Papers

We consider evolution of piecewise linear curves with boundary by linear hyperbolic curvature flows. Given appropriate boundary behaviours we find corresponding self-similar solutions. More generally, given any specific boundary trajectories we write down general solutions using finite Fourier expansions. Finally we consider the problem of evolving one piecewise linear curve to another using nonhomogeneous linear hyperbolic flow.

Let $C_k(q)$ be the least cardinality of a family of nonzero polynomials over $\mathbb F_q$ whose associated codimension-$k$ partial-circulant kernels cover the full coefficient space. Chow and Rimani'c conjectured that $C_k(q)=1+q+\cdots+q^k$. We disprove the unrestricted conjecture by constructing thirteen monic polynomials over $\mathbb F_2$ whose $k=3$ kernels cover $\mathbb F_2^7$; in particular, $C_3(2)\le 13<15$. For a general covering family of size $N=q^k+S$ and $\mathbb F_q$-linear rank $d$, we prove $S\gg d^{2/3}\left(\frac{\log(2q)}{\log(eNq^k/S)}\right)^{2/3}$. Consequently, for every fixed $k\ge 2$ and all sufficiently large $q$, $C_k(q)\ge q^k+c_kq^{2/3}$. When $k=2$, an integer-multiplicity refinement of the second-moment covering argument yields $\liminf_{q\to\infty}(C_2(q)-q^2)/q\ge \widetilde c_2$, where $\widetilde c_2$ is an explicit one-variable variational constant with numerical value $\widetilde c_2=0.5829944375\ldots$. We also classify triples admitting two independent low-degree polynomial syzygies and prove a conditional packet-free lower bound of size $q^k+(1/2-o(1))q^{k-1}$.

Generalising Hölder's classical group enumeration for squarefree orders (1895), we provide an exact formula for the number of isomorphism types of groups of a given cubefree order. After more than 130 years, this is the first such formula that covers significantly more orders than the squarefree ones (83% versus 61% of all integers). Like Hölder's formula, ours is combinatorial: it can be evaluated from the prime factorisation of the order by arithmetic operations and table look-ups, without constructing a single group. The structure of our formula leads to counting formulas for natural subclasses of cubefree groups, with applications in computational group theory. We also derive new asymptotic results. Blackburn et al. (2007) conjectured that the number gnu(n) of groups of cubefree order n satisfies gnu(n)<n^2. We show that gnu(n)\leq n^{2+o(1)}, which improves the bound gnu(n)<n^8 recorded in their survey, and we prove that the exponent 2 is best possible, that is, gnu(n)\geq n^{2-o(1)} for infinitely many cubefree n. Lastly, we show that a much stronger form of the conjecture holds for almost every cubefree order, namely, \gnu(n)\leq (\log n)^{(\log\log n)^{O(1)}}.

For a finite group $G$, the representation dimension $\delta(G)$ is the least dimension of a faithful complex representation of $G$. We prove that $\delta(H\times K) = \delta(H) + \delta(K)$ whenever $H$ and $K$ are $p$-groups for a fixed prime $p$, and show by example that this additivity can fail more generally for nilpotent groups. We also determine $\delta(G)$ for several important classes of non-abelian $p$-groups, viz. VZ groups, Camina groups, and metacyclic groups; and compute $\delta(G)$ for all groups of order $p^6$ ($p\geq 5$), organized by their isoclinism family.

In the last years, a number of proofs of the fact that $O_2$ is a multiple context-free grammar (MCFG) were given. Such results can be exploited in the fields of both computational linguistics and of computational algebra. Here, we focus on a recent such proof spelled in terms of factorizations of string tuples, and give a new result with a stronger characterization of such factorizations than in existing theorems.

This work establishes uniform error estimates for classical numerical schemes applied to the generalized Korteweg-de Vries (gKdV) equation in the dispersionless limit regime, specifically before the development of dispersive shock wave (DSW). We analyze the Crank-Nicolson (CN) and a Lawson-type Runge-Kutta (Lawson-RK) methods, when discretized in space via the Fourier pseudo-spectral method. We prove that both fully discrete schemes achieve optimal second-order temporal accuracy and spectral spatial accuracy, with error constants being uniform in the vanishing dispersion parameter $\varepsilon$. The analysis also addresses the unique solvability of the CN scheme in the dispersionless limit regime. Theoretical findings are supported by numerical experiments, demonstrating the $\varepsilon$-uniform accuracy and the necessity of step size restriction for CN. The study in general validates the classical methods for solving the dispersionless gKdV equation before DSW.

This paper introduces Tensor Field Models (TFMs), realization-level Mathematical Structures in which a learned Operator maps a product of admissible component-section families to a prescribed family of time-dependent tangent sections on a Generative State Manifold. Analytic and dynamical restrictions are encoded through the choice of admissible families rather than imposed by the root definition. Constructed, component-separable, and Tensor Bundle TFMs provide structured refinements of this common object. In the conditional realizations considered here, a structured condition $c=(c_1,\ldots,c_n)$ is mapped componentwise to a reusable collection $\mathbf H_c=(H_{c_1}^{(1)},\ldots,H_{c_n}^{(n)})$. In the architectures evaluated here, the component representations remain distinct and are combined only by the Field Operator to produce the generated Vector Field. All learned models are trained using Flow Matching. Experiments show that TFMs can improve performance and that amortized sampling enabled by reusable condition representations can accelerate generation.

In this article we determine the toric representation type of the Veronese surface $(\mathbb{P}^2,\mathcal{O}_{\mathbb{P}^2}(d))$. Based on Klyachko filtrations, we introduce an explicit criterion for a toric vector bundle of arbitrary rank to be arithmetically Cohen–Macaulay. For $d \geq 3$, suitable configurations of partial flags produce stable toric $d$-aCM bundles corresponding to imaginary non-isotropic Schur roots of star-shaped quivers. Their self-extensions give an exact representation embedding of $\operatorname{mod}\mathbb{C}\langle x,y\rangle$, proving that the Veronese surface is toric-wild precisely for $d \geq 3$, while it is toric-finite for $d=1,2$. For $d=3,4$, suitable twists of the basic stable bundles are Ulrich, and the same construction proves that the corresponding Veronese surfaces are toric Ulrich-wild.

For $\tfrac12<\alpha<1$ we construct a closed realization of $\mathcal L u=\mathcal D_1^\alpha D_0^\alpha u$ in $L^2(0,1)$ whose domain contains the singular mode $x^{\alpha-1}$. The domain, defined via the distributional Caputo composition, is characterized by $I_0^{1-\alpha}u\in H^1(0,1)$ and $D_0^\alpha u-c\in\operatorname{ran}I_1^\alpha$; we prove a Volterra representation for its elements, graph-norm completeness, and that $\mathcal L$ is the adjoint of its zero-trace restriction. Four bounded endpoint traces give an ordinary boundary triplet, so all self-adjoint extensions correspond to Lagrangian planes in $\mathbb C^4$, including coupled endpoint conditions. For each we give an exact zero-eigenvalue criterion and, when $0\in\rho(\mathcal L_\omega)$, an inverse of the form $I_0^\alpha I_1^\alpha$ plus an explicit finite-rank correction. Sharp singular-value asymptotics $s_n(\mathcal L_\omega^{-1})=(\pi n)^{-2\alpha}\bigl(1+O(n^{-1})\bigr)$ yield $\mathcal L_\omega^{-1}\in\mathfrak S_p\iff p>1/(2\alpha)$, with weak-Schatten endpoint and Weyl asymptotics — now for every invertible boundary plane, not only the positive reference problem. As an application we obtain a mild-solution theorem for a Caputo-time diffusion equation in $C([0,T];L^2(0,1))$.

The classical theory of nonlinear observer error linearization—the nonlinear observer canonical form—has attracted sustained attention since its inception in the 1980s. Under the existing theoretical framework, the verification and construction for a nonlinear system to achieve observer error linearization admits a systematic, efficient implementation, severely limiting the applicability of the theory to high-dimensional systems. To address this issue, inspired by the definition of high-order fully measured systems, this paper proposes the Pascal-Hessian condition for single-output systems. This condition equivalently converts the necessary and sufficient condition for observer error linearization into a structural test on the Hessian matrix of the nonlinear term in high-order fully measured systems: the coefficients in the upper-left corner of Hessian matrix form a Pascal's triangle, while the lower-right corner vanishes identically. Simultaneously, we provide explicit integral formulas for all output-dependent univariate functions in the canonical form, eliminating the need to solve partial differential equations. Compared with the existing theory, our method reduces the computational complexity of condition verification from $O(n^4)$ to $O(n^2)$, and replaces the intricate process of solving partial differential equations with explicit indefinite integral for canonical form construction. We further extend the result to multi-output systems with equal observability indices. Beyond its computational advantages, this work reveals a fundamental structural connection between the nonlinear observer canonical form and Pascal's triangle—a link that has remained unnoticed since the inception of the theory in the 1980s. Numerical examples validate the effectiveness of the proposed method.

We provide explicit examples showing that the existing assumptions in literature on the force are sharp for the stated properties of strong solutions to the 2D Navier-Stokes equations. If they fail to be satisfied, finite time blow-up may occur in different forms.

We study radio node (RN) placement for indoor enterprise networks. Using stochastic geometry (SG), we derive the meta-distribution (MD) of the SINR for a test user equipment (UE), with and without cooperation from outdoor macro base stations (MBSs), and compare these results with an integer linear programming (ILP) approach. SG provides an estimate of the required number of RNs but not their locations, while ILP can yield inaccurate local optima and requires high computational power. To address this, we investigate clustering-based algorithms for initializing RN locations using UE location distributions. Along with standard methods, we propose a weighted $k$-harmonic means (WKHM) clustering strategy tailored to maximize SINR. We then introduce a constrained sequential minimum cut algorithm, SeqMinCut, to merge multiple RNs into larger cells and further improve SINR. This is the first work that integrates SG-based statistical analysis, optimization, and clustering to obtain system design insights, dimensioning rules, and planning strategies for enterprise 5G.

Let $W$ be a henselian valuation ring with fraction field $L$, residue field $k$, and value group $\Gamma_W$. Let $N=\ell^\nu$ be invertible in $W$. Choose an ordered $\mathbf Z/N$-basis $B$ of $\Gamma_W/N\Gamma_W$. Products of suitable classes define an equivalence of complete filtered spectra $$ \bigoplus_{\substack{J\subseteq B\\J\text{ finite}}} \Sigma^{|J|}\operatorname{Fil}_{\mathrm{mot}}^{\bullet-|J|} K(k;\mathbf Z/N) \xrightarrow{\simeq} \operatorname{Fil}_{\mathrm{mot}}^{\bullet} K(L;\mathbf Z/N) $$ The summand indexed by $\varnothing$ is the generic restriction map, after the rigidity equivalence $K(W;\mathbf Z/N)\simeq K(k;\mathbf Z/N)$, and is therefore split injective. Independently of this splitting, excision yields a lifting theorem for regular henselian pairs. In particular, this yields finite-coefficient Gersten injectivity for noetherian henselian regular local rings. Applications to completions along regular primes give relative and sometimes nonhenselian examples. More generally, if $P$ is a Prüfer ring and $R$ is a henselian local ind-smooth $P$-algebra, then $R$ is a domain and $K_n(R;\mathbf Z/N)\to K_n(\operatorname{Frac}(R);\mathbf Z/N)$ is injective for every $n$, provided $N\in R^\times$. These injectivity consequences extend from prime-power to arbitrary finite invertible coefficients by primary decomposition.

This paper addresses morning commute congestion caused by concentrated school-related trips in urban networks. We propose a bi-level optimization framework for regulating school start times in a multi-region urban network characterized by Macroscopic Fundamental Diagrams (MFDs), explicitly coupling system-level regulation with multi-class user-equilibrium-based departure-time choices. The Upper-Level problem jointly minimizes total time spent and deviations from current school schedules, while the Lower-Level problem models commuter behavior through a deterministic dynamic multi-class user equilibrium formulation incorporating alpha-beta-gamma preferences for travel time, earliness, and lateness costs. To address the computational challenges arising from the bilevel structure, non-convex traffic dynamics, and endogenous demand responses, an iterative algorithm alternating between the Upper- and Lower-Level problems is developed. The Upper-Level problem is approximated through a formulation solvable with standard mathematical programming solvers, while an iterative algorithm provides an approximate solution to the Lower-Level equilibrium problem. Numerical results demonstrate substantial congestion reductions and characterize the trade-off between school start-time flexibility and traffic efficiency. Sensitivity analyses further examine the effects of MFD uncertainty and scheduling preferences.

We consider the question of the optimal timing of the sale of an asset with stochastic dynamics. Our analysis is based on the method of the distribution builder introduced by Sharpe, Goldstein and Blythe [SGB00] for the purpose of optimal portfolio selection. Instead of specifying a utility function or risk aversion coefficient, this tool directly elicits the target distribution of the investor. We show how the problem of an optimal asset sale is in this setting linked to the problem of finding a Skorokhod embedding of a distribution into a diffusion process. In the case where the asset process follows a geometric Brownian motion and a specific family of distributions is targeted, one can observe a risk-return tradeoff.

In 1996, Baranov and Kleshchev classified maximal ideals in the finitary symmetric group algebra over fields of characteristic not 2. The corresponding question in characteristic~2 has remained open since. In the current paper we solve the problem by establishing part of a conjectural connection between prime ideals in this group algebra and the recently defined higher Verlinde categories. To achieve this we investigate thick tensor ideals of tilting modules of the general linear group in characteristic 2.

Let $g:\mathbb{D}\to\mathbb{D}$ be a hyperbolic holomorphic self-map of the unit disk with Denjoy–Wolff point $\tau\in\partial\mathbb{D}$ and angular derivative $\alpha\in(0,1)$. We say that $g$ has an extremal hyperbolic rate if its forward iterates satisfy the sharp metric asymptotic \begin{equation*} \rho_{\mathbb{D}}(g^{\circ n}(z),w) = n\log\frac{1}{\alpha} + O(1) \quad\text{as }n\to\infty, \end{equation*} for every $z,w\in\mathbb{D}$. Using the Herglotz–Nevanlinna representation, we prove that $g$ has an extremal hyperbolic rate if and only if the associated boundary measure $\sigma$ satisfies \begin{equation*} \int_{\partial\mathbb{D}\setminus\{\tau\}} \log\frac{1}{|\zeta-\tau|} \,d\sigma(\zeta) < \infty. \end{equation*} We further show that this condition is equivalent to a non-degenerate angular asymptotic of the Koenigs linearization: for any conformal map $\phi_\tau:\mathbb{D}\to\mathbb{H}$ with $\phi_\tau(\tau)=\infty$, \begin{equation*} 0< \left| \angle\lim_{z\to\tau} \frac{h(z)}{\phi_\tau(z)} \right| < \infty, \end{equation*} where $h$ is a Koenigs function. We then extend the extremal-rate theory beyond the unit disk. For finitely connected hyperbolic planar domains, the disk characterizations transfer to ordinary boundary points of the associated deck transformation group. For holomorphic self-maps of the unit ball $\mathbb{B}^n$ and for $K$-quasiconformal self-maps, where no comparable Herglotz representation is available; we establish sufficient boundary regularity conditions that guarantee the extremal hyperbolic rate. The quasiconformal result is further extended to finitely connected planar domains at ordinary boundary points. These results show that, across the settings considered here, extremal hyperbolic growth is governed by the non-degeneracy of the boundary linearization at the Denjoy–Wolff point.

The Ramsey number $R(m,n)$ is the smallest order at which every red-blue edge coloring of a complete graph must contain a blue clique (a complete subgraph) of size $m$ or a red clique of size $n$. Determining these numbers exactly is extremely hard, and even certifying a lower bound requires exhibiting an explicit coloring that avoids both cliques. We develop an integer programming framework for certifying such lower bounds, restricting the search to circulant graphs, whose rotational symmetry lets us reformulate the problem in a projected distance space, reducing the number of binary variables from quadratic to linear in the graph order. We strengthen this projected model through coefficient reduction and solve it with a branch-and-cut algorithm whose separation routine exploits the common neighborhood structure of circulant graphs, combining heuristic and exact maximum-clique algorithms. In an extensive computational campaign on circulant graphs with up to 410 vertices, we improve the best lower bounds previously obtained by other methods by up to 11 points for 25 values of $R(3,n)$ with $24\le n\le49$ and $n\neq27$, each backed by an explicit graph certificate that can be independently verified with a stand-alone exact clique solver. To the best of our knowledge, our method also provides the first reproducible optimization-based procedure for certifying circulant Ramsey numbers $R_C(m,n)$, which we use to establish eight new values of $R_C(3,n)$ with $13\le n\le20$. Our framework, graph certificates, and stand-alone checker are provided as supplementary material to support independent verification and reuse.

The classical AT1 phase-field model contains an intrinsic energy barrier for crack nucle ation, which makes the predicted strength depend on the fracture toughness and the regularization length. For tensile-dominated brittle fracture, this barrier is shifted by mapping the Rankine criterion, evaluated on the effective stress, onto a state-dependent active-energy threshold. The prescribed tensile strength then controls crack nucleation, while the AT1 crack-density functional, stiffness degradation, and degraded stress response remain unchanged. Since the threshold depends on the current stress state, the field equations are derived from a restricted variational principle. A microforce formulation identifies the barrier shift as a dissipative resistance and provides the corresponding lower bound on the regularization length. In one-dimensional tension, closed-form solutions recover the prescribed peak strength and give a cosine-type localization profile that ap proaches the classical AT1 profile as the shift vanishes. Numerical examples show that, within the admissible range, the nucleation load is nearly insensitive to the regularization length and the predicted multiaxial nucleation states follow the Rankine envelope. Under overall compression, crack nucleation remains associated with local tensile stress concen trations. The formulation also captures the transition from strength-controlled failure for small flaws to the LEFM limit for large cracks.

We show that small planar perturbations of families of unimodal maps have maximal (equal to the intersection of the iterates of their trapping region) attractors at parameters near homoclinic tangency of one of the fixed points. The results hold for general unimodal maps exhibiting certain hyperbolic properties. Hence, admissible families considered include Hénon-like families, Lozi-like families, or perturbations of border collision normal forms. Therefore, our results generalize results of Viana in the case of Hénon-like families and Glendinning and Simpson in the case of border collision normal forms.

By a short adaptation of an argument of Singerman, we show that for every prime power $q$, every orientably regular map (or hypermap) whose orientation-preserving automorphism group is isomorphic to PGL(2,q) is reflexible.

For every $d\geq3$ we construct a connected graph of order $2\binom{3d+1}{d}$ that is locally $K(2d+1,d)$. Fix a block $A$ and put $a'=\min(|A|,3d+1-|A|)$. Among the loopless binary voltages on the fixed labeled base $K(3d+1,d)$ that depend only on the intersection types with $A$, the local-neighborhood-preserving assignments form, modulo gauge, an $\mathbb F_2$-space of dimension $[q^{a'-5}]\binom{d}{3}_q$, with explicit canonical coordinates. Adjacent-line rigidity holds on $2d+2\leq n\leq3d$: every loopless fixed-block type-invariant local-neighborhood-preserving binary voltage is gauge trivial. Dropping type invariance over the fixed labeled $K(10,3)$, we classify all local-neighborhood-preserving binary voltage assignments and find that $H^1(M_3(10);\mathbb F_2)$ has dimension 42. Under base relabeling by $S_{10}$ these classes form 1,245,395 orbits, of which 1,245,394 consist of connected covers; an explicit class has orbit size 126 and stabilizer $S_5\wr C_2$. The proofs combine a triangle-voltage criterion and cohomology with a simplicial collapse and Burnside enumeration.

We define a new matrix associated with a hypergraph. A study of this matrix is carried out, in particular the study of its spectrum, which shows the relevance of the construction. We demon- strate that this adjacency matrix is characterisitic of the hypergraph, that is to say, two isomorphic hypergraphs have similar matrices. Furthermore, starting from this matrix, we can reconstruct the hypergraph. Starting from this matrix, we introduce new graphs associated with the hypergraph.

Extreme value theory provides a fundamental framework for modeling rare and extreme events; however, threshold selection remains a persistent challenge, particularly in the presence of inliers such as instantaneous or early failures. Such observations commonly arise in applications including reliability studies and environmental data, where clusters of observations near the origin or at the origin can substantially distort classical threshold selection procedures and tail inference. In this paper, we propose a robust modeling framework that accounts for inliers, extremes, and the tail proportion. Parameter estimation is carried out using maximum likelihood. The proposed methodology is compared with classical numerical and graphical diagnostic tools, including the mean excess plot, parameter stability plot, Hill plot, and Pickands plot, as well as existing extreme value mixture models. The theoretical properties of the proposed model are established, and its performance is evaluated through extensive Monte Carlo simulations and real-data applications. The results demonstrate that the proposed methodology provides more accurate threshold estimation and more reliable extreme-value inference in the presence of inliers compared with existing classical approaches. Overall, the proposed methodology provides more accurate threshold estimation and tail inference in the presence of inliers, addressing key limitations of existing methods.

We extend the notion of Plancherel convergence, introduced in the study of compact quotients of locally compact groups, to finite-area hyperbolic surfaces and establish a quantum ergodicity theorem for Plancherel sequences. We show that the expectation values of integral operators whose kernels are uniformly bounded and have uniformly bounded propagation become asymptotically equidistributed, on average, for eigenvalues in a fixed compact interval. Our result recovers the case of multiplication operators studied by Le Masson and Sahlsten, corresponding to the degenerate case of propagation bound zero and kernels supported on the diagonal. Moreover, our result allows the systole to shrink without a uniform lower bound, with the allowed degeneration constrained by Plancherel convergence. This identifies Plancherel convergence as a natural spectral framework for quantum ergodicity on degenerating hyperbolic surfaces.

Thomassen (J. Combin. Theory Ser. B 108 (2014), 81-91) showed that every graph admitting a nowhere-zero $3$-flow also admits an $S^1$-flow. He also proved that the converse holds for cubic graphs, but constructed counterexamples showing it fails in general. Wang et al. (SIAM J. Discrete Math. 29 (2015), 2166-2178) presented a couple of sufficient conditions under which the existence of an $S^1$-flow guarantees the existence of a nowhere-zero 3-flow. In this paper, we first prove that a graph with maximum degree at most four admits a nowhere-zero $3$-flow if and only if it admits an $S^1$-flow. We then develop some reduction techniques for $S^1$-flows based on graph operations including bull-growth, $2$-sums, and contractions. Finally, we apply those techniques to characterize triangularly connected graphs and graphs containing a spanning triangle-tree that admit $S^1$-flows, respectively.

For a ring $R$, let $\mathcal{GP}(R)$, $\mathcal{GF}(R)$, and $\mathcal{PGF}(R)$ denote the classes of Gorenstein projective, Gorenstein flat, and projectively coresolved Gorenstein flat left $R$-modules, respectively. We answer negatively the question whether $\mathcal{GP}(R)=\mathcal{PGF}(R)$ for every ring. More precisely, over every field $k$ we construct a left and right coherent central $k$-algebra $T$ and a strongly Gorenstein projective left $T$-module which is not Gorenstein flat; hence $\mathcal{PGF}(T)\subsetneq\mathcal{GP}(T)$.

Stacked intelligent metasurfaces (SIM) provide an efficient architecture for integrated sensing and communication (ISAC) with few radio-frequency (RF) chains. However, diagonal SIM provide only element-wise phase control, so balancing multiuser communication and sensing performance may require additional layers. In this letter, we propose a beyond-diagonal SIM (BD-SIM) architecture for ISAC, enabling controllable intra-layer coupling through reconfigurable impedance networks, thereby enhancing wave-domain processing flexibility. We develop a unified alternating optimization framework applicable to fully-connected, group-connected, and diagonal SIM architectures. Within this framework, we derive a closed-form power allocation rule and propose an effective variable separation algorithm for multi-layer phase-shift design. Simulation results show that the proposed BD-SIM achieve a better communication-sensing trade-off and require fewer layers to attain performance comparable to conventional SIM.

Motivated by the need for reliable traffic management under fluctuating travel demand, we study the problem of determining the worst-case congestion in a multi-commodity traffic network subject to demand uncertainty. To this end, we stress-test a given network by identifying demand realizations and corresponding travelers' route choices that maximize congestion. The users of the traffic network are assumed to act according to one of the two Wardrop principles, the user equilibrium or the system optimum, so that the resulting congestion models can be seen as bilevel problems with a single leader and multiple followers. To address uncertain travel demand, we consider different models such as ellipsoidal or budgeted uncertainty sets and the hose polyhedron. We present single-level mixed-integer nonlinear reformulations of the congestion models that exploit binary variables and big-M constants, prove the existence of optimal solutions, derive valid big-Ms, and propose several enhancement techniques to further strengthen the formulations. An extensive computational study on instances of the Sioux Falls network and instances from the SNDlib demonstrates the computational effectiveness of the proposed techniques and provides insight into the impact of different congestion measures and uncertainty models on the resulting worst-case congestion.

This paper introduces a generalization of classical Galois points by extending the ambient space from standard projective spaces to weighted projective spaces. Specifically, the study focuses on smooth weighted hypersurfaces of degree 6 in $\mathbb{P}(1,1,1,3)$ (which are $K3$ surfaces) to determine the defining equations and the number of weighted Galois points. Furthermore, it characterizes these $K3$ surfaces with weighted Galois points in terms of their automorphisms.