Papers

Triangulations, i.e., well-structured decompositions of geometric objects into triangle-like pieces, are central objects in many domains of mathematics and physics. In particular, fine, regular, and star triangulations (FRSTs) of 4D reflexive polytopes give rise to smooth Calabi-Yau threefolds, which are of significant interest in string theory. However, the high dimensionality and combinatorial complexity of triangulations make them particularly challenging to model with classical numerical methods or machine learning. In this work, we show that transformers, equipped with an appropriate encoding scheme, can be effectively trained to representatively generate new FRSTs across a range of polytope sizes. Moreover, these models can also self-improve through retraining on their own output. This opens the door to both concrete applications to the classification of Calabi-Yau manifolds and further research in physics, combinatorics and algebraic geometry.

We study self-similar singularity formation for the one-dimensional weak-advection Hou–Li model, a reduced model motivated by the axisymmetric Euler equations. In the periodic setting, we construct exact finite-time self-similar blowup solutions for $2/3<a<1$, with profiles that are neither focusing nor expanding. In the whole-space setting with a Neumann condition, we construct exact finite-time self-similar blowup solutions for the full range $0<a\leq1$, with profiles of focusing, non-expanding/non-focusing, or expanding form depending on the sign of the self-similar scaling parameter. The construction is based on a fixed-point formulation near the origin, followed by an ODE extension argument. We also establish regularity, asymptotic behavior, monotonicity properties of the profiles, and uniqueness up to the natural scaling invariance.

In this paper, we study Nyström subsampling for vector-valued regression in vector-valued reproducing kernel Hilbert spaces. Standard kernel methods often suffer from prohibitive computational costs due to the construction and inversion of large kernel matrices, which limits their scalability to large datasets. To overcome this bottleneck, we propose an efficient operator learning algorithm based on Nyström subsampling that accommodates functional outputs. Under general source conditions characterized by index functions-extending beyond the classical Hölder-type and operator-monotone frameworks-we establish minimax-optimal convergence rates for the proposed estimator. As an application of the proposed framework, we consider function denoising problems. Unlike classical denoising methods, which are typically tailored to specific signal representations or noise models, our approach formulates denoising within a general operator learning framework. Numerical experiments on signal denoising, real-time audio denoising, image denoising, inverse Radon transform reconstruction, and energy-efficiency prediction confirm that the proposed method achieves performance comparable to full kernel methods while substantially reducing computational cost.

Residential batteries can reduce the carbon emissions of households with photovoltaic (PV) generation, but how much depends on how they are operated under uncertain load, PV generation, and grid carbon intensity. We cast the operation of a residential PV-battery system as a sequential decision problem and compare, under a common and explicit evaluation protocol, a family of linear policies that differ in the information available at decision time, in how they anticipate the future, and in their adaptivity: a rule-based self-consumption policy, an open-loop stochastic schedule, affine feedback policies, receding-horizon policies with deterministic, scenario-based and two-stage lookahead models, and perfect-information benchmarks. Scenarios are generated by a joint principal component model of the three uncertain quantities, which supports conditional sampling during the day. Experiments on Italian data at 15-minute resolution comprise single-day evaluations on 10000 test scenarios in three seasons and a closed-loop simulation over more than eight months of real data. When the state of charge is treated consistently across policies, the self-consumption rule is within 3.2% of the perfect-foresight emissions in closed loop, whereas open-loop schedules and receding-horizon policies that decide before measuring the current period are about 20–35% above it. With real-time measurements, scenario-based receding-horizon control consistently outperforms its deterministic counterpart and, for households that rely more on the grid, reduces emissions by about 5% with respect to the rule. The benefit of emission-aware optimization is therefore determined mainly by the information structure of the controller and by the size of the system relative to the demand.

We are interested in the local higher integrability of solutions to elliptic equations with linear growth of the form $$-\text{ div}A(x,Du)=f(x). $$ Under a Besov regularity on the partial map $x \mapsto A(x,\xi)$ and a summability condition on the datum $f$ in the scale of Lebesgue spaces, we prove that the solutions are almost Lipschitz continuous, i.e. their gradients belong locally to $L^q$, for any finite exponent $q$. In turn, solutions are locally $\gamma$-Hölder continuous, for every $\gamma \in (0,1)$. The difficulty arising from the lack of an explicit second variation for the problem is overcome by testing the equation with a function proportional to a power of the finite difference quotient of the solution. To the best of our knowledge, this technique is used in this context for the first time. We also provide an example showing the sharpness of our result in the scale of Lebesgue spaces.

In network vulnerability analysis, it is crucial to evaluate the robustness of $k$-cores against vertex removals. A $k$-core is often fragile since removing a few vertices can trigger a large reduction in the core size, a phenomenon known as core collapse. In this paper, we study the problem of enumerating all minimal removable sets (MinRSs) of a given $k$-core, where a MinRS is a minimal nonempty set of vertices whose removal results in a smaller $k$-core graph. We consider this problem within a general mathematical framework based on monotone systems. We show that, for a monotone system that is given with an underlying graph $G=(V,E)$, all MinRSs of a solution can be enumerated in $O((n+m)n\tau_\omega)$ time, where $n=|V|$, $m=|E|$ and $\tau_\omega$ denotes the computation time of evaluating the monotone function of the system. Furthermore, if the system satisfies the newly defined in-dominating seed property, the complexity drops to $O((n+m) \log n \cdot \tau_\omega)$ time. We prove that standard $k$-cores in undirected graphs satisfy this property, enabling MinRS enumeration in $O((n+m)\log n)$ time, a significant improvement over the baseline. We also extend our framework to enumerate all solutions in a given monotone system. This yields an $O((n+m)\log n)$-delay algorithm for all $k$-core subgraphs, outperforming an algorithm given by [Boley et al., Theoretical Computer Science, 2010]. Our framework is applicable to various $k$-core extensions, including weighted $k$-cores, multi-layer $\boldsymbol{k}$-cores, and $(k,\ell)$-cores.

We show that there is an explicit resurgent duality between weak and strong coupling expansions when one of the expansions has zero radius of convergence and the other has infinite radius of convergence. This complements the situation where the convergent expansion has finite radius of convergence, or when both expansions have zero radius of convergence. We illustrate this phenomenon for the Airy and Pearcey catastrophe integrals, and we apply it to two physical examples: the weak and strong coupling expansions of Dyson-Schwinger equations in zero-dimensional scalar $\phi^4$ theories, and the short and long time expansions of the heat kernel trace for the fluctuation operator of the kink-antikink crystal saddle configuration in the Gross-Neveu model.

We consider the stochastic reaction-diffusion equation on the whole space: \begin{align*} \left\{ \begin{aligned} du(t,x) &=\frac{1}{2}\partial_{xx} u(t,x) dt+b(u(t,x))dt+ \sigma(u(t,x)) W(dt,dx),\quad t\geq 0,\ x\in \mathbb{R},\\ u(0,x)&=u_0(x), \quad x\in \mathbb{R}, \end{aligned} \right. \end{align*} where $W(dt,dx)$ is a space-time white noise, $b$, $\sigma$ are measurable coefficients. We first show that the solution is not strong Feller, and then establish the existence and uniqueness of invariant measures, exponential mixing as well as irreducibility for the solutions. To overcome the difficulties caused by the unbounded domain, we design special controls and controlled equations to prove the irreducibility. To obtain the exponential mixing property under the dissipative condition $$(b(x)-b(y))(x-y)\leq -\alpha (x-y)^2,$$ the obstacle is the lack of the Itô formula/energy equality. To circumvent the problem, we manage to find a new way to fully exploit comparison principles, which we believe could be useful for other type of stochastic partial differential equations driven by multiplicative space-time noise. We note that the dissipative condition allows the coefficients to be of polynomial, even exponential growth. There exist plenty of models that satisfy the dissipative condition, including the Allen-Cahn type equations. To the best of our knowledge, this is the first paper to establish the ergodicity, exponential mixing and irreducibility of stochastic reaction-diffusion equations (SRDEs) driven by multiplicative space-time noise on unbounded domains. The results on exponential mixing are also new for (SRDEs) driven by multiplicative space-time noise on bounded domains.

We investigate a class of nonlocal equations whose leading operator is modeled on either the fractional $p$-Laplacian or the regional fractional $p$-Laplacian, $p \in (1,\infty)$. We prove local self-improving properties of weak solutions to the fractional $p$-Laplacian in the case $p\in(1,\infty)$ with non-integrable right-hand side, as well as to the regional fractional $p$-Laplacian in the subquadratic case $1 < p < 2$, by extending the nonlinear commutator estimates developed by Schikorra (Math. Ann. 366 (1-2):695–720, 2016).

In this manuscript, we consider finitely many maps defined on a compact probability space; each of which preserves the probability measure. We consider two aspects of growth of dynamical systems in this scenario; the first setting where we let some infinite lettered word determine the path through which the orbit of points evolve, while the second setting where we consider all possible paths through which the orbit of a set may evolve. Our aim in this paper is to provide a full description of the celebrated multiplicative ergodic theorem, originally due to Oseledets, for such dynamical systems that is point-valued (as in the first case) and set-valued (as in the second case).

A matching covered graph is a brick if it is 3-connected and bicritical. An edge of a matching covered graph G is a forcing edge if it lies in precisely one perfect matching of G. We prove that every vertex of a simple brick G is incident with a forcing edge if and only if G is an odd wheel or is isomorphic to one of the following three graphs: the triangular prism \overline{C_6},\overline{C_6}^+ or the bicorn H_8. Here \overline{C_6}^+ is obtained from \overline{C_6} by adding an edge between two nonadjacent vertices.

Selecting a fixed number of representative points from a finite Pareto-front approximation is a fundamental post-processing task in multiobjective optimization. This paper studies this problem for the integral R2 indicator in three objectives, where the indicator is defined as the integral of the lower envelope of weighted Tchebycheff scalarizations over the two-dimensional weight simplex. We provide two complementary algorithmic results. On the positive side, we show that the integral R2 improvement with respect to any fixed baseline is a monotone submodular set function. For the usual ideal-point based R2 indicator, with the ideal point fixed, this yields a direct gap-reduction guarantee: greedy selection closes at least a $(1-1/e)$-fraction of the maximum possible R2 gap between a fixed dominated anchor value and the best cardinality-$k$ value. We also give a tested greedy implementation that evaluates exact integral R2 values by subdivision, with worst-case running time $O(n^6)$. On the negative side, we prove that exact fixed-cardinality subset selection is NP-hard already in three objectives. The hardness proof uses a perspective transformation that maps Tchebycheff-shadow improvements to a weighted anchored-box union problem with density $(x_1+x_2+x_3)^{-4}$, and then adapts the three-dimensional anchored-box construction of Bringmann, Cabello, and Emmerich. Together, these results separate the tractable two-objective case from the three-objective case while identifying a principled approximation route based on submodular optimization.

There are many formulas in the literature providing roots of a real cubic that avoid some of the well-known pathologies of Cardano's formulas. Among these, we identify two that consistently provide the unique roots of a depressed cubic that have the greatest and smallest absolute value, whenever those exist. We call these the longest and shortest roots. The existence conditions are elementary and are in terms of the signs of the coefficients and the discriminant. Our proofs use two algebraic identities satisfied by hypergeometric functions; once the standard real branches are fixed, the root comparisons are entirely real. As an application, the longest-root formula gives an explicit factorization of all but a vanishing proportion of depressed real quartics.

We extend graph theoretic characterizations of time structurable extensive games from finite informational digraphs to countable and continuum digraphs built on continuum rooted real trees. Introducing the symmetric class quotient poset of information sets as the canonical object an external clock must order, we develop a theory of transitive reduction for these quotients and prove existence and uniqueness under an interval finiteness condition. Under natural topological regularity assumptions like closed reachability classes, order density, and sigma compactness, we give necessary and sufficient conditions for real valued time labelings and construct measurable selection procedures in Polish spaces. We also provide constructive algorithms and a differential game example illustrating applications and computational implications.

For a time dependent Schrödinger equation, the scattering map is the map sending the asymptotic profile of a solution as $t \to-\infty$ to its asymptotic profile as $t\to+\infty$. In this paper we show that, for a certain class of metrics, the scattering maps and Poisson operators associated to two Schrödinger operators on the same curved space only differ by a compact operator on a critical level if and only if the two potentials are equal.

Synthetic electrocardiogram (ECG) generation can support algorithm development and robustness evaluation, but simulated signals must preserve interpretable activation, recovery, and morphology properties. We present a graph-based ECG synthesis framework that combines activation-consistency certification with diagnostics-aware morphology curation. A unified heart graph supports an eikonal-template backend (ET) and a pseudo-diffusion reaction–eikonal backend (RE). We formulate graph Eikonal activation as a Bellman fixed-point problem and use the Bellman residual as a computable certificate for activation-time consistency. Each simulated ECG is evaluated by a two-stage diagnostics pipeline that separates metric computation from experiment-specific acceptance policies. On the cardiac graph, RE-derived activation times showed near-millisecond agreement with the Eikonal backbone and achieved $R^2=0.99876$ after causal predecessor filtering. Recovery experiments showed that endo-epicardial APD gradients determined the main T-wave morphology window, whereas the diffusion strength $\kappa$ provided secondary repolarization smoothing. In final balanced multi-lead curation, RE accepted 658/2000 samples versus 578/2000 for ET and increased per-model morphology coverage from 0.09248 to 0.09888. The framework provides a conservative basis for controllable and curated synthetic ECG generation.

It is well known that every regular Cayley map $M = \CM(G,X,p)$ on a finite group $G$ with respect to an inverse-closed generating set $X$ of $G$ and a specified cyclic permutation $p$ on $X$ corresponds to a skew morphism $\varphi$ on $G$ such that the restriction of $\varphi$ to $X$ is $p$. The skew-type of the map $M$ is defined as the index $[G:\Ker \varphi]$, which equals the number of distinct values in $\mathbb{Z}_{|\varphi|}$ taken by the associated power function $\pi$ of the skew morphism $\varphi$. In this paper, we develop a covering theory of skew morphisms and as an application we provide a classification of regular Cayley maps of skew-type three on the semidihedral groups.

The classical existence result of Nash guarantees that every finite noncooperative game admits an equilibrium in mixed strategies, but it leaves open the question of when pure strategy equilibria exist. This paper develops a structural approach to that question by exploiting properties of the best-response correspondence on finite strategy sets. Building on recent work, we derive new sufficient conditions for the existence of pure strategy Nash equilibria in finite games. We introduce several broad classes of finite games for which pure equilibria are guaranteed, including a class of generalized potential games that generalizes unilaterally competitive games and a class characterized by the existence of an aggregate-payoff maximizer over an ordered set. Our results clarify the role of acyclicity, and aggregation in producing pure equilibria and connect disparate sufficient-condition results in the literature into a unified framework.

We consider the problem of distinguishing sets of quantum product states with local operations and classical communication (LOCC). Recent work has used graph theory to identify sets of product states distinguishable with one-way LOCC. We extend these efforts to full two-way LOCC, with the first significant analysis of the set of graphs corresponding to bipartite product states that can be distinguished with two-way protocols after finitely many steps. We derive basic closure properties of the set of distinguishable graphs and identify some classes of graphs that guarantee local distinguishability and some graphs that do not. We also include several examples and forward-looking comments.

We study the lifespan of mild solutions to the fractional semilinear parabolic Cauchy problem with a Hardy–Hénon-type weight \[ u_t + (-\Delta)^s u = |x|^{-\gamma}\,|u|^p, \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N, \qquad u(0,x)=\varepsilon\,u_0(x), \] where $0<s<1$, $0\le\gamma<\min(2s,N)$, $p>1$ and $u_0\in L^1\cap L^\infty$ with $\int_{\mathbb{R}^N}u_0(x)\,dx>0$. Setting \[ p_F \;:=\; 1+\frac{2s-\gamma}{N}, \] we prove that the lifespan $T_\varepsilon$ obeys, for every sufficiently small $\varepsilon>0$, \[ T_\varepsilon \;\approx\; \begin{cases} \varepsilon^{-\,\beta^{-1}},& 1<p<p_F,\\[1mm] \exp\!\big(C\,\varepsilon^{-(p-1)}\big),& p=p_F,\\[1mm] +\infty,& p>p_F, \end{cases} \qquad \beta \;=\;\frac{(2s-\gamma)-N(p-1)}{2s(p-1)}. \] The lower bound rests on fractional heat-kernel estimates and an $L^1$–$L^\infty$ Hardy-type interpolation inequality; the upper bound is obtained by testing the equation against the backward fractional heat kernel, a globally defined positive weight for which $(-\Delta)^s$ is controlled everywhere and the linear terms cancel identically by self-adjointness. This circumvents the compactly supported cutoffs of the classical test-function method, which are incompatible with a nonlocal operator. The exponent $\beta$ is sharp; for $\gamma=0$ it reduces to the fractional Lee–Ni exponent $\frac{1}{p-1}-\frac{N}{2s}$. To the best of our knowledge, these results are new even for $\gamma=0.$ We also establish a large-data lifespan law, sharp lower bounds on the blow-up rate together with a conditional Type-I upper bound, a conditional self-similar profile result.

This paper develops the intrinsic geometry of the categorified spectral object $\mathfrak{Spec}(A)$ associated with an admissible operator-semantic system $A$ in the Categorified Spectral Duality (CSD) framework. We prove that the tangent complex, singular locus, inertia stack, and contextual curvature class are canonically determined by the duality adjunction between $\mathfrak{Spec}$ and the global sections functor, together with the reconstruction theorem identifying $A$ with the global sections of $\mathfrak{Spec}(A)$. The Canonical Geometry Theorem establishes that any CSD-compatible geometric structure is induced by the canonical datum consisting of $\mathfrak{Spec}(A)$, its tangent complex, its singular locus, its inertia stack, and its contextual curvature class; hence the geometry of $\mathfrak{Spec}(A)$ is intrinsic to the semantic structure of $A$. We prove that the tangent complex controls the deformation theory of $\mathfrak{Spec}(A)$ and satisfies a Hochschild realization, establishing a direct bridge between geometry and algebra. The assignment sending $A$ to its canonical geometric datum is functorial and Morita invariant, with explicit computations for the complex numbers and matrix algebras demonstrating that noncommutativity, detected by the inertia stack, is distinct from contextuality, which requires additional structures. Thus semantics determines geometry, and the intrinsic geometry of $\mathfrak{Spec}(A)$ provides a canonical geometric encoding of $A$ up to Morita equivalence.

We prove sharp $k$-linear $L^p$ estimates for Carleson–Sjölin oscillatory integral operators with arbitrary separated frequency scales for all $k\ge 2$ and $1\le p\le \infty$. The estimates are sharp, including the endpoint logarithmic behavior for general Carleson–Sjölin phases. Moreover, we obtain log-free endpoint bilinear spectral cluster estimates on every closed three-dimensional Riemannian manifold, resolving a problem of Burq–Gérard–Tzvetkov. As a consequence, we establish sharp $k$-linear $L^p$ spectral cluster estimates for all $k\ge 2$ and $1\le p\le \infty$.

This paper studies the problem of identifying the optimal server assignment policy in single-server queues with customer abandonment. We consider a system with Poisson arrivals and exponentially distributed patience times. We show that when service times follow either an Erlang-$K$ or a hyperexponential distribution and the decision maker can observe the phase of a customer's service time, the Shortest Remaining Expected Processing Time (SREPT) policy maximizes the long-run average throughput, independent of the abandonment rate.

In this article, we consider the anisotropic Calderón problem of determining the potential from boundary measurements of the time-independent Schrödinger equation on compact Riemannian manifolds with boundary. We prove a global uniqueness theorem for small perturbations of the Euclidean metric.

This paper introduces a family of natural generalizations of the pentagram map from polygons to (twisted) polyhedra and proves their integrability through the partial difference operators. A canonical special case, which corresponds to the discrete Laplace transformation of discrete conjugate nets, is investigated in detail. We first establish a canonical bijection between the projective equivalence classes of these polyhedra in $\mathbb{RP}^3$ and the spectral data of doubly periodic partial difference operators modulo the gauge actions. Furthermore, we prove the complete integrability of these pentagram-type maps by explicitly identifying them with the refactorization maps on the Poisson-Lie group of pseudo partial difference operators. This algebraic identification naturally yields an explicit Lax representation and an $r$-matrix induced Poisson bracket for the geometric dynamics.

In 2017, Suda, Tanaka and Tokushige conjectured that if $1>p_1\ge\cdots\ge p_n>0$ with $p_3\le \frac{1}{2}$, then every intersecting family $\mathcal A\subseteq 2^{[n]}$ satisfies $\mu_{\mathbf{p}}(\mathcal A)\le p_1$, where $\mu_{\mathbf{p}}$ is the non-uniform product measure defined by $\mu_{\mathbf{p}}(\mathcal{A})=\sum_{A\in\mathcal{A}} \prod_{i\in A} p_i \prod_{j\in [n]\setminus A}(1-p_j)$. In addition, if $p_1 > p_3$ or $p_1 < \frac{1}{2}$, then equality holds if and only if $\mathcal{A}$ is a star centered at some $i \in [n]$ with $p_i = p_1$. In this paper, we prove this conjecture in the following stronger $t$-intersecting form: for any $t\ge 1$, if $p_{t+2}\le \frac{1}{t+1}$, then every $t$-intersecting family $\mathcal{A} \subseteq 2^{[n]}$ satisfies $\mu_{\mathbf{p}}(\mathcal A)\le \prod_{i=1}^t p_i$. Moreover, when $p_{t+2}<\frac{1}{t+1}$, equality holds if and only if $\mathcal{A}=\{A\subseteq [n]: T\subseteq A\}$ for some $T\in \binom{[n]}{t}$ with $\prod_{i\in T} p_i=\prod_{i=1}^t p_i$. Our result unifies and generalizes the classical theorems of Fishburn-Frankl-Freed-Lagarias-Odlyzko and Friedgut.

We study qualitative properties of positive singular solutions to a weakly coupled elliptic system with critical exponents in a punctured ball. We give a sharp criterion on the removablity of the isolated singularity. We prove that semi-singular solutions (i.e., solutions with one component bounded near the singularity and the other component unbounded near the singularity) do not exist for the dimensions $N\geq 4$ but do exist for $N=3$. Asymptotic symmetry and sharp pointwise estimates are also proved for singular solutions. These generalizes some classical results of (Caffarelli, Gidas and Spruck, Comm. Pure Appl. Math, 1989) to the weakly coupled system. Moreover, for the weakly coupled system, we demonstrate a novel phenomenon that does not arise in the scalar equation.

This paper investigates the long-time behavior of a semilinear beam equation in a domain $\Omega \subset R^{n}$, with memory and fractional damping of the form $(-\Delta)^{\alpha}u_{t}$ ($\alpha \in [0,2]$ the dissipation index). Two critical growth indices of the nonlinear term are determined for smooth and $C^2$ boundaries respectively, concerning the existence of the associated semigroup. We prove the existence of global attractor for the semigroup by showing that it possesses a bounded absorbing set and asymptotic compactness. Furthermore, we find out a new way to obtain, for all $\alpha$, higher regularities than anticipated for the attractors, and the regularity result indicates an interesting phenomenon that even much weaker damping can produce regularity that is infinitely close to that in the case of strong damping ($\alpha =2$). As a consequence, our regularity result deepens and extends the existing related ones for the case when the memory is absent.

The Grone–Merris conjecture, proved by Bai in~2011, states that the spectrum of the graph Laplacian $\Delta_0 = -\operatorname{div}\operatorname{grad}$ is majorized by the conjugate of the vertex degree sequence. Duval and Reiner proposed a simplicial complex analogue of this statement. On a graph, where triangles serve as $2$-simplices, their conjecture reduces to the assertion that the spectrum of $\operatorname{curl}^*\operatorname{curl}$ is majorized by the conjugate of the second-order degree sequence, which records the number of triangles containing each vertex. We prove that the sum of the two largest eigenvalues of $\operatorname{curl}^*\operatorname{curl}$ does not exceed the sum of the first two entries of that conjugate sequence. This confirms the first two majorization inequalities predicted by Duval and Reiner for $\operatorname{curl}^*\operatorname{curl}$. As a corollary, we obtain upper bounds for the two largest eigenvalues of the full graph Helmholtzian $\Delta_1 = -\operatorname{grad}\operatorname{div} + \operatorname{curl}^*\operatorname{curl}$. The same result extends to the up-Laplacian of any $3$-family, yielding a concrete step towards the Duval–Reiner conjecture in dimension~$1$.

The generalized Yang-Baxter equations are multi-site versions of the standard Yang-Baxter equation. When spectral parameters are included, such equations are expected to lead to integrable Hamiltonians with local interactions involving multiple degrees of freedom. In this work we characterize both the continuous and discrete symmetries of these equations required to establish an equivalence class of solutions. We find that the set of such symmetries depends on the number of sites on which the equation is supported. In several cases there are more symmetries than the standard Yang-Baxter equation, thus placing heavy constraints on the number of inequivalent solutions and the associated integrable models. As an application we show how to restrict the generic $16\times16$ ansatz to the 30-dimensional subspace invariant under a chosen discrete symmetry subgroup. An explicit $16\times 16$ solution is also written down.