Papers

The Markov linear recurrence sequence in a random environment (MRSRE) is a particular case of Markov chain on non-negative integers. It is a natural generalization of several models with branching. We consider MRSRE with absorption at zero and study the survival time. We introduce a classification of subcritical MRSRE corresponding to that of branching process in a random environment. We show that general R-positivity theory can be applied to this model in a strongly subcritical case and prove the analogues of the Kolmogorov theorem and the Yaglom theorem. In other words, we describe the asymptotic behavior of the survival probability over a long time and the conditional distribution of the sequence, conditioned on the survival event. The results are applied to particular branching models.

We provide the spectral decompositions of $(\star,\epsilon_1,\epsilon_2)$-structured matrix polynomials $P(\lambda)$ in the unified form by a standard pair $(X, J)$ and a parameter matrix $\Gamma$. Using the recursive relationship between the coefficient matrices of $P(\lambda)$, equivalent expressions of these coefficient matrices are provided. When $J$ is assumed to be a block diagonal matrix, we show that the parameter matrix $\Gamma$ has a special structure.

Generalised Bayesian inference tempers the likelihood by a learning rate $\eta$ to mitigate model misspecification, and the choice of $\eta$ is consequential. Zafar and Nicholls (2024) proposed selecting $\eta$ by a posterior predictive check (PPC): one chooses the smallest $\eta$ at which a log-likelihood PPC $p$-value is not rejected. An exact, finite-sample analysis of this selector on the Gaussian linear model is given. With known variance and a flat prior, the PPC $p$-value equals $P(\chi^2_n > \mathrm{RSS}/\sigma_0^2)$ for every $\eta$, so the selector is $\eta$-invariant; under variance misspecification it is two-sided non-identifying. With unknown variance and the reference prior, the $p$-value depends only on $(n,d,\eta)$ and not on the realised data or the data-generating process. Consequently the selector's output is fixed before any data are seen, typically collapsing to the smallest grid value, which over-tempers and inflates predictive intervals relative to held-out selection. The phenomenon is a pivotality property specific to the Gaussian scale–location family and the reference prior; it disappears under informative priors. These results delineate the selector's scope, identify a canonical class on which it cannot identify the learning rate, and motivate a cheap, data-free pre-screening diagnostic.

In this brief note, we present a cyclic proof system developed specifically for the equational version of Primitive recursive arithmetic and establish the equivalence of the two systems. A distinctive feature of our approach is that it does not rely on automata-theoretic methods but is implemented primarily using the tools of structural proof theory.

We discuss the numerical approximation of the stochastic Cahn–Hilliard equation with a singular double-obstacle potential and multiplicative conservative noise. We propose a regularised fully discrete finite element approximation scheme for the problem and show that it satisfies stability estimates which are uniform with respect to the discretization parameters. We show convergence of the approximation for vanishing discretization parameters towards a regularised version of the singular stochastic Cahn–Hilliard equation by monotonicity arguments. Hence, thanks to a uniform $H^1$-estimate for the regularised problem we show that the regularised solution converges towards the pathwise unique probabilistically strong solution of the original singular stochastic Cahn–Hilliard equation. We conclude by presenting numerical simulations where we compare the regularised numerical approximation to its unregularised counterpart and illustrate the effect of the conservative noise.

We give a new proof of Webb's theorem on maximal central hyperplane sections of the regular \(n\)-simplex \(\Delta_n\), viewed in its standard embedding in \(\mathbb R^{n+1}\). A similar method also yields sharp maximal estimates for non-central sections of \(\Delta^n\) whose distance \(d\) from the barycenter is small, namely $d< \sqrt{\frac{1}{(n+1)(2n+1)}}.$ Moreover, we obtain sharp volume estimates for central hyperplane sections of the cross-polytope \(B_1^n\) that pass through the barycenter of a facet.

In the setting of a complete, smooth properly immersed mean curvature flow, we assume uniformly bounded $|H|$ and $|\nabla H|$ on $M^n\times[0,T)$ and some bounded initial geometry to get local spatial $L^p$ estimates for the second fundamental form with $p\in[4,\infty)$. For $p>n+2$, this leads us to a local space time $L^\infty$ bound for the second fundamental form which allows us to smoothly extend the flow $F:M^n\times [0,T) \rightarrow \mathbb{R}^{n+1}$ past the singular time $T<+\infty$ for a short time.

Shallow water equations (SWE) are fundamental models in fluid dynamics that are essential for studying a wide range of geophysical and engineering phenomena. In many practical applications, uncertainties arising from initial conditions and bottom topography must be taken into account, motivating the development of stable and accurate numerical methods for stochastic SWE. Building on the hyperbolicity-preserving stochastic Galerkin formulation for SWE [Dai, Epshteyn, Narayan, 2021 SISC] and a stochastic extension of the entropy stable discontinuous Galerkin methods for skew-symmetric SWE [Fu, 2022 JSC], we develop a structure-preserving, entropy conservative, and entropy stable discontinuous Galerkin–stochastic Galerkin method for the stochastic shallow water system, with the well-balanced property. We demonstrate the accuracy, applicability, and robustness of the proposed structure-preserving algorithms through several numerical experiments.

Physics-informed learning is increasingly used for partial differential equation (PDE)-governed inverse problems, but its reliability remains difficult to certify. This paper develops a no-harm certification-and-selection framework for physics-informed inverse learning. A learned reconstruction is accepted only when its residual-calibrated radius is no worse than the baseline radius, namely when $$R_{\mathrm{learn}}\le R_{\mathrm{base}}+\varepsilon_{\mathrm{safe}};$$otherwise, the method returns the baseline. The certificate combines data, physics, boundary or initial-condition, and optimization residuals. Under a conditional stability estimate, these residuals yield an a posteriori reconstruction-error bound and a deterministic uncertainty radius. A high-probability certificate is also derived for physics residuals estimated from independent random collocation points. Numerical tests on Poisson source recovery, inverse heat reconstruction, limited-angle tomography, elliptic coefficient identification, and stochastic residual validation show that the selector accepts certified improvements, rejects shifted, hallucinated, or unfinished candidates, and becomes conservative in strongly ill-posed regimes. The framework is therefore a certification-and-selection layer, not another reconstruction architecture.

In this paper we present a construction for unstructured splines over quadrilateral meshes by iterative averaging and refinement. We represent the spline as a multi-patch B-spline, where the degrees of freedom are those B-spline coefficients on the quadrilateral patches that are not associated with interior edges and vertices of the mesh, i.e., their corresponding Greville points lie inside the patches. In every averaging step, we replace the remaining B-spline coefficients associated with interior edges and vertices by suitable averages of neighboring degrees of freedom. In the refinement step we apply regular splits to all patches by knot insertion. This process results in a subdivision scheme that, for degree $p=2$, is similar to the almost-$C^1$ spline construction from (Takacs, Toshniwal. CMAME, 2023) and behaves similar to Doo-Sabin subdivision, cf. (Doo, Sabin. CAD, 1978), and that can be defined for arbitrary degrees and regularities inside the patches. We derive two families of spline constructions, based on simple and coplanar averaging, respectively, and analyze their spectral properties when interpreted as subdivision schemes. Using this interpretation, we show that they are $C^1$ in the limit. Moreover, the coplanar averaging scheme produces splines that are $C^1$ at all vertices for every level of refinement, whereas the simple averaging is $C^1$ only in the limit. For both constructions, we have control over the subdominant eigenvalue, which has multiplicity two and can range between $\frac{1}{4}$ and $1$, with $\frac{1}{2}$ often being the desired option. The resulting basis functions form a partition of unity. Moreover, they form a non-negative partition of unity for suitably selected averaging parameters.

The well-known Stein–Tomas restriction theorem gives the sharp range of $p$ for which $L^p\to L^2$ restriction estimates hold for the surface measure on the sphere. This was generalised to arbitrary measures satisfying certain Fourier decay and Frostman conditions by Mockenhaupt, Mitsis, and Bak–Seeger, with the most general version now a fundamental result in harmonic analysis. The Frostman condition essentially asks for uniform control on the measure of small balls and is the endpoint of a continuum of more nuanced conditions which describe the local fluctuations of the measure. This analysis gives rise to the $L^q$-dimensions of a measure and these are a central concept in fractal geometry and a crucial tool in multifractal analysis and the theory of large deviations. In this paper we prove a new Fourier restriction theorem which uses the $L^q$-dimensions instead of the Frostman condition, thus providing a continuum of estimates which recover Stein–Tomas at the endpoint. Our proof gives the endpoint estimate for all values of $q\in(1,\infty]$ via Stein's complex interpolation. In particular, in the case $q=\infty$ this partially resolves a question raised by Bak and Seeger. We explore when our theorem improves on Stein–Tomas, that is, when the range is not optimised at $q=\infty$, and show that this is the case quite generally, including for certain Mandelbrot cascade measures and measures with multifractal behaviour. On the way to proving our main theorem we obtain a novel description of the $L^q$-dimensions based on certain convolution norms, which may be of interest in its own right.

In 1992, Pukhlikov and Khovanskii provided a description of the cohomology ring of toric variety as a quotient of the ring of differential operators on spaces of virtual polytopes. Later Kaveh generalized this construction to the case of cohomology rings for full flag varieties. In this paper we extend Pukhlikov-Khovanskii type presentation to the case of K-theory of toric and flag varieties. First, we study the Frobenius algebras obtained as quotients of the group algebra of free abelian group (possibly of infinite rank). Then we apply this construction to define a K-ring associated to a linear family of (virtual) polytopes. We study in detail two examples of such families: the family of integer (virtual) polytopes with a fixed normal fan and the family of (virtual) Gelfand-Zetlin polytopes. We show that the K-theory of toric and flag varieties can be realized as K-rings of the above families and use this to get natural set of relations in the above K-rings. Further, we describe the classes of structure sheaves of toric orbit closures and Schubert varieties in type A flag varieties. Finally, we show that our results also hold true in T-equivariant setting.

We survey the dimension theory of sets of real numbers with regular continued fraction expansion restricted to a non-empty and possibly infinite subset $\mathcal{D}\subset\mathbb{N}$.

We give an alternative presentation of the small version of the reflection equation algebra associated to $GL_N$ at both odd and even roots of unity, and use our presentation to classify its irreducible representations. We then describe a family of algebras generalizing the small reflection equation algebra, and consider their application to the study of module categories over $U_q(sl_N)$ fusion categories.

We improve the known upper bound for the extremal number of Berge-$C_4$-free $3$-uniform hypergraphs. More precisely, we prove that every $n$-vertex $3$-uniform hypergraph with no Berge cycle of length four has at most \[ \frac{n^{3/2}}{2+\sqrt2}+O(n) \] hyperedges. This improves the previous best-known leading constant $1/\sqrt{10}$ to $1/(2+\sqrt2)$.

We study the minimax estimation error for distributed covariance matrix estimation in the vertical-split (feature-split) setting, where two agents each observe different coordinates of~$m$ i.i.d.\ sub-Gaussian samples and communicate a limited number of bits to a central server. While \cite{rahmani2025fundamental} established nearly tight bounds for dense (unstructured) cross-covariance matrices, we investigate whether imposing elementwise $s$-sparsity on the cross-covariance $C_{21}$ can reduce the required communication and sample complexity. We show that sparsity can reduce the communication required for cross-covariance estimation in the vertical split at a fixed target distortion. Specifically, for sufficiently large $d_1d_2/s'$ and $0<\varepsilon<\sigma^2\sqrt{s'}/32$, any scheme achieving expected Frobenius distortion at most $\varepsilon$ must satisfy $B_k = \Omega(\sigma^4 d_k\, s' \log(d_1 d_2/s')/\varepsilon^2)$ and $m = \Omega(\sigma^4\, s' \log(d_1 d_2/s')/\varepsilon^2)$ for cross-covariance estimation, where $s' = s \wedge d_{\min}$. For the $1$-sparse case, our achievable scheme reduces the $d_1d_2$ factor in the dense communication rate to $\log(d_1d_2)$, up to polylogarithmic factors, for the cross-covariance communication component in the matching regime. Our lower bounds are established via Fano's method with an explicit sparse packing using a Varshamov–Gilbert-type argument for signed partial permutation matrices combined with the Conditional Strong Data Processing Inequality of \cite{rahmani2025fundamental}. We show that the communication lower bound is tight up to polylogarithmic factors under the conditions of Remark 4.9, using an achievable scheme based on covering-net quantization and entry-wise hard thresholding.

We develop a unified DeepONet framework for logarithmically stable inverse problems between infinite-dimensional function spaces, with inverse acoustic scattering as a model application. The framework is formulated at the operator level by separating the learned inverse map into measurement encoding, finite-dimensional neural approximation, and functional reconstruction components. For inverse maps satisfying a logarithmic stability estimate, we establish quantitative a priori error bounds that separate the encoder, finite-dimensional neural approximation, and reconstruction contributions. For prescribed encoder and reconstruction ranks, we obtain a network-size-dependent bound for the finite-dimensional neural approximation error, together with rank-dependent bounds for the encoding and reconstruction errors. For comparison, we also record the corresponding Lipschitz-stable estimate arising from the same error decomposition. The quantitative inverse-scattering analysis is then specialized, in three dimensions, to the recovery of a medium contrast from fixed-frequency far-field measurements. Numerical experiments in two and three dimensions, using both function-space and finite-dimensional priors, illustrate the reconstruction performance and empirical sensitivity to synthetic measurement noise.

We generalize the Einstein condition for Kropina metrics obtained in the positive definite setting by Zhang, Shen and others to all signatures. As examples of Einstein-Kropina metrics $L=L_{a,b}$, we construct new explicit positive definite ones and the first ones with Lorentzian signature. Next, we classify all Einstein-Kropina solutions to the vacuum equation of Finsler gravity by Pfeifer and Wohlfarth, in arbitrary dimension and including a possible cosmological constant $\Lambda$. For the Lorentzian and positive definite cases, the local picture is as follows. In dimension 4 or lower, only the trivial solution exists: a Minkowski or Euclidean space, essentially. In dimension 5 and higher, the solutions are those $L_{a,b}$ for which the metric $a$ is a product of the real line with a Ricci-flat metric (Lorentzian or Riemannian) and the vector field $b$ is the unique unit vector on the first factor. As a very surprising rigidity phenomenon, all Einstein-Kropina solutions to the $\Lambda$-vacuum equation are Berwald and Ricci-flat, and the cosmological constant necessarily vanishes.

Next-generation wireless networks, including satellite-to-Open RAN systems, demand agile and intelligent resource management capable of handling dynamic multi-user interference under stochastic quality of service constraints. This paper introduces DIFFRACT, a neuralized utility maximization framework that leverages differentiable programming to integrate deep learning with optimization in wireless networks. Central to our approach is the exploitation of the mathematical structure of standard interference functions, which are foundational in wireless power control. By developing a duality theory for these functions, we map iterative interference management algorithms into differentiable neural network architectures via algorithm unrolling. This enables distributed, end-to-end gradient-based learning at the network edge, supporting real-time adaptation to interference in both terrestrial and non-terrestrial environments. DIFFRACT allows for scalable and robust utility maximization by modeling complex channel dynamics and leveraging the expressiveness of differentiable models. Experimental results confirm the framework's theoretical soundness and practical effectiveness for next-generation wireless systems.

We establish an equivalence between the existence of Costas polynomials and the existence of a special kind of orthomorphism such that their compositions are also orthomorphisms. Computations are easier over these orthomorphisms. We provide a lower bound for the number of Costas polynomials and derive some of their properties. We show that Costas polynomials, by virtue of being multiplicative analogs of planar polynomials, can also be used to construct complete families of mutually orthogonal Latin squares.

We prove the lack of selection in the zero-noise limit for solutions to SDEs driven by a divergence-free, Hölder continuous vector field with exponent $\alpha\in(0,1)$, arbitrarily close to $1$ but fixed. The result applies to a broad class of regularizing additive noises, including fractional Brownian motion and stable Lévy processes. The proof combines pathwise Lagrangian arguments, based on the analysis of the deterministic flows associated to mixing velocity fields, with probabilistic estimates coming from the stochastic sewing lemma. This allows to show that lack of selection happens simultaneously on a large set of initial data, whose complement has arbitrarily small Lebesgue measure.

Stochastic constrained decision-making requires optimizing performance objectives while enforcing statistical requirements such as safety or fairness. However, standard primal–dual methods struggle to update multipliers robustly under stochastic mini-batch feedback, as the noise of mini-batch gradients and constraint estimates can be directly accumulated into the multiplier memory. To address this issue, we propose Residual-Controlled Multiplier Learning (RCML), which reformulates multiplier updating as projected-pressure feedback. The central idea is to decompose the projected multiplier into an effective pressure signal for primal descent and a pressure-memory residual for finite-gain multiplier tracking. To handle heterogeneous and noisy observations, we further augment this residual-integral backbone with modular stochastic stabilization components. For the convex-affine backbone, we establish finite-gain convergence, derive a stochastic residual bound under mini-batch feedback, and show that the residual feedback law admits a local KKT-residual interpretation near regular KKT points of nonconvex problems. Experiments across optimization, allocation, and fair-ranking tasks show that RCML improves feasibility control and multiplier stability while maintaining competitive objective performance. Code is released at https://anonymous.4open.science/r/RCML-3114/.

In this paper, we develop an inferential framework with sharp asymptotic optimality guarantees for Ising models on inhomogeneous random graphs in the subcritical parameter regime. We begin by characterizing the asymptotic distribution of the maximum likelihood (ML) estimate of the natural parameter, based on a single sample from the underlying model, covering both sparse and dense network regimes. Next, to overcome the computational intractability of the ML method, we propose a simple closed-form estimate obtained from a one-step approximation to the likelihood equation. We show that this estimate attains the same asymptotic distribution and variance as the ML estimate, thereby yielding a computationally efficient and asymptotically valid confidence interval for the natural parameter. We complement these inferential results by establishing a Hájek–Le Cam-type local asymptotic minimax theorem, showing that the proposed estimate achieves the smallest possible asymptotic maximum risk, both in rate and in leading constant, over shrinking neighborhoods of the true parameter. We also derive the corresponding limit of experiments. To the best of our knowledge, these are among the first sharp asymptotic optimality results for network-dependent data. Finally, we study goodness-of-fit testing for the natural parameter, deriving the local power of the likelihood ratio test and minimax detection rates. Our analysis relies on new fluctuation results for the sufficient statistic (Hamiltonian) and for the random partition function of Ising models on inhomogeneous random graphs, which are of independent interest.

We study an inverse source problem for the Monge–Ampere equation \[ \det D^2u=f(x) \] on a bounded smooth uniformly convex domain. In the smooth classical regime, we prove that the Dirichlet-to-Neumann map associated with convex solutions determines the positive source uniquely. The proof uses a family of large boundary values and reduces the inverse source problem to the injectivity of the Euclidean X-ray transform.

This paper provides a (Laurent) polynomial reduction to $q$-holonomic sequences $F_k(q)$. We first characterize Laurent polynomials $\tilde{p}(x)$ such that the product $\tilde{p}(q^k)F_k(q)$ is summable. Then the reduction framework is given to decompose any given Laurent polynomial into a summable part and a remainder with lower degree. Finally, we introduce a power-partible reduction for $q$-holonomic sequences of which the recurrence relation satisfies a certain symmetry condition. The advantage is that it can not only simultaneously eliminate the highest-degree and lowest-degree terms of a Laurent polynomial satisfying a symmetry condition, but also guarantee the symmetry of the remainder. As applications, we apply the reduction to $q$-central-Delannoy numbers to derive new $q$-identities and $q$-congruences.

Variational autoencoders (VAEs) learn low-dimensional latent representations of high-dimensional data. When the data lies on a manifold with non-Euclidean topology, the standard Gaussian prior introduces a topological mismatch that degrades reconstruction quality and prevents faithful representation. We present a constructive mathematical framework that resolves this mismatch for all manifolds that admit a product covering space. These are manifolds expressible as products of elementary factors (circles, intervals, or lines) or as quotients of such products by a finite symmetry group. The class includes cylinders, tori, Möbius strips, Klein bottles, and real projective spaces. Factorized distributions over the elementary factors yield product topologies with closed-form, decoupled KL divergences, so that each latent factor can be shaped independently while keeping training tractable. We catalogue reparametrizable encoder-prior pairs for periodic, bounded, and unbounded supports, and provide coordinate transformations that allow standard neural networks to output non-Euclidean parameters with smooth gradients. For quotient manifolds, the decoder receives group-invariant features of the covering-space coordinates, so that identified points produce identical outputs. Anchor constraints fix the coordinate system relative to the data or create soft topological holes. Experiments on synthetic manifolds and real-image datasets (rotated and cyclically shifted MNIST) confirm that a topology-matched prior aligns KL regularization with the data manifold. The resulting topology-aware models outperform the Gaussian baseline at all practically relevant regularization strengths. The code is available at https://github.com/JvHulst/VAE-Topology.

We study quantum anomalies associated with $U(1)$ 1-form symmetries from the perspective of invertible phases and bordism. We compute the oriented and spin bordism groups of the Eilenberg-Mac Lane space $K(\mathbb{Z},3)$ up to degree 8 using the Atiyah-Hirzebruch spectral sequence, resolving the relevant extension problems by geometric arguments and identifying both bordism invariants and geometric generators. We then relate these invariants to perturbative and global anomalies, and discuss physical examples and top-down constructions of the corresponding anomaly terms. For 5-dimensional theories, we find a new mixed perturbative anomaly between the $U(1)$ 1-form symmetry and spacetime diffeomorphisms, while for 7-dimensional theories we find a new $\mathbb{Z}_2$-valued discrete anomaly intrinsic to the $U(1)$ 1-form symmetry. We also discuss their boundary realizations and give new physical interpretations of these anomalies.

In a previous paper \cite{BT} we studied the asymptotic behavior of $\| \sum_{k=1}^N a_k X_{n_k}\|_p$ for lacunary sequences $(X_{n_k})$ of random variables in $L_p$ and used the result to give a necessary and sufficient condition for the first alternative in the Kadec-Pełczynski theorem in the case $1\le p<2$. In the present paper we extend this result for nonlinear functionals $f_k (a_1 X_{n_1}, \ldots, a_k X_{n_k})$, establishing a uniform version of the subsequence principle of Aldous \cite{ald}. Moreover, we prove Kadec-Pełczynski type theorems in Orlicz spaces $L_\psi$.

We study thermal quantum correlations and coherence in Heisenberg $XY$ model with anisotropic interactions under a uniform magnetic field $ B $. Using concurrence $C$, local quantum uncertainty (LQU), Bell-Clauser-Horne-Shimony-Holt (CHSH) nonlocality $ \mathbb{B}$, and coherence $C_l$ as quantifiers, we analyze how magnetic anisotropy $ \delta_m $, coupling anisotropy $ \delta_c $, Dzyaloshinskii-Moriya (DM) interaction $ D $, temperature $ T $, and magnetic field $ B $ modulate quantum resources. At low temperatures and relevant magnetic fields, the entanglement is maximized, but exhibits sudden death for $ \delta_m = 0 $, which turns into a smooth decay as $ \delta_m $ increases, highlighting its stabilizing role. LQU shows that stronger anisotropy suppresses quantum correlations, while $ \mathbb{B} $ induces a non-monotonic response peaking at a critical field $ B_c $. Bell-CHSH nonlocality violations ($ \mathbb{B} > 2 $) persist below $ B_c $, but thermal noise ($ T \geq 1 $) suppresses them. Coherence $ C_l $ is most robust to thermal fluctuations, especially for high \( \delta_m \), which also dampens abrupt quantum phase transitions. The DM interaction is essential for entanglement generation, with $ D $ and anisotropy synergistically enhancing correlation resilience. We identify a hierarchy of thermal degradation: nonlocality ($ \mathbb{B} $) vanishes first, followed by entanglement ($ C $), then general quantum correlations (LQU), while coherence $ C_l $ persists the longest. These results demonstrate tunable control of quantum resources via anisotropy and external parameters, providing insights for the design of robust spin-based quantum technologies.

The present work develops the results of the series of papers by Buchstaber and Terzić on the standard actions of the compact torus $T^n = (S^1)^n$ on the complex Grassmann manifolds $G_{n,2}$. In those works, a hyperplane arrangement in $\mathbb{R}^n$ was introduced that determines the chamber decomposition of the hypersimplex $\Delta_{n,2}$ for the $T^n$-action on $G_{n,2}$. We introduce a notion of admissible graph for the standard action of the torus $T^n$ on the complex Grassmannian $G_{n,2}$. In terms of admissible graphs, we give a complete inductive description (with respect to $n \ge 4$) of the admissible polytopes in $\Delta_{n,2}$, as well as of the toric varieties arising as closures of $(\mathbb{C}^*)^n$-orbits on $G_{n,2}$ under the standard $(\mathbb{C}^*)^n$-action. We consider the $T^n$-equivariant Plücker embedding $G_{n,2} \hookrightarrow \mathbb{C}P^{N_2}$, where $N_2 = \binom{n}{2}-1$. Using admissible graphs, for the considered $T^n$-actions, we describe hyperplane arrangements in $\mathbb{R}^n$ that determine the chambers in $\Delta_{n,2}$ for the $T^n$-actions on $G_{n,2}$ and $\mathbb{C}P^{N_2}$. Gel'fand, Kapranov, and Zelevinsky introduced the notions of secondary polytopes and secondary fans in connection with the problem of describing triangulations of a given convex polytope, which is closely related to the Newton polytopes of discriminants and resultants. For the $T^n$-action on $\mathbb{C}P^{N_2}$, we show that the cones in $\mathbb{R}^n$ with vertex at the origin spanned by the chambers form the secondary fan of the cone spanned by the vertices of $\Delta_{n,2}$.