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1. Gravitational Collapse for Polytropic Gaseous Stars: Self-Similar Solutions

   https://doi.org/10.1007/s00205-022-01827-8  

   Guo, Yan; Hadžić, Mahir; Jang, Juhi; Schrecker, Matthew (November 2022, Archive for Rational Mechanics and Analysis)

   Abstract In the supercritical range of the polytropic indices$$\gamma \in (1,\frac{4}{3})$$ $\gamma \in (1,\frac{4}{3})$we show the existence of smooth radially symmetric self-similar solutions to the gravitational Euler–Poisson system. These solutions exhibit gravitational collapse in the sense that the density blows up in finite time. Some of these solutions were numerically found by Yahil in 1983 and they can be thought of as polytropic analogues of the Larson–Penston collapsing solutions in the isothermal case$$\gamma =1$$ $\gamma =1$. They each contain a sonic point, which leads to numerous mathematical difficulties in the existence proof. 

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2. Sparse representation of vectors in lattices and semigroups

   https://doi.org/10.1007/s10107-021-01657-8  

   Aliev, Iskander; Averkov, Gennadiy; De_Loera, Jesús_A; Oertel, Timm (May 2021, Mathematical Programming)

   Abstract We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the$$\ell _0$$ ${\ell }_0$-norm of the vector. Our main results are new improved bounds on the minimal$$\ell _0$$ ${\ell }_0$-norm of solutions to systems$$A\varvec{x}=\varvec{b}$$ $Ax=b$, where$$A\in \mathbb {Z}^{m\times n}$$ $A\in Z^{m\times n}$,$${\varvec{b}}\in \mathbb {Z}^m$$ $b\in Z^m$and$$\varvec{x}$$ $x$is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In certain cases, we give polynomial time algorithms for computing solutions with$$\ell _0$$ ${\ell }_0$-norm satisfying the obtained bounds. We show that our bounds are tight. Our bounds can be seen as functions naturally generalizing the rank of a matrix over$$\mathbb {R}$$ $R$, to other subdomains such as$$\mathbb {Z}$$ $Z$. We show that these new rank-like functions are all NP-hard to compute in general, but polynomial-time computable for fixed number of variables. 

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3. Global existence and singularity formation for the generalized Constantin–Lax–Majda equation with dissipation: the real line vs. periodic domains

   https://doi.org/10.1088/1361-6544/ad140c  

   Ambrose, David M; Lushnikov, Pavel M; Siegel, Michael; Silantyev, Denis A (December 2023, Nonlinearity)

   Abstract The question of global existence versus finite-time singularity formation is considered for the generalized Constantin–Lax–Majda equation with dissipation $-{\mathrm{\Lambda }}^{\sigma }$, where $\widehat{{\mathrm{\Lambda }}^{\sigma }}=|k{|}^{\sigma }$, both for the problem on the circle $x\in [-\pi ,\pi ]$and the real line. In the periodic geometry, two complementary approaches are used to prove global-in-time existence of solutions for $\sigma \text{⩾}1$and all real values of an advection parameterawhen the data is small. We also derive new analytical solutions in both geometries whena = 0, and on the real line when $a=1/2$, for various values ofσ. These solutions exhibit self-similar finite-time singularity formation, and the similarity exponents and conditions for singularity formation are fully characterized. We revisit an analytical solution on the real line due to Schochet fora = 0 andσ = 2, and reinterpret it terms of self-similar finite-time collapse. The analytical solutions on the real line allow finite-time singularity formation for arbitrarily small data, even for values ofσthat are greater than or equal to one, thereby illustrating a critical difference between the problems on the real line and the circle. The analysis is complemented by accurate numerical simulations, which are able to track the formation and motion of singularities in the complex plane. The computations validate and build upon the analytical theory. 

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4. Arctic curves of the T-system with slanted initial data

   https://doi.org/10.1088/1751-8121/ad65a5  

   Di_Francesco, Philippe; Vu, Hieu Trung (July 2024, Journal of Physics A: Mathematical and Theoretical)

   Abstract We study theT-system of type $A_{\mathrm{\infty }}$, also known as the octahedron recurrence/equation, viewed as a $2+1$-dimensional discrete evolution equation. Generalizing earlier work on arctic curves for the Aztec Diamond obtained from solutions of the octahedron recurrence with ‘flat’ initial data, we consider initial data along parallel ‘slanted’ planes perpendicular to an arbitrary admissible direction $(r,s,t)\in {\mathbb{Z}}_{+}^3$. The corresponding solutions of theT-system are interpreted as partition functions of dimer models on some suitable ‘pinecone’ graphs introduced by Bousquet–Mélou, Propp, and West in 2009. TheT-system formulation and some exact solutions in uniform or periodic cases allow us to explore the thermodynamic limit of the corresponding dimer models and to derive exact arctic curves separating the various phases of the system. This direct approach bypasses the standard general theory of dimers using the Kasteleyn matrix approach and uses instead the theory of Analytic Combinatorics in Several Variables, by focusing on a linear system obeyed by the dimer density generating function. 

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5. Regularity Criteria for the Kuramoto–Sivashinsky Equation in Dimensions Two and Three

   https://doi.org/10.1007/s00332-022-09828-3  

   Larios, Adam; Rahman, Mohammad Mahabubur; Yamazaki, Kazuo (September 2022, Journal of Nonlinear Science)

   Abstract We propose and prove several regularity criteria for the 2D and 3D Kuramoto–Sivashinsky equation, in both its scalar and vector forms. In particular, we examine integrability criteria for the regularity of solutions in terms of the scalar solution$$\phi $$ $\varphi$, the vector solution$$u\triangleq \nabla \phi $$ $u\triangleq \nabla \varphi$, as well as the divergence$$\text {div}(u)=\Delta \phi $$ $\text{div}\left(u\right)=\Delta \varphi$, and each component ofuand$$\nabla u$$ $\nabla u$. We also investigate these criteria computationally in the 2D case, and we include snapshots of solutions for several quantities of interest that arise in energy estimates. 

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