TIFR Centre for Applicable Mathematics

Colorectal cancer is a leading cause of cancer-related mortality; early, accurate histopathological assessment is pivotal. We present a two-stage framework—GR-HDUNET plus a transformer ensemble—that unifies boundary-aware segmentation with multi-model classification. The segmentation stage combines a DenseNet encoder, multi-scale attention, residual decoding, and cGAN refinement to produce anatomically faithful masks, achieving class-average Dice = 0.895, Boundary F1 ≈ 0.82, HOG-similarity ≈ 0.92 on EBHI-Seg. These masks guide a three-branch classifier (Vision Transformer, Swin Transformer, ConvNeXt) whose predictions are fused by weighted averaging. On the six-class EBHI-Seg dataset, weighted averaging attains 95.1% accuracy, 94.2% F1, and micro-AUC = 0.95, surpassing the best single backbone by ≥1.0 percentage point. Qualitative overlays confirm crisp gland delineation, and ROC curves show clear class separation. Importantly, segmentation is trained with pixel-level masks, while classification uses slide-level labels. The pipeline is computationally tractable (≤1.6 s per 224×224 image on a single RTX 4090) and readily reproducible. By coupling cGAN-refined, boundary-aware segmentation with complementary transformer features, our approach delivers robust colorectal-tissue segmentation and classification on EBHI-Seg and offers a transferable template for digital pathology.

Abstract In this article, we investigate necessary and sufficient conditions on the perturbation ρ for the existence of positive least energy solutions of the critical singular semilinear elliptic equation $ -\Delta u = \frac{|u|^{2^{*}(s)-2}}{|x|^s}u + \rho(u) $ with Dirichlet boundary condition in a bounded smooth domain in $\mathbb R^n$ containing the origin, where $2^*(s)=\frac{2(n-s)}{n-2}$ , $0\leq s \lt 2 \lt n$ . We show that the almost necessary and sufficient condition obtained for the case s = 0 in [1] differs conceptually when $0 \lt s \lt 2$ .

Using the AI approaches and the quadratic programming driven multiobjective optimisation, this paper proposes a universal framework for smart microgrid energy management. It considers its operational preconditions such as the loads demand and the renewable power source availability in order to reduce the microgrid operational cost and emission. An artificial neural network constructed models the predicted loads requirement, one-hours wind power output and solar generation for 24 hours. The simulated machine learning possesses good generalisation capacity and an excellent learning structure. Managing batteries or auxiliary devices in order to maximise microgrid operating efficiency runs counter to the traditional optimisation laws. This study used fuzzy logic advisor network to schedule the battery. Both the fuzzy environment of the microgrid operation as a whole and the uncertainty among the predicted parameters are manageable in the suggested solution. In comparison with microgrid energy management methods acquired from the literature, namely opportunity recharging and inductive schematic alternating battery management, the experimental results yield significant operational cost and emission reduction.

This paper deals with the problem of identification of a Robin coefficient (also known as impedance coefficient) in a parabolic PDE from terminal observations of the temperature distributions. The problem is ill-posed in the sense that small perturbation in the observation may lead to a large deviation in the solution. Thus, in order to obtain stable approximations, we employ the Tikhonov-regularization. We propose a weak source condition motivated by the work of Engl and Zou (2000) and obtain a convergence rate of O(δ12), where δ is the noise level of the observed data. The obtained rate is better than some of the previous known rates. Moreover, the advantage of the proposed source condition is that the above mentioned convergence rate is obtained without the need for characterizing the range space of modelling operator and its Fréchet derivative, which is in contrast to the general convergence theory of Tikhonov-regularization for non linear operators.

This paper is concerned with recovering the solution of a final value problem associated with a parabolic equation involving a non linear source and a non-local term, which to the best of our knowledge has not been studied earlier. It is shown that the considered problem is ill-posed, and thus, some regularization method has to be employed in order to obtain stable approximations. In this regard, we obtain regularized approximations by solving some non linear integral equations which is derived by considering a truncated version of the Fourier expansion of the sought solution. Under different Gevrey smoothness assumptions on the exact solution, we provide parameter choice strategies and obtain the error estimates. A key tool in deriving such estimates is a version of Grönwalls’ inequality for iterated integrals, which perhaps, is proposed and analysed for the first time.

This paper reports the preparation of ZnFe2O4/ZnO metal oxide nanocomposite by solution combustion synthesis. Zinc nitrate and ferric nitrate were utilized as oxidizers in this work, while glycine served as fuel. The powder X-ray diffraction pattern indicated that the nanocomposite consisted of ZnFe2O4 and ZnO having spinel and wurtzite phases respectively. The efficiency of the nanocomposite in the photocatalytic degradation of Congo Red (CR) dye from its aqueous solution was studied. The effects of CR starting concentration, photocatalyst dose, and irradiation duration were investigated. More than 90 % degradation of 10 ppm CR solution was achieved for a photocatalyst dosage of 1.0g/litre of the dye solution in 40 minutes. The excellent photocatalytic activity of the nanocomposite was considered as the result of the synergistic mechanism between its constituent phases, significantly reducing electron-hole recombination. It was noticed that the photocatalyst after the first regeneration was about 85 % efficient compared to the original one. The water contamination with human activity can be reduced by the usage of ZnFe2O4/ZnO metal oxide nanoparticles.

This paper is concerned with identification of a spatial source function from final time observation in a bi-parabolic equation, where the full source function is assumed to be a product of time dependent and a space dependent function. Due to the ill-posedness of the problem, recently some authors have employed different regularization method and analysed the convergence rates. But, to the best of our knowledge, the quasi-reversibility method is not explored yet, and thus we study that in this paper. As an important implication, the Hölder rates for the apriori and aposteriori error estimates obtained in this paper improve upon the rates obtained in earlier works. Also, in some cases we show that the rates obtained are of optimal order. Further, this work seems to be the first one that has broaden the applicability of the problem by allowing the time dependent component of the source function to change sign. To the best of our knowledge, the earlier known work assumed the fixed sign of the time dependent component by assuming some bounded below condition.