upper-diagonal entries given by complex fractional Brownian motions of Hurst index $H\in (0,1)$. We follow the approach of [A. Deya and R. Schott: On the rough paths approach to non-commutative stochastic calculus, JFA (2013)] to define a natural integral with respect to the HfBm when $H>\frac13$, and identify this interpretation with the rough integral with respect to the $d^2$ entries of the matrix. Using this correspondence, we establish a convenient It{\^o}--Stratonovich formula for the Hermitian Brownian motion. Finally, we show that at least when $H\geq \frac12$, and as the size $d$ of the matrix tends to infinity, the integral with respect to the HfBm converges (in the tracial sense) to the integral with respect to the so-called non-commutative fractional Brownian motion.