Abstract In this article, we consider a weakly coupled <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p -Laplacian system of Bernoulli-type free boundary problem, through minimization corresponding functional. We prove various properties any local minimizer and the boundary.

We study the null controllability properties of somedegenerate/singular parabolic equations in a bounded interval ofR. For this reason we derive new Carleman estimatewhose proof is based on Hardy inequalities.

Abstract Given is a bounded domain <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:math> {\Omega\subset\mathbb{R}^{n}} , and vector-valued function defined on <m:mo>∂</m:mo> <m:mo>⁡</m:mo> {\partial\Omega} (depicting temperature distributions from different sources), our objective to study the mathematical model of physical problem enclosing with specific volume...

In this paper we consider a weakly coupled $p$-Laplacian system of Bernoulli type free boundary problem, through minimization corresponding functional. We prove various properties any local minimizer and the boundary.

Abstract In this paper we study the following parabolic system $$\begin{aligned} \Delta \mathbf{u }-\partial _t }=|\mathbf{u }|^{q-1}\mathbf{u }\,\chi _{\{ |\mathbf{u }|&gt;0 \}}, \qquad }= (u^1, \cdots , u^m) \ \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msup> <mml:mo>|</mml:mo>...

Abstract This paper presents a mathematical framework for the prognosis of glioblastoma brain tumor growth on patient-specific basis, employing heterogeneous image-driven methodology. The approach utilizes reaction-diffusion model to capture diffusion and proliferation dynamics cell density, integrated with an inverse problem based PDE-constrained formulation that links medical images. We establish theoretical forms robust foundation our proposed Then numerical algorithm is introduced...

The existing algorithms for identification of neurons responsible undesired and harmful behaviors do not consider the effects confounders such as topic conversation. In this work, we show that can create spurious correlations propose a new causal mediation approach controls impact topic. experiments with two large language models, study localization hypothesis adjusting effect conversation topic, toxicity becomes less localized.

We investigate the use of in-context learning and prompt engineering to estimate contributions training data in outputs instruction-tuned large language models (LLMs). propose two novel approaches: (1) a similarity-based approach that measures difference between LLM with without provided context, (2) mixture distribution model frames problem identifying contribution scores as matrix factorization task. Our empirical comparison demonstrates is more robust retrieval noise learning, providing...

Abstract. In this work we address graph based semi-supervised learning using the theory of spatial segregation competitive systems. First, define a discrete counterpart over connected graphs by direct analogue corresponding system. This model turns out doesn’t have unique solution as expected. Nevertheless, suggest gradient projected and regularization methods to reach some solutions. Then focus on slightly different motivated from recent numerical results reaction-diffusion case show that...