Papers

In this paper, we present a structure-preserving rotational discrete gradient method of the full Ericksen-Leslie model for nematic liquid crystal flows with general anisotropic Oseen-Frank elasticity and full Leslie stress. The main difficulty resides in preserving both the pointwise unit-length constraint on the director field and the energy-dissipation at the discrete level, which involves high nonlinearities in the interplay between the length constraint, anisotropic elastic energy, and hydrodynamic couplings. To address this issue, we first reformulate the Ericksen-Leslie system into an equivalent rotational form in which the director evolution is intrinsically tangent to the unit sphere and the coupling terms in the Leslie and Ericksen stresses are reorganized so that the energy exchange and dissipation structure become explicit. Based on this reformulation, we construct rotational discrete gradient schemes that preserve the director length and satisfy an unconditional discrete energy law, together with a fully discrete scheme based on an exact divergence-free spectral approximation. Numerical experiments verify the accuracy, unit-length preservation and energy-stability of the proposed method, and illustrate dynamical effects induced by anisotropic elastic coefficients and shear flow in the full Ericksen-Leslie model.

The paper studies the Q-tensor model for nematic liquid crystals, a system that couples a Navier-Stokes equation with an evolution equation for the order parameter tensor Q. The first goal of the paper is to establish the local well-posedness of the system in $\mathbb R^N$ and $\mathbb R^N_+$ for $N=2,3$ in the $L^2$ framework, improving existing results in the literature, where the existence of local strong solutions was obtained only under smallness assumptions on the initial data. Fundamental is the application of the energy method, which shows a cancellation phenomena on the nonlinear terms, allowing the use of a contraction argument to prove existence and uniqueness of solutions. Finally, with the same approach we establish global well-posedness in the three-dimensional case for small initial data.

We consider complex-weighted graphs, whose edge weights belong to a sector in the complex plane. We show that the corresponding Dirichlet Laplacian is $m$-sectorial, and, hence, generates a contractive holomorphic $C_0$-semigroup. Further, it is shown that every sectorial complex-weighted graph can be extended to an electrical network, where by electrical networks we mean graphs, whose edge weights are holomorphic functions, arising from physical admittances. This result allows us to establish convergence results for the infinite complex-weighted graphs, e.g. convergence of solutions of Dirichlet problems and convergence of complex-valued capacities on a finite exhaustion. Finally, we define a recurrence for complex-weighted graphs, and, using the convergence results, give its characterizations in terms of functional spaces, capacity, Green's function, resolvents of the Dirichlet Laplacian and properties of the Neumann Laplacian.

We study linearly constrained composite convex optimization with a smooth term and a proximable nonsmooth term. We develop a unified augmented primal-dual framework with primal-dual hybrid gradient-type and augmented Chambolle-Pock-type metric choices, including a fully augmented Chambolle-Pock-type family that retains the augmented quadratic term. The exact scheme admits a degenerate proximal-point form; the linearized scheme admits a preconditioned forward-backward form. These representations allow reflected Halpern acceleration to be analyzed directly in primal-dual variables. For the shadow iterates, we prove convergence to Karush-Kuhn-Tucker (KKT) points and nonergodic O(1/k) bounds for the KKT residual and objective gap, with a scalar worst-case example. We show that finite identification belongs to the shadow sequence rather than to the anchored Halpern state. After identification, an affine-face model yields an exact reduced residual identity and a local-sharpness criterion. Finally, we prove linear convergence of restart anchors under fixed-point sharpness on the visited restart set, with local or tail convergence when sharpness follows from local error bounds. Experiments on linear and convex quadratic programs illustrate augmentation and linearization.

We study the one-dimensional cubic nonlinear Schrödinger system \[ u_i''+2\left(\sum_{k=1}^N u_k^2\right)u_i=-\mu_i u_i \quad \mbox{in } \ \mathbb R,\ \ i=1,2,\cdots,N, \] where $u=(u_1,\cdots,u_N)\in \big(H^1(\mathbb{R})\big)^N$, $\mu_1\leq\mu_2\leq\cdots\leq\mu_N<0$, and $N\geq 2$ is arbitrary. In this paper, we prove the following results for any $N\ge 2$: (i). All nontrivial solutions of the system can be completely classified; (ii). The linearized operator at any nontrivial solution of the system is non-degenerate; (iii). For all $i=1, 2,\cdots, N$, the exact $L^2$-mass identity of $u_i$ is derived in terms of $2\sqrt {|\mu_i|}$, which yields a complete characterization of normalized solutions satisfying $\int_{\mathbb{R}}u_i^2dx=1$. These settle some conjectures of [R. Frank, D. Gontier and M. Lewin, CMP, 2021] and [Y. Guo, Y. Luo and J. Wei, APDE, 2026], where the system was addressed specially for $N=2$ and $N=3$, respectively.

The replication conjecture [Conforti and Cornuéjols, 1993] states that every clutter with the packing property has the MFMC property. If true, this conjecture would have far-reaching consequences from integer programming and combinatorial optimization to commutative algebra. In this paper, we set out to verify the conjecture for the cuboid of a set-system in which the Hamming graph induced on the infeasible points has degree at most $\delta$. The family of cuboids of degree at most $\delta$ contains a rich source of clutters with the packing property, including all clutters over a ground set of size at most $\delta$. We prove that any minimal counterexample must have dimension at most $\delta$, thus making the target search space finite. We then use a state-of-the-art SAT solver to verify the replication conjecture for cuboids of degree at most $9$, and for clutters over at most $10$ elements. Our computational verification relies crucially on another theoretical result, that to verify the MFMC property of a clutter over $n$ elements, it suffices to check finitely many weight vectors, namely $w\in \left\{0,1,\ldots,t\right\}^n$ where $t\leq\max\{\lceil n/2\rceil, \lfloor n-\sqrt{4n+1}+1\rfloor\}$. The upper bound of $t$ improves the previous best upper bound by algebraists, which could be exponential in $n$.

This paper explores a new context for operadic calculus and Koszul duality arising in the study of complex manifolds or, more abstractly, of bidifferential bigraded algebras up to pluripotential weak equivalence. In particular, we establish a Pluripotential Homotopy Transfer Theorem. This provides a homotopical framework for the theory of Massey products in Bott-Chern and Aeppli cohomologies, defined for any complex manifold.

In this paper, we study the Lin–Lu–Yau Ricci curvature of strongly connected locally finite digraphs through an explicit optimal-coupling construction. For an arc of a digraph, we derive a computable curvature formula by constructing a coupling between the probability measures at its tail and head, and by proving its optimality using a suitable $1$-Lipschitz function. The formula is not only effective for direct computation, but also unifies several known results: in particular, it recovers the Lin–Lu–Yau Ricci curvature formula for Cayley graphs of Right-Angled Artin–Coxeter Hybrid groups as a special case and gives shorter proofs of curvature results arising from matching-type conditions. We then characterize arcs with zero Ricci curvature through perfect distance matching and perfect distance partitions. We further prove that, under suitable assumptions, such arc curvature in directed Cayley graphs increases when an inverse generator or a new generator is added to the generating set. As applications, we compute the curvature of directed Cayley graphs of dihedral groups and generalized quaternion groups, including $\Gamma(D_n,\{a,b\})$, $\Gamma(Q_{4m},\{a,b\})$, $\Gamma(Q_{4m},\{a,a^{-1},b\})$ and $\Gamma(Q_{4m},\{a,b,b^{-1}\})$. Finally, we provide an algorithm for computing the Lin–Lu–Yau Ricci curvature of Cayley graphs of finitely generated groups with prescribed generating sets, together with complete curvature tables for several important families of finite groups.

We present a canonical geometric quantization scheme for closed contact 3-manifolds $(M,\xi)$ using the corresponding explicit embedding $M \hookrightarrow \C^3$ as we constructed in our earlier papers. The contact structure becomes holomorphic along the chosen link $L \subset M$ of complex tangents, of which we may also consider this link as the binding of a supporting open book for $\xi$. Stein extension yields a unique holomorphic line bundle $L_\xi \to \C^3$ whose restriction to $L$ will define the quantum Hilbert space $\Hilb_\xi = H^0(L, L_\xi|_L \otimes \kappa^{1/2})$, which we find will be finite-dimensional. The Reeb vector field $R_\alpha$ of a chosen compatible contact form $\alpha$ is shown to be geodesic with a time Killing Reeb field under further assumption that the manifold is Sasakian, and under this assumption will model Einstein gravity over $M$. This construction depends only on $(M,\xi,\alpha)$ and a choice of supporting open book for $\xi$. This establishes a unified geometric framework in which the contact structure encodes quantum mechanics and whose Reeb field will encode gravity assuming that the manifold is Sasakian. In addition, we use my previous paper in which we get a related invariant $\mu_M(\xi) = [L_\xi] \in \Pic_\C(M)$ which is useful in that it distinguishes between different tight contact structures on $\T^3$ in a novel way. We also show how this will have implications for the quantum model and serve as a quantum invariant there-in.

This study proposes a Petrov-Galerkin based Variational Physics-Informed Neural Network (VPINN) for efficiently solving two-dimensional singularly perturbed problems (SPPs) with one and two small perturbation parameters. The approach employs neural networks to construct the trial solution space, while tensor-product hat functions are adopted as test functions to enforce the variational form. To accurately resolve of sharp boundary layers, the variational form is implemented using a Petrov-Galerkin formulation. Dirichlet boundary conditions are imposed directly, while the source terms are computed using automatic differentiation. Computational experiments on standard two-dimensional problems demonstrate that the proposed method achieves high accuracy in both the maximum and L_2 norms. These results confirm the efficiency and robustness of the Petrov-Galerkin VPINN approach in accurately capturing the multiscale features of two-dimensional SPPs.

Let $\mathcal M$ be a semifinite von Neumann algebra and $T : \mathcal{M} \to \mathcal{M}$ be a positive $L_\infty-L_1$ contraction in the sense of Junge-Xu, of which the numerical range, when viewed as an operator on $L_2(\mathcal M),$ is contained in a closed polygon with vertices on the unit circle. In this article, we prove that there exists a positive constant $C_p(T)$ such that \begin{equation}\label{abstract1stin} \Big\|\sup_{n \ge 0}\!^{+} T^n x \Big\|_p \le C_p(T)\, \|x\|_p \end{equation} for all \( x \in L_p(\mathcal{M}) \), $1<p<\infty$ extending some noncommutative maximal ergodic inequalities proved by Junge-Xu \cite{junge-Xu} and later generalized by Bekjan \cite{Bekjan2008}. In the commutative setting, similar inequalities as in \eqref{abstract1stin} hold for arbitrary $L_\infty-L_1$ contractions with the same condition in the numerical range, yielding a vast generalization of a classical maximal ergodic theorem of Stein \cite{Stein-ergodic-theorem} proved in 1960s. Moreover, we establish a noncommutative weak-type maximal inequality for convolution powers which was proved by Calderón and Bellow \cite{Bellow-Calderon} in the classical setting, complementing our strong type noncommutative maximal ergodic inequalities. Our method relies on several new polynomial identities, suitable square function estimates tailored to fit our setting and generalization of Stein's method of embedding maximal function into analytic family of operators. However, we show that even in the classical setting, the variational inequality extending \eqref{abstract1stin} holds for arbitrary operators described above, precisely when the spectrum meets the unit circle only at $1.$

We develop an anti-concentration method for volume-constrained nonlocal shape optimization. Precisely, we apply it to prove the nonexistence of minimizers of an optimal design problem driven by a Hartree-type energy in the large mass regimes. The proof refines and extends the ideas from [Lu-Otto, CPAM 2014], where Lu and Otto dealt with nonexistence of minimizers for the Thomas-Fermi-Dirac-von Weizsäcker energy. We also establish the asymptotic behavior of the associated isoperimetric profile of the energy.

We investigate homomorphisms between Specht modules for level 1 cyclotomic quiver Hecke algebras of affine type C. For hook partitions with a `short leg' or a `short arm', we give the full quiver Hecke algebra action and we show that every non-zero homomorphism from an arbitrary Specht module is determined by sending the cyclic generator to a single standard basis element. We use this action to investigate homomorphisms to these Specht modules, giving a full classification in the short leg case. This provides the first investigation of homomorphisms between Specht modules in affine type C and extends methods previously used in affine type A.

We study the topological dynamics of random circle endomorphisms of degree one over an ergodic measure-preserving dynamical system. Under an integrability assumption, we prove that the random rotation set is almost surely the compact interval whose endpoints are the mean random rotation numbers of the associated lower and upper random maps. We also show that the natural orbitwise versions of the random rotation set agree almost surely, and on the same full-measure set, every value in this interval is realised as the asymptotic average displacement along an individual orbit. In addition, every closed subinterval of the random rotation set is realised, on a full-measure set, as the set of accumulation values of the displacement averages along a single orbit. Finally, we prove that a positive length of the random rotation set implies a positive random topological entropy, in contrast to random monotone maps, which have zero random topological entropy. We illustrate the theory by computing the random rotation set and random topological entropy for a piecewise linear example.

We prove two $q$-congruence conjectures of Guo on truncated basic hypergeometric series. The first result strengthens a congruence obtained from Jackson's terminating very-well-poised ${}_8\phi_7$ summation from the modulus $\Phi_n(q)^4$ to the modulus $\Phi_n(q)^5$ when $n\equiv3\pmod 5$. The second result proves a cyclotomic congruence modulo $\Phi_n(q)$ when $n\equiv1\pmod4$, by a specialization of Andrews' terminating ${}_4\phi_3$ summation.

We study the continuous-time Lyapunov equation under the assumption that the state matrix is a Hurwitz companion matrix. The standard Lyapunov theory implies that the unique solution $X$ is positive semidefinite. Motivated by positive systems, we investigate the question of whether $X$ is entrywise nonnegative. We prove that this is the case when the companion matrix has only real eigenvalues. The proof reduces each entry of $X$ to a quadratic form associated with a class of Cauchy-like matrices whose entries are expressed in terms of elementary symmetric polynomials. The required nonnegativity then follows from the positive semidefiniteness of these Cauchy-like matrices. We also discuss a stronger total-positivity property: total nonnegativity does not hold in general, but it is recovered under an additional sign condition on the expansion of the forcing vector in the eigenbasis of $A^\top$.

We derive logarithmic large-deviation asymptotics for randomly stopped sums with nonnegative summands satisfy a weak form of regular variation. Our approach is based on a count-scale condition that controls the logarithmic order of the random number of summands and does not require relative concentration around its expectation. A main advantage of the framework is that it applies to every renewal count, without moment, regular-variation, or domain-of-attraction assumptions on the interarrival distribution. In particular, it covers infinite-mean interarrival regimes in which relative concentration may fail. It is proven that the mean renewal count is an admissible count scale even when relative concentration fails. We further apply the results to finite heterogeneous insurance portfolios, obtaining dependence-robust asymptotics for unions and weighted aggregate losses, as well as explicit joint shortfall costs under block independence.

Electricity prices result from the complex interplay of supply and demand, which depends on variable renewable energy production, fuel costs, CO$_2$ price, and grid bottlenecks.Between 2020 and 2024, COVID-19 shifted demand and disrupted supply chains and operations in Europe, while Russia's invasion of Ukraine constrained gas supply, causing exceptional volatility in gas prices. It remains unclear whether optimal dispatch models can reliably replicate historical hourly prices during crises, given the rapid fluctuations in fuel prices and operators' limited foresight. In this work, we ask whether an optimal dispatch model, parametrised with historical data on demand, fuel, and CO$_2$ prices, can reproduce the observed market outcomes during this period. We perform hourly hindcasts of electricity generation in 35 countries from 2020 to 2024 using PyPSA-Eur and compare the nodal marginal electricity prices with historical ENTSO-E market prices using the Symmetric Mean Absolute Percentage Error (SMAPE). The scenarios compare static vs dynamic assumptions on fuel and CO$_2$ prices, as well as perfect foresight vs a two-week rolling-horizon optimization. Combining high-resolution fuel and CO$_2$ price time series with limited-foresight substantially improves hindcast accuracy, yielding an average SMAPE of 20.76% based on the daily load-weighted average price for the entire Europe. While improvements relative to the scenario with perfect foresight and static price inputs occur, they are most pronounced during periods of high fuel-price volatility, when marginal-cost swings propagate to electricity prices. In 2022, the optimal generation mix in most countries shows substantially less natural gas and more coal than historically observed. Other discrepancies can be attributed to the model's omission of real-world policy interventions, other dispatch constraints, and generator outages.

Koopman operators have shown significant potential in designing linear model predictive control (MPC) schemes for nonlinear systems on a lifted observable space. Recent advances have tackled the robust Koopman MPC design issue in the presence of modeling errors, relying on the prior estimation of the modeling uncertainty set. However, deriving a robust positively invariant set using a precalculated uncertainty set can be conservative because the uncertainty set bound is time-varying and dependent on the state and control. Additionally, no existing Koopman MPC design has addressed the closed-loop robustness challenge for nonlinear time delayed systems. Thereby, this article presents a robust adaptive Koopman MPC approach with online updates of uncertainty sets for a class of nonlinear time delayed systems. The unknown nonlinear time delayed system is first modeled in a data-driven manner to derive a lifted time delayed Koopman model in the feature space. By analyzing fundamental properties such as controllability and observability, a robust tube-based MPC algorithm is designed for the time delayed Koopman model. The robust adaptive Koopman MPC algorithm with online updates of the uncertainty sets is then presented to reduce conservatism. Closed-loop robustness under exogenous disturbances and asymptotic convergence in the nominal scenario are proven. Finally, numerical examples verify the effectiveness of the proposed approach.

We introduce a geometric decomposition of the hard-sphere partition function. Using a close-packing-inspired geometric bound on the available insertion volume, made rigorous when a corresponding local density certificate is available, we establish a reference upper bound $Q^\ast$ on the configurational integral. Factoring this upper bound out of the statistical geometric partition function found by Speedy yields a new form for the $d$-dimensional partition function, $Q(N,V,T)=Q^\ast \exp(-N \mathcal{J})$, where $\mathcal{J}$ depends strictly on the boundary-to-volume ratio of the voids and the close-packing density. Overall, this work provides an exact reformulation of hard-sphere thermodynamics, thus deepening our statistical geometric understanding of hard spheres.

We study Euler characteristics of scaled toric varieties arising from second hypersimplices. In algebraic statistics, these are closely connected to maximum likelihood (ML) degrees of toric models. We establish a correspondence between delta-matroids and the non-vanishing factors of the principal $A$-determinant, providing an explicit connection between delta-matroid theory and algebraic statistics. Using this framework, we show that a conjectured minimum ML degree is realizable by a suitable embedding of the variety. Furthermore, for second hypersimplices up to order six, we prove that this value is minimal among all embeddings, as conjectured by Clarke et al. (2024).

Let $G$ be a graph. The binding number of $G$, denoted by $\mbox{bind}(G)$, is defined as $$ \mbox{bind}(G)=\min\left\{\frac{|N_G(S)|}{|S|}:\emptyset\neq S\subseteq V(G) \ \mbox{and} \ N_G(S)\neq V(G)\right\}. $$ If $\mbox{bind}(G)\geq r$, then $G$ is called $r$-binding, where $r$ is a positive real number. The adjacency matrix of $G$ is denoted by $A(G)$. The largest eigenvalue of $A(G)$, denoted by $\rho(G)$, is said to be the spectral radius of $G$. A spanning subgraph $F$ of $G$ is called an odd-even factor $F=F_W$ if $d_F(u)\in\{1,3,\ldots,k\}$ for every $u\in W$ and $d_F(v)\in\{0,2,\ldots,k+1\}$ for every $v\in V(G)-W$, where $k$ is a positive odd integer and $W$ is any set of even number of vertices of $G$. In this paper, we propose a tight sufficient condition based on the spectral radius to guarantee that a connected 1-binding graph $G$ contains an odd-even factor $F=F_W$ such that $d_F(u)\in\{1,3,\ldots,k\} \ \mbox{for all} \ u\in W$ and $d_F(v)\in\{0,2,\ldots,k+1\} \ \mbox{for all} \ v\in V(G)-W$.

In massive MIMO systems, phase noise can spoil the performance of the usual receiver techniques. The problem arises because of the aging of phase-noise information based on pilots. In this paper, in a realistic 5G uplink scenario, we quantify the impact of information aging and we propose an iterative receiver based on expectation-maximization (EM). Simulation results show that the iterative receiver is robust to information aging related to phase noise.

This article develops a geodesic interpolation framework for data on the Grassmann manifold. The motivation is that many matrix-valued data sets represent subspaces rather than fixed bases: if the columns of two matrices differ only by a right orthogonal transformation, then they describe the same point on $Gr(r,m)$. Interpolation should therefore be invariant under this basis ambiguity. We first introduce $GLERP$, a Grassmann analogue of spherical linear interpolation defined by the Grassmann exponential and logarithm maps. The method follows the constant-speed geodesic joining two nearby subspaces and is second-order accurate for smooth Grassmann-valued curves under the usual normal-neighborhood condition. We then define $GIDER_n$, a recursive higher-order construction obtained by replacing each affine interpolation step in Neville's algorithm by $GLERP$. The resulting interpolant matches $n+1$ subspace data exactly, is basis-invariant, and is locally of order $n+1$ for sufficiently smooth curves. We compare the construction with tangent-space interpolation and projection-matrix interpolation, discuss intrinsic and extrinsic error measures, and present numerical tests. The results confirm the expected convergence orders and show that $GIDER_n$and tangent-space interpolation are nearly indistinguishable in a smooth local regime, while projection-matrix interpolation provides a useful extrinsic baseline. We also outline how the recursive construction can be used as the basis of a Grassmann ENO procedure.

Quasilinear parabolic boundary value problems are considered. Sharp conditions on the data ensuring existence and uniqueness of solutions in Sobolev classes are exposed. They are smoothness and consistency conditions on the data and some inequalities ensuring the maximum principle for a solution. The proof relies on a priori bounds and the method of continuation in a parameter.

We investigate a $q$-deformation of the shifted topological recursion, extending the construction of Belliard-Bouchard-Kramer-Nelson to the context of quantum algebras. Through the study of highest weight vectors in $q$-deformed of $\mathcal{W}$-algebra representations, we derive a $q$-analogue of the topological recursion and show it yields $q$-deformed quantum curves. This framework unifies various approaches to quantum integrability and provides new insights into the geometry of $q$-deformed moduli spaces.

We study the heat equation in the half-space with nonhomogeneous Dirichlet boundary data. For the caloric extension $v$ of the boundary data $g$, we prove maximal regularity estimates in mixed Lebesgue norms $L^p_tL^q_x$ for any order derivative of $v$ in terms of mixed Besov and Lizorkin–Triebel type norms of $g$. We also establish the corresponding reverse inequalities, which are caloric trace estimates recovering the boundary regularity of $g$ from the mixed-norm regularity of $v$. As a model case, our results show that the natural \[\dotc W^{1,p}\big(\R;L^q(\R^d_+)\big)\cap L^p\big(\R;\dotc W^{2,q}(\R^d_+)\big)\] regularity norm of $v$ is controlled by the \[\dotc {F}^{1-\frac{1}{2q}}_{p,q}\big(\R;\,L^{q}(\R^{d-1})\big)\cap L^{p}\big(\R;\,\dotc{B}_{q,q}^{2-\frac{1}{q}}(\R^{d-1})\big)\] norm of $g$. The maximal regularity estimate holds for $1\leq p,q<\infty$, while the caloric trace estimate holds for $1<p<\infty$ and $1\leq q\leq\infty$. In particular, the endpoint cases $p=1$ or $q=1$ in the maximal regularity estimate are included and appear to be new. These endpoint estimates may be useful in the analysis of free-boundary Navier–Stokes problems with small initial data, whereas the caloric trace estimates may be relevant to the construction of Stokes or Navier–Stokes flows exhibiting strong boundary singularities.

This paper introduces a new theoretical framework for the diagonalization of nonlinear functions defined in finite-dimensional real Euclidean spaces. Extending classical results from linear algebra, we provide a unified setting to determine when a nonlinear map can be represented in diagonal form via a change of basis. Due to the complexity of the equations involved, the final part of the paper focuses primarily on the two-dimensional case, for which clear characterizations can be obtained. We also illustrate how these theoretical findings can be applied to classical contexts, such as differential equations, dynamical systems, and the explicit computation of higher-order compositions and inverses of functions.

We study the theory of right representations of Novikov algebras. We introduce the notion of a modular right ideal of a Novikov algebra and prove that every irreducible right representation of a Novikov algebra is isomorphic to the quotient by a maximal modular right ideal. We prove that the class of Novikov algebras with no irreducible representations forms a hereditary radical. We introduce the notion of a primitive Novikov algebra and show that a Novikov algebra is primitive if and only if it admits an almost faithful irreducible representation. We also introduce the notion of the quasi-kernel of a representation, defined as the largest ideal contained in its kernel. Finally, we prove that the constructed radical coincides with the intersection of the quasi-kernels of all its irreducible representations.

The large time behavior of non-negative solutions to the absorption-diffusion equation $\partial$\textsubscr{t} u = $\Delta$ u\textsuperscript{m} - |x|\textsuperscript{$\sigma$ }u\textsuperscript{m} in (0,$\infty$) x \textsuperscript{N}, with m > 1 and $\sigma$> $\sigma$\textsubscr{0}\,:= N (m-1)/(m+1) is identified. It is shown that all solutions approach a unique stationary solution in self-similar variables, which also provides a universal upper bound (friendly giant ), strongly contrasting to the standard case $\sigma$ = 0. On the one hand, the convergence proof exploits the variational structure of the equation and a suitable Caffarelli-Kohn-Nirenberg inequality, along with the Bénilan-Crandall homogeneity regularizing effect. On the other hand, the detailed study of the stationary problem combines elliptic estimates, Moser iteration and techniques from ordinary differential equations.