Papers

The mean first-passage time (MFPT) provides a universal temporal measure of transport, reaction, search, and switching processes in physical, chemical, and biological systems. Understanding how MFPTs respond to perturbations is therefore crucial for prediction and control, yet a systematic theory has been lacking. We establish a compact theoretical framework for linear and nonlinear MFPT response in continuous-time Markov processes. The key tool is an exact correspondence that maps the intrinsically transient response of MFPTs onto the steady-state response of an auxiliary system. This correspondence yields exact and universal response relations for MFPTs between arbitrary state pairs, expressed entirely in terms of unperturbed MFPTs and steady-state probabilities. We then obtain a factorized physical decomposition of the MFPT response into linear upstream, linear downstream, and nonlinear contributions. Further corollaries include response-curve inference rules, fundamental bounds on MFPT responses, analytical expressions for higher-order responses of MFPTs and steady-state probabilities, and multi-rate response formulas. Additionally, our result offers computational advantages in calculating both MFPTs and steady-state distributions. Finally, a biologically motivated folding network is analyzed, and a recently reported paradox on MFPT is clarified.

In this article we investigate the (non)-coincidence of critical parameters for various related percolation problems. More precisely, for the random walk loop soup we show that on $\mathbb Z^d$, $d\ge 5$, the critical parameters for the percolation problems differ on the discrete graph and the respective metric graph. Moreover, on trees we deduce an analogous statement as well as the coincidence of the critical parameters for percolation and susceptibility for a more general class of percolation problems, the so-called Poisson zoo. Along the way we develop the useful notion of sensitivity to Bernoulli enhancements of such percolation problems with long range correlations, which builds on previously developed enhancement ideas.

AI agents are increasingly used in mathematics research, but it is often unclear how to use them effectively. Towards this, we present an extensive case study of how AI was used to improve bounds on the Grothendieck constant $K_G$, which captures the hardness between combinatorial problems and their continuous relaxations. Specifically, while the precise value of $K_G$ is not known, we recently tightened the best known bounds to \[ \frac{6\pi}{11} \;\le\; K_G \;\le\; \frac{\pi}{2\log(1+\sqrt2)} - 10^{-4}. \] Crucially, these improvements were achieved using an AI research system that could arrive at insights deemed novel by domain experts. We give a detailed discussion of our experience using AI for mathematics research, particularly touching upon its strengths and weaknesses, as well as our experience with creating ideal conditions for AI to arrive at breakthrough insights.

In this work we give a counterexample to maximal $\mathrm{L}^2$-regularity in Lions' problem for divergence-form differential operators. On a bounded interval, we construct a bounded, uniformly elliptic, real scalar diffusion coefficient that is $\frac{1}{2}$-Hölder continuous in time with values in spatial $\mathrm{L}^\infty$. It can be chosen arbitrarily close to the constant coefficient of the heat equation. For zero initial data and a forcing term that is continuous in time with square-integrable spatial values, the unique Lions variational solution has a time derivative that is not square integrable in space-time. Thus $\frac{1}{2}$-Hölder continuity alone does not imply maximal $\mathrm{L}^2$-regularity, even for arbitrarily small scalar perturbations of the heat equation. The construction is based on a lacunary family of oscillatory trigonometric modes localised on shrinking time intervals. The spatial profile and the oscillatory modes, together with their first spatial derivatives, vanish at both endpoints. This permits zero extension of the counterexample to the real line. Tensorisation and localisation by parabolic rescaling then yield real symmetric isotropic counterexamples on $\mathbb{R}^d$ and on every bounded domain $\Omega\subset\mathbb{R}^d$, for all $d\ge1$.

In the first part of this work, we have, for a not necessarily semisimple modular category, generalized the action of the mapping class groups of surfaces on the spaces of conformal blocks to the so-called derived block spaces. In the second part presented here, we compute this action explicitly in the case of Drinfel'd doubles of finite groups over fields of positive characteristic. To do that, we connect Lyubashenko's approach to mapping class group representations with the theory of representation varieties. In this way, we are able to show that the mapping class group representations on the derived block spaces are in general different from those on the ordinary block spaces.

We introduce the notion of "$k$-thresholds” and show that Talagrand's discrete convexity conjecture is equivalent to the assertion that, for some universal integer $k \ge 2$, the $k$-threshold of every increasing family is at most a universal constant times its expectation threshold. We prove a reduction theorem that bounds the $k$-threshold of any increasing graph property in terms of ordinary thresholds of graphs in suitable decompositions of its members. As a consequence, we determine, up to a constant factor, the $k$-threshold of every fixed graph in terms of a natural $k$-density parameter. We also prove that $k=2$ suffices for several classical spanning graph containment properties. More generally, we establish the conjectured comparison between $k$-thresholds and expectation thresholds for broad classes of graph containment properties whose target graphs have low degeneracy.

Persistent tensors form a recursively defined class adapted to the substitution method and provide nontrivial lower bounds on tensor rank. We study the behavior of symmetric persistent tensors under Kronecker products. We establish a global differentiation identity expressing the Hessian matrix of a Kronecker product of homogeneous polynomials in terms of the Hessian matrices of its factors; although no corresponding determinant identity holds globally, the Hessian polynomials factor exactly at decomposable points. This yields a multiplicative formula for the distinguished Hessian coefficients of isobaric forms and closure under Kronecker products for symmetric persistent tensors that are isobaric of the distinguished weight, including iterated products and powers. Combined with the classification in small dimensions, this implies closure when both factors have dimension at most three, and for persistent cubics when both have dimension at most four. We further give sufficient closure criteria via simultaneous strict triangularizability of normalized Hessian spaces, for cubics and then arbitrary degree, and show that the distinguished isobaric class satisfies this condition. Finally, we show that symmetric persistence is not preserved in general by constructing a persistent cubic $f\in\operatorname{Sym}^3\mathbb{C}^{12}$ such that $f\boxtimes f\in\operatorname{Sym}^3\mathbb{C}^{144}$ is not persistent. The obstruction occurs at a nondecomposable point, showing that the Hessian factorization on the Segre variety does not extend to the full tensor product space.

In this work, we derive selection rules imposed by Liouville integrability conditions for monomial charge densities with arbitrary powers. For a certain monomial Hamiltonian system, the selection rules reduce to a negative Pell equation, and its solutions generate an infinite set of integrals of motion that are mutually in involution. Furthermore, we study the correspondence between combinatorial polynomial sequences and Liouville integrable Hamiltonian field theories in 1+1 dimensions. We show that the Motzkin system coincides with the dispersionless limit of the Levi system, while the binomial system is equivalent to the dispersionless derivative nonlinear Schrödinger equation. Additionally, we show that the binomial Hamiltonian model admits a reduction to the inviscid Burgers equation and its higher-order charges generate generalized Burgers-type conservation laws.

We study the near-origin behavior on $[0,T]$ of the singular diffusion whose transition density is given by a Doob transform of the integral kernel of the semigroup generated by the $d$-dimensional Schrödinger operator $L^{\gamma}$ with a one-point potential at the origin, where the driving family is the ground state of $L^{\gamma}$ and $d\in\{2,3\}$. We construct a submartingale whose increasing component grows only at times when the diffusion visits the origin. Using this submartingale, we show that the diffusion hits the origin with positive probability and that, conditionally on hitting the origin by time $T$, the first hitting time has a truncated generalized inverse Gaussian (GIG) distribution. We further study the dynamics under conditioning to avoid the origin: under the conditional law, the diffusion is not a standard Brownian motion, but instead admits a representation in terms of a regularized drift and a continuous martingale. While these properties are known in dimension two, the present submartingale-based approach provides an alternative verification and treats dimensions two and three in a unified manner.

A Lotka-Volterra type system with porous diffusion, which can be used as an alternative model to the classical Lotka-Volterra system, is under study. Multiparameter families of exact solutions of the system in question are constructed and their properties are established. It is shown that the solutions obtained can satisfy the zero Neumann conditions, which are typical conditions for mathematical models describing real-world processes. It is proved that the system possesses two stable steady-state points provided its coefficients are correctly-specified. In particular, this occurs when the system models the prey-predator interaction. The exact solutions are used for solving boundary-value problems. The analytical results are compared with numerical solutions of the same boundary-value problems but perturbed initial profiles. It is demonstrated that the numerical solutions coincide with the relevant exact solutions with high exactness in the case of sufficiently small perturbations of the initial profiles.

We prove several bounds on the largest and smallest eigenvalues of the combinatorial Hodge Laplacian $\Delta^H_k$ of a finite simplicial complex $\Sigma.$ As a consequence, we obtain new vanishing criteria for cohomology groups $H^k(\Sigma,\mathbb{R)}$ and confirm a conjecture of O on the dimensional monotonicity of the largest eigenvalue.

We introduce and start the study of a variant of packing functions in graphs. Given a graph $G$ with vertex set $V$ and nonnegative integer vectors $\mathbf{k}=(k_v)_{v\in V}$, $\boldsymbol\ell=(l_v)_{v\in V}$ and $\mathbf{u}=(u_v)_{v\in V}$, a function $f : V \rightarrow \mathbb{Z}_0^+$ is a Released $( \mathbf{k}, \boldsymbol\ell, \mathbf{u})$-packing function of $G$ if $l_v\leq f(v)\leq u_v$ for every $v\in V$ and the sum of the values of $f$ over the closed neighborhood of vertices $v$ with $f(v) = u_v$ is at most $k_v$. The weight of $f$ is the value $f(V) = \sum_{v\in V} f(v)$. We study the associated decision problem (RPP), which asks, given $G$, $\mathbf{k}$, $\boldsymbol\ell$, $\mathbf{u}$ and an integer number $x$, whether $G$ admits a Released $( \mathbf{k}, \boldsymbol\ell, \mathbf{u})$-packing function of weight at least $x$. We relate RPP to the $r$-dependent set problem, derive several NP-hardness results, model RPP as a compact (polynomial in size) Integer Linear Program, and take the first steps of a polyhedral study.

A technique for realizing a universal set of fault-tolerant quantum operations is the code switching method, which leverages two quantum codes with complementary sets of transversal gates. To date, the application of this technique has been largely limited to families of color codes supporting a logical $T$ gate. No analogous code switching protocols exist for many other prominent families, such as rotated surface codes, or for finer $Z$-rotation gates. In this work, we first utilize the doubling technique as a unified framework to construct a class of quantum color codes encoding a single logical qubit with an arbitrarily large minimum distance, enabling the transversal realization of arbitrary small logical $Z$-rotation gates. We investigate the structural properties of this code family, demonstrating that they improve upon the parameters of state-of-the-art triorthogonal codes, achieve lower qubit overhead compared to certain known color codes, and admit single-shot decoding of $Z$-syndromes via meta-checks. Furthermore, we show that this framework extends beyond color codes; specifically, it enables the generation of $r$-orthogonal quantum codes, $r \ge 2$, that inherit the local geometry of rotated surface codes. We then provide an overhead optimization protocol alongside several candidate codes tailored for realizing logical $Z$-rotation gates within rotated surface codes. Finally, we extend the fault-tolerant code switching protocol based on transversal CNOT gates to incorporate fault-tolerant realization of $Z$-rotation gates at any level of the Clifford hierarchy for geometries compatible with rotated surface codes. We present the first demonstration of fault-tolerant magic state preparation by means of code switching within a distance-three rotated surface code using a total footprint of only 45 physical qubits, and evaluate its performance through a simulation.

We show that the von Neumann entropy of an unknown $d$-dimensional quantum state $\rho$ can be estimated to within additive error $\varepsilon$ using \[ \Theta\!\left(\frac{d^2}{\varepsilon\log^2(d)(1+\varepsilon\log^2(d))} + \frac{\log^2(d)}{\varepsilon^2}\right) \] samples of $\rho$, matching the lower bound due to Wang (2026). Our estimator is based on a high-degree approximation polynomial and the main technical challenge is bounding its variance despite the large polynomial degree. We overcome this using a quantum Hoeffding decomposition, which preserves cancellations between different polynomial terms and reduces the analysis to derivative bounds for the approximation polynomial.

The class of Weierstrass functions $W_{a,b}$ is one of the first class of examples of continuous and nowhere differentiable real functions. A challenging line of research has been to determine the various dimensions of graphs $G(W_{a,b})$ of such functions. For instance, the Hausdorff dimension of $G(W_{a,b})$ was only recently determined by Shen in 2018, after a long series of partial results by many different authors. While the Assouad dimension of $G(W_{a,b})$ remains an open problem, also posed as a question by J. M. Fraser, there have been many indications that it might be equal to $2$. Such indications include the graph of Wiener processes, the graphs of almost all Hölder functions in the Baire category sense, and the graphs of Weierstrass functions after a series of countably many reflections all having Assouad dimension equal to $2$. In this paper we show that this is not the case, providing a quantitative upper bound on the Assouad dimension of $G(W_{a,b})$ that is strictly less than $2$. In particular, we show that such a bound is true for a class of generalized Weierstrass functions $W_{a,b}^\phi(x) = \sum_{j=0}^\infty a^j\phi(b^j x)$, which includes $W_{a,b}$ and the class of Takagi functions. The latter fact is used to also answer in the negative a conjecture of H. Yu on the Assouad dimension of graphs of Takagi functions for parameters $a\in (0,1)$, $b\in (1/a, \infty)\cap \mathbb{Z}$.

Voltage certificates for droop-controlled radial distribution networks are often constructed from worst-case quantities. In heterogeneous radial networks, this approach can hide where voltage deviations are most likely. Furthermore, such certificates become increasingly conservative as the network sensitivities accumulate. Leveraging the structure of the linearized DistFlow model and slope-restricted droop controllers, we derive tighter deterministic voltage certificates that retain heterogeneous network, disturbance and inverter characteristics of each bus. The certificates reveal the buses and local limitations that dominate certified voltage performance. Worst-case bounds are recovered as a special case. Although tighter, the heterogeneous certificates can still deteriorate downstream due to the network structure. To address this, we leverage our certificates as design variables for coordinated voltage control. Specifically, we develop a virtual droop architecture with coordination and affine feedforward compensation to reshape the effective voltage sensitivity, and formulate a linear program that jointly synthesizes the controller and minimizes its heterogeneous voltage certificates under operational, communication and inverter placement constraints. The resulting controller guarantees the voltage and inverter bounds for all admissible disturbances. Evaluation on two network benchmarks, a five-customer residential feeder and a 26-customer rural network comprising four feeders, demonstrates improved certificates and voltage behaviour under inverter limits.

We determine the smallest number of active questions per player at which a four-player binary exclusive-or (XOR) game of commuting-operator value one need not admit a Greenberger–Horne–Zeilinger (GHZ) equatorial realization. Such a realization uses the four-qubit GHZ state and equatorial qubit observables, reducing perfect play to additive phase equations. We prove that every four-player XOR game with commuting-operator value one and at most three active questions per player has a perfect GHZ-equatorial strategy. Conversely, we construct a uniform eight-clause game with four active questions per player whose commuting-operator value is one but whose phase equations are inconsistent. Thus four is the sharp local-question threshold. The positive result follows by lifting every integral incidence obstruction to an ordered noncommutative refutation, using primitive circuits, forest matchings, and ternary Hamming geometry. For the separating game, a Klein four-group incidence relation obstructs the phase system, while an even-subgroup normal form and degree-one and degree-two Magnus coefficients exclude refutations of arbitrary length.

We study the geometry of simplicial cochains on Vietoris-Rips complexes built from i.i.d. samples of a compact embedded manifold. By integrating differential forms over affine simplices, we define a cochain map from smooth forms to simplicial cochains and equip the latter with kernel-weighted inner products determined by ambient pairwise distances. At scales where the sampled complex recovers the homotopy type of the manifold, we prove quantitative high-probability convergence of these inner products and, under uniform sampling, of the associated codifferential energies to their continuum counterparts. As consequences, we obtain spectral upper bounds for the discrete Hodge Laplacian and, for orientable manifolds, convergence of harmonic representatives of discretized harmonic one-forms and consistency of harmonic smoothing for circular coordinates.

We propose a comprehensive framework for solving overlapping-generations (OLG) models in continuous time with both idiosyncratic and aggregate risk. Our general characterization of equilibrium through the master equation operates on the joint distribution over the continuous idiosyncratic states, age and wealth. Our computational strategy is to take a finite-dimensional representation of this distribution as an input of a neural net which in turn outputs a finite-difference representation of the (conditional) value function. This idea can be applied generally to heterogeneous agent models with aggregate risk, and we call it finite-difference neural operator. Our method combines advantages from modern neural nets and traditional finite-difference methods: It is grid-free in the high-dimensional distribution, and retains control on boundary conditions in low-dimensional state variables. Moreover, our method is able to enforce shape constraints. We showcase its flexibility by solving a continuous-time OLG model with aggregate risk alone where we characterize the distribution by its supporting function; and to an OLG model with both types of risk.

For $s\ge1$, let $G_s$ be the bipartite graph between the $s$-subsets and the $(s-1)$-subsets of $[2s]$, where adjacency means disjointness, and let $w(s)$ be the maximum number of $s$-subsets on an induced path in $G_s$. We prove $w(s)\ge \frac{4^s}{2048s^{5/2}}$ for all $s\geq 6$. This implies $\sup_{s\ge1}w(s)^{1/s}=4$, as conjectured by Kohayakawa (1991). His recursive construction then gives induced paths of order $\Omega(4^r/r^{5/2})$ in the Kneser graph $KG(2r+1,r)$ and yields \[ \max\{\cc(\overline{P_n}),\ \cc(\overline{C_n})\} \le \log_2 n+\frac52\log_2\log_2 n+O(1). \] Together with the known lower bounds, this settles a conjecture of de Caen, Gregory, and Pullman (1985) and gives \[ \cc(\overline{P_n})=\log_2 n+\Theta(\log_2\log_2 n), \qquad \cc(\overline{C_n})=\log_2 n+\Theta(\log_2\log_2 n). \] We also give an independent proof of the latter order estimates. It uses a Hamiltonicity result of Kneser graphs and a key lemma proved by the Lovász local lemma.

We study posterior contraction in positive-order Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families. We embed the natural parameter in a Hilbert scale and model it via a Gaussian series prior expanded in the eigenbasis generating the scale. Under a two-sided link condition on the Fisher information and suitable local regularity assumptions, we show that smoothness-matching priors achieve minimax-optimal posterior contraction rates in any Hilbert scale norm up to the regularity of the ground truth. Our analysis builds on the novel approach to posterior contraction based on the Wasserstein distance recently introduced by Dolera et al. (2024, Probab. Theory Relat. Fields). It combines Laplace-type approximations for infinite-dimensional integrals associated to the posterior kernels with a mixed-geometry estimate controlling their stability under fluctuations in the data, itself resting on a tailored Poincaré inequality for posterior distributions conditioned on neighbourhoods of the truth. We apply the general theory to density estimation under the logistic parametrisation, Poisson intensity estimation under the exponential link, and mildly ill-posed linear inverse problems observed in Gaussian white noise. The resulting minimax rates yield optimal recovery of density score functions, derivatives of Poisson intensities and, in a concrete elliptic inverse problem, the unknown source function.

The work is a natural continuation of that published in Stud Appl Math. 2024; 153:e12737. All possible two-components evolutions systems of (1+2)-dimensional second-order PDEs admitting an infinite-dimensional Lie algebra are constructed. It is shown that a natural generalisation of this Lie algebra to the higher-dimensional case does not lead to a more general result because the infinite-dimensional symmetry is broken. The recently derived system, which is related to Ricci flows, is identified as a very particular case among the evolution systems obtained. All possible stationary solutions of this system in the radially symmetric case are constructed using the surprisingly rich Lie algebra of the reduced system of ODEs. Moreover, it is proved that this Lie algebra is reducible to the fifteen-dimensional algebra of the simplest system of two second-order ODEs. Several time-dependent exact solutions in the radially symmetric case are constructed as well. It is shown that the solutions obtained are bounded and smooth provided arbitrary parameters are correctly specified.

A discrete group $\Gamma$ has the \emph{$C^*$–ISR property} if every unital $\Gamma$–invariant $C^*$–subalgebra $A\subseteq C_r^*(\Gamma)$ is of the form $C_r^*(N)$ for a normal subgroup $N\triangleleft\Gamma$. We show that $\text{PSL}_n(\mathbb{Z})$ satisfies the $C^*$–ISR property for every $n\geq3$.

We prove that for every integer $N\geq 3$ and $\alpha\geq \frac{1}{2}$, Beckner's inequality \begin{equation*} \frac{\alpha}{2}\int_{\mathbb{S}^N}u(P_{N}u) dw+(N-1)!\int_{\mathbb{S}^N}u dw-\frac{(N-1)!}{N}\ln\int_{\mathbb{S}^N}e^{Nu} dw\geq 0 \end{equation*} holds for any axially symmetric $u\in H^{\frac{N}{2}}(\mathbb{S}^N)$ whose center of mass is at the origin. The proof is mainly based on a weighted $\ell ^2$ estimate on Gegenbauer coefficients and a rigidity theorem for stable critical points. Hence, we answer the generalized Chang-Yang conjecture positively in the axially symmetric case for every integer $N\geq 3$.

Active-quiescent switching occurs in biological populations in which growth is confined to a proliferative active state, while cells may reversibly enter a nonproliferative quiescent state. Experiments often observe only part of this process, through active-state markers, aggregate population measurements, or aggregate data supplemented by sparse active-state observations. Such measurements arise when marker panels are limited, active-state assays require fixation or endpoint sampling, or only total cell number, optical density, tumor burden, or aggregate fluorescence is reported. These observation choices complicate sparse regression methods such as sparse identification of nonlinear dynamics (SINDy), because the measured variable may satisfy a different equation from the underlying active-quiescent system. A library chosen for the wrong observable may therefore fit a trajectory without preserving mechanistic interpretation or transferability. We study this issue using a two-compartment ordinary differential equation model. We define observable reduction as the elimination of hidden states to obtain the differential equation satisfied by the measured variable. For several biologically relevant growth laws, we derive observation-specific reductions and use them to construct sparse-regression libraries. Using synthetic data, we compare these structured libraries with standard polynomial SINDy. Polynomial libraries can match training trajectories while failing coefficient-relation and transfer tests, whereas reduction-guided libraries recover interpretable coefficient maps when the observed regime is informative. These results show that interpretable equation learning in hidden-compartment systems requires matching both the regression target and candidate library to the observation process.

The first aim of this paper is to show that there is merit to the question posed in the title. Indeed, for certain function classes, approximation by rational functions and by piecewise polynomials are surprisingly similar. Between these two, polynomials are more widely studied and more widely used. One setting in which both approaches are equally well understood, and equally practical in use, is that of approximating univariate functions with point singularities. In this context we can fully address the question. We review classical literature on the topic which shows that both approaches do indeed achieve similar convergence rates. However, rational approximations come with significantly smaller constants. Owing to recent advances in practical rational approximation, we can augment the discussion with a comparison of modern numerical techniques that achieve the optimal rates. We end by showing numerically that the difference becomes even more pronounced in several variables.

We prove that if $X$ and $Y$ are first-countable perfect spaces such that the free Abelian topological groups $A(X)$ and $A(Y)$ are topologically isomorphic, then $X$ is a hereditarily Baire space if and only if $Y$ is hereditarily Baire as well. We also establish that for any Tychonoff space, if there exists a continuous linear surjection of the space $C_p(X)$ onto the space $C_p(Y)$ and the space $X$ is either strongly $\sigma$-scattered or has property $(\kappa)$, then $Y$ also satisfies those properties. Additionally, we obtain the following result: if $X$ and $Y$ are Tychonoff spaces in which every closed set has a $W$-point, and if the free Abelian topological groups $A(X)$ and $A(Y)$ are topologically isomorphic, then $X$ is scattered if and only if $Y$ is scattered.

Consider the two-dimensional semiclassical Schrödinger operator $P_\hbar=-\frac{\hbar^2}{2}\Delta+V$ on $\mathbb R^2$, where $V$ has a nondegenerate well at the origin, its harmonic frequencies are rationally independent, and $V$ is $\mathbb Z_2$-symmetric. We prove that, generically, the first three layers of the quantum Birkhoff normal form (QBNF) determine the full Taylor series of $V$ at the origin, up to the unavoidable spatial inversion $V(x)\mapsto V(-x)$. The generic condition depends only on the jet of $V$ through order ten. Consequently, for real analytic potentials with a unique isolated global well at the origin, the low-lying semiclassical spectrum generically determines $V$ up to spatial inversion.

We compare two mirror-theoretic degeneration pictures attached to odd-dimensional projective space. If \(\PP^{2n-1}\) is viewed as a toric variety, one obtains the classical toric mirror. If it is viewed as the type \(C_n\) homogeneous space \(Sp_{2n}/P_1\), Rietsch's Lie-theoretical construction gives a superpotential on the dual side. We compute the type \(C_n\) boundary \(D_C\), while on the dual-side Lusztig torus we compute the corresponding Laurent polynomial. We construct Newton–Okounkov bodies, and in each case the degree-augmented value semigroup is identified with the lattice-point semigroup of the cone over the polar dual of the corresponding Newton polytope. This gives two toric degeneration pictures attached to the same projective space. We also exhibit a rank-one weight degeneration connecting \(D_C\) with the standard toric boundary while keeping the ambient projective space fixed.

In this paper, we establish a system-wide $\ell^1$-norm Harnack-type inequality for positive solutions of weakly coupled nonlocal diffusion systems on $\mathbb{R}$, without assuming cooperativity or irreducibility of the coupling matrices. As a primary application, we integrate this analytical tool with a rescaling argument to establish the existence of the minimal wave speed for traveling waves for a class of non-cooperative nonlocal diffusion systems with network structures. Furthermore, we derive the precise asymptotic behavior of wave profiles at negative infinity. This is achieved by combining Harnack-type inequalities and Ikehara's theorem with Riesz projections to analyze singularities of vector-valued Laplace transforms.