Papers

We construct hyperbolic horseshoes for piecewise-analytic homeomorphisms of the circle with multiple break-type singularities, under the assumptions of bounded type rotation numbers and bounded geometry – provided the sizes of the breaks are uniformly small. As a consequence, we obtain a $C^{1+\alpha}$-rigidity result for such maps and prove that rigidity classes are analytic submanifolds.

We present some conditions in case $b\in L_{(p,q),\text{loc}}$ sufficient to guarantee the existence of a strong Markov diffusion process with drift $b$ and uniformly nondegenerate bounded matrix-valued diffusion. An example is given to illustrate how close these conditions are to be necessary.

We consider nonparametric regression with a Banach-space-valued predictor where the covariate takes values in a general separable Banach space. Our primary objective is to develop a unified framework for estimating the regression operator without relying on an inner-product structure, thereby extending classical and functional nonparametric regression beyond Euclidean and Hilbert spaces in the presence of measurement error. We establish the fundamental properties of the proposed estimator and develop its large-sample theory under conditions formulated in terms of the geometry and local concentration of the Banach-valued predictor. In particular, we study pointwise estimation and inference and discuss the construction of pointwise confidence intervals and uniform confidence bands. Our proposed framework is further extended to accommodate error-in-variables settings, where the predictor is observed with contamination. We also investigate related inverse problems arising from the construction of inverse weighting functions and consider partial contamination models. These developments provide a general framework for nonparametric regression and inference with complex infinite-dimensional predictors and identify several statistical and mathematical challenges that arise when the usual Hilbert-space structure is unavailable.

We study sampling from a distribution supported on an unknown compact $d$-dimensional $C^2$ manifold $M\subset\mathbb{R}^D$, observed only through i.i.d. uniform points from $M$. We reconstruct the constraint using an adaptive local-convex-hull estimator and target an ambient distribution penalized by squared distance to the reconstruction. Although the reconstructed set may be nonsmooth or fail to be a manifold, its Hausdorff accuracy alone suffices to control the Wasserstein error. We quantify the tradeoff between reconstruction accuracy and penalty strength and show that optimal tuning gives an error proportional to the square root of the Hausdorff error. A planar example proves that this dependence is sharp for the proposed scheme. For a $d$ dimensional constraint reconstructed from $N$ observations, the error becomes $\mathcal{O}( (N/N)^{1/d})$. Finally, we prove uniform geometric ergodicity of a Gaussian random-walk Metropolis–Hastings sampler and combine reconstruction, approximation, and mixing into an explicit finite-time guarantee.

We prove that for all $\Delta \geq 2$ and $\gamma > 0$, there exists a constant $C = C(\Delta, \gamma)$ such that for $p\geq C(\log n/n)^{1/\Delta}$, asymptotically almost surely, every spanning subgraph $G$ of $G(n,p)$ with minimum degree at least $(1-1/(2\Delta)+\gamma)pn$ contains every $n$-vertex graph $H$ with maximum degree at most $\Delta$ and with at least $Cp^{-2}$ vertices not in any triangles of $H$. This is a 'sparse local resilience version' of a classical theorem of Sauer and Spencer. The condition that $H$ should contain some vertices not in triangles is necessary, and in fact, the quantity $p^{-2}$ is asymptotically best possible. A key feature of our result is that $H$ is allowed to be an expander graph, distinguishing it from previous results of similar nature, which dealt with, e.g., graphs of sublinear bandwidth. Our proof makes use of regularity arguments, with the sparse blow-up lemma for random graphs being a key tool.

Talagrand asked for a construction of a large convex subset of a bounded Minkowski sum of a large Gaussian set. We first prove a stronger nonsymmetric existential statement: if $A\subset\mathbb{R}^n$ is measurable and $\gamma_n(A)>2/3$, then $A+A+A$ contains a compact convex set of Gaussian measure at least $1/2$. Let now $A$ be closed with $\gamma_n(A)\ge7/8$, let $\Phi$ be the standard Gaussian distribution function, and put $a_0=\Phi^{-1}(3/4)$. For every $\Lambda>1$ and every $0<p<2\Phi(\Lambda a_0)-1$ we construct a centrally symmetric finite polytope $C\subset\Lambda(A+A+A)$ with $\gamma_n(C)\ge p$. Thus measure $3/4$ is obtained for every $\Lambda>1.705510\ldots$. We give matching upper and lower bounds for the optimal high-measure dilation and show that the dilation profile used by the construction is sharp among all symmetric half-measure cores. Finally, if $\gamma_n(A)\ge5/6+\eta$, we construct, for every $0<\varepsilon\le1/2$, a centered ellipsoid $E_\varepsilon$ with $\gamma_n(E_\varepsilon)\ge1-\varepsilon$ and \begin{equation} \frac{c}{\Phi^{-1}(1-\varepsilon/2)}\sqrt{\frac{\log n}{n}}\,E_\varepsilon\subset A+A+A. \end{equation} Both the dimension dependence and the dependence on $\varepsilon$ are optimal up to constants. The construction also gives a six-summand theorem for balanced sets and a second finite construction based on subgaussian tests.

Let $k$ be an algebraically closed field of characteristic $2$ and let $q=2^h$, $h\ge3$. We construct ordinary bielliptic curves $X$ of genus $q+1$ for which \[ \Aut(X)\cong \Dih(C_q)\times C_2, \qquad |\Aut(X)|=4q=4(g(X)-1). \] These curves realize case (ib) in the classification of Giulietti–Korchmáros and give an infinite family answering a problem posed by Korchmáros. More generally, we prove that every curve in case (ib) arises from the same construction. Case (ib) occurs exactly in genera $g=2^h+1$ with $h\ge2$. For $g\ge9$ we determine the full automorphism group; in genus $5$ we determine its Sylow $2$-subgroup but do not claim the full automorphism group. For every fixed $q\ge8$, the isomorphism classes in case (ib) of genus $q+1$ are parametrized bijectively by $(k^\times)^2$. The construction is described in terms of an ordinary elliptic curve and an invariant differential. We determine the short orbits and ramification, the unramified cyclic quotients, and the quotients by the central involutions. For $q=8$ an explicit plane model over $\F_2$ is given.

Given a real square matrix $A$ and a nonempty closed target set of complex matrices $\mathcal{E}$, invariant by complex conjugation and containing real matrices, does the subset of $\mathcal{E}$ consisting of the matrices nearest to $A$ in the Frobenius distance always contain a real matrix? We give negative answers when $\mathcal{E}$ is either the set of normal matrices (solving a question posed by N. Higham) or the set of matrices whose eigenvalues lie in the closed left half-plane (solving a question posed by the first author and F. Poloni). For the latter problem, we also argue that the ratio between the real and complex distances is unbounded in every dimension $n\geq 3$, and this holds for every distance induced by a unitarily invariant norm. For both problems and $n \geq 3$, we show that a real input $A$ uniformly drawn from the unit sphere $\|A\|_F=1$ has no real minimizer over $\mathcal{E}$ with probability strictly between $0$ and $1$. Beyond these two cases, we prove for an arbitrary nonempty closed target sets $\mathcal{E}$ that, for almost every real input, either $\mathcal{E}$ contains a unique nearest matrix which is real, or every nearest matrix in $\mathcal{E}$ is nonreal.

Inspect the vertices of a finite simple graph in uniformly random order, accepting each vertex if none of its neighbors has previously been accepted. Let $X_G$ be the number of accepted vertices. For every triangle-free graph with $n\ge2$ vertices and $e$ edges, we prove $\operatorname{Var}(X_G)\le e((n-2)/n)^2$, with equality precisely for edgeless graphs and connected stars. In particular, among trees of a given order, the star uniquely maximizes the variance, with value $(n-1)(n-2)^2/n^2$. Known expected vertex-deletion stability already yields the elementary baseline $\operatorname{Var}(X_G)\le e$. We obtain the sharp finite-order refinement by combining a stronger centered first-choice estimate with a triangle-free edge-count identity in the law of total variance.

We prove the transcendence over $\mathbb{Q}$ of the image of Montgomery reduction factor $R'$ in the ring $\mathscr{A}$ of integers modulo infinitely large primes. Here, for an odd prime number $p$, $R'$ is defined as the modular inverse of a power $R$ of $2$ modulo $p$ satisfying $2^{-k} R \leq p < R$ for a fixed constant $k \in \mathbb{N}_{> 0}$ typically given as the standard bit size $32$ of an integer type, and is the element of $\mathbb{Z}/p \mathbb{Z}$ representing Montgomery reduction regarded as a $\mathbb{Z}/p \mathbb{Z}$-linear homomorphism.

We show that every countable group $G$ that has Ozawa's property $\rm PHP$ and contains a non-abelian free subgroup has the following property: the group ring $KG$ is primitive for any field $K$. As a consequence, we deduce that all non-trivial countable linear groups with trivial amenable radical have this property. Along the way, we prove that the class of groups with $\rm PHP$ enjoys some permanence properties that are of independent interest.

This paper introduces an online evolve–filter–relax (EFR) strategy for long-term stability of Operator Inference (OpInf) reduced-order models (ROMs) of complex fluid flow simulations. The main novelty of the new EFR-OpInf strategy is the use of online (i.e., at the learned ROM online evaluation level) spatial filtering inspired from large eddy simulation to significantly improve long-term stability and predictive performance of standard OpInf. Furthermore, the EFR-OpInf strategy reduces, and in some cases even eliminates, the need for standard $L^2$ regularization, while providing physical interpretability for the OpInf hyperparameters. The EFR-OpInf framework is modular, readily integrated into existing OpInf workflows, and accommodates a user-selected ROM filtering strategy. We demonstrate the EFR-OpInf's effectiveness using a fully non-intrusive projection-based ROM filter, a ROM differential filter, and a hybrid projection-differential ROM filter. The new EFR-OpInf models are evaluated on a high-Péclet-number convection–diffusion–reaction problem that embeds the variation in one parameter and two unsteady Navier–Stokes problems that focus on predictions beyond a training horizon: a transitional two-dimensional flow past a cylinder and a three-dimensional turbulent minimal channel flow. Across these three scenarios, EFR-OpInf can reduce prediction errors by up to an order of magnitude relative to standard OpInf. Moreover, EFR-OpInf remains stable over long prediction horizons in cases where standard OpInf diverges. Depending on the ROM filter used, EFR-OpInf's computational cost is comparable to that of standard OpInf.

Fox's conjecture famously asserts that the absolute values of the coefficients of the Alexander polynomial of alternating links are trapezoidal. In the setting of MOY graphs, where a different notion of Alexander polynomial appears, the equivalent result to Fox's conjecture is known to hold. In this paper, we relate the two polynomials in the case of special alternating links. More precisely, we show that their degrees, first, and last coefficient agree, and that the MOY polynomial coefficients always dominate the classical Alexander polynomial.

Let $\F$ be a codimension-one holomorphic foliation of degree $d$ on $\PP^n$, $n\geq3$, admitting an invariant hyperplane $H$. We study the extremal situation in which $S=(H\cap\Sing(\F))_{\rm red}$ is an irreducible hypersurface of $H$ of degree $d+1$. When $d+1$ is a power of a prime, we prove that, in suitable homogeneous coordinates with $H=(t=0)$, \[ \Omega=Q\,dt-\frac{t}{d+1}\,dQ , \] where $Q$ is homogeneous of degree $d+1$. Thus $Q/t^{d+1}$ is a rational first integral. The proof reduces the Frobenius equation to a twisted closedness equation on a plane section and uses Zariski's theorem on the Alexander polynomial of an irreducible plane curve. We also prove a complementary rigidity theorem for an arbitrary smooth invariant hypersurface $D\subset\PP^n$: if the reduced singular divisor on $D$ is smooth, irreducible, and of maximal degree, then the same normal-form phenomenon holds, with no arithmetic hypothesis on its degree; in the low-weight range the smoothness assumption on the singular divisor can be dropped. Finally, we show that the principal hypotheses are sharp. Dropping the maximal-degree condition yields a family with irreducible reduced singular support and no non-constant rational first integral. For every $d+1$ which is not a prime power we construct a global counterexample with irreducible maximal-degree singular support, and a final family shows that irreducibility of the reduced support is also genuinely necessary.

Let $f:\; M \to {\mathbb C}P^n$ be a Lagrangian fibration on a compact hyperkahler manifold. We prove that $f$ has no multiple fibers in codimension 1. We prove that this condition is equivalent to the primitivity of the fundamental class of $\pi^{-1}(H)$, where $H \subset {\mathbb C}P^n$ denotes a hyperplane divisor.

Many semantic localization tasks require determining a contextually meaningful region or state rather than minimizing the error of a full Cartesian position estimate. In such settings, aggregate localization accuracy need not align with the accuracy of the semantic decision. This paper develops a Cramér–Rao bound (CRB)-based framework that uses the local geometry of a known semantic map to guide uplink receive beamforming. Using local boundary crossing as a surrogate for semantic misclassification, we derive normal information maximization (NIM), which minimizes the CRB of the position component along the local semantic boundary normal. Under the considered single-path line-of-sight model, an optimal receive codebook can be restricted to the subspace spanned by the matched steering vector and its angular derivative, reducing the design to an allocation of measurement resources between matched and derivative spatial modes. We derive the resulting allocation for a general smooth boundary, with geofencing and intrusion detection arising as radial and tangential limiting cases. The framework is further extended to multi-region semantic maps through a distance-normalized minimax criterion and to uncertain prior locations through worst-case and prior-weighted robust formulations. Numerical results with finite-slot implementations and observation-level Monte Carlo simulations using profile maximum-likelihood (ML) estimation show that NIM allocates measurements according to the task-relevant boundary geometry and can substantially reduce semantic error relative to the classic squared position error bound (SPEB) design in the considered scenarios. The empirical results also closely follow the local CRB-based boundary-crossing approximation in the studied operating regime.

An equilateral set is a set of points in a metric space whose pairwise distances are all equal. Kusner conjectured that the maximum cardinality of an equilateral set in $\mathbb{R}^n$ with the $\ell_p$ metric is $n+1$ for every $1<p<\infty$. We disprove the conjecture for all $p>4$. In particular, for every such $p$, we define $m=m(p)$ and construct an equilateral set of $8m$ points in $\mathbb{R}^{8m-2}$. Previously, Swanepoel disproved the conjecture for $1<p<2$, while Ge, Xu, and Zhou showed that it holds for $2\le p\le 4$. Combining these prior results with the paper's main theorem resolves Kusner's conjecture for all $1<p<\infty$.

We study torsional rigidity as a function of the labeled vertices of a convex polygon. Starting from the distributed second shape derivative, we derive the Hessian with respect to vertex coordinates. At a regular polygon, dihedral symmetry makes this matrix block circulant in radial-tangential coordinates, reducing its spectrum to the eigenvalues of Hermitian matrices of order two. We also derive an exact second-variation Galerkin identity and guaranteed functional residual majorants. Finite elements approximate the PDE solutions entering the Hessian, and FLINT/Arb provides the interval arithmetic needed for certification. In the scale-invariant setting, we certify exactly four zero eigenvalues generated by similarities and $2n-4$ strictly negative eigenvalues for $5\leq n\leq25$. The regular polygons in this range are therefore strict local maximizers, modulo similarities, of torsional rigidity divided by area squared. We prove broken regularity of the first and second material derivatives at each fixed regular polygon, and deduce an $O(h^2)$ error bound for the exact Galerkin Hessian and its eigenvalues on meshes fitted to the coarse fan.

We present a generalization of the Fast Fourier and fast Walsh transform algorithms to tensor products of $n$ arbitrary $q\times q$ matrices over a commutative ring, reducing the cost of applying such a tensor product from $q^{2n}$ to $nq^{n}$ ring operations. We then give an explicit construction of square orthogonal matrices over a commutative unitary ring, starting from a prescribed first row, and specialize this construction to ${\mathbb Z}/256{\mathbb Z}$. Combining these two ingredients, we propose a symmetric cryptosystem in which the secret key determines~$n$ orthogonal matrices over ${\mathbb Z}/256{\mathbb Z}$ whose tensor product encrypts a block of $q^n$ bytes; encryption and decryption both exploit the fast tensor product algorithm. We give three examples of implementations.

We calculate the quotient of the mod-two cohomology of a product of symmetric and alternating groups by the ideal of negligible elements, namely those that restrict trivially to any order-two subgroup. This quotient arises in the study of cohomological invariants of fields, as by work of Serre the ideal of negligible elements coincides with the ideal of elements that pull back trivially in Galois cohomology. Our calculations answer questions recently posed to us by J-P. Serre, whose recent results we build on. It is a pleasure to share these results on the occasion of his 100th birthday.

We propose a decentralized privacy-preserving learning algorithm in which each agent holds a single private sample and a shared model. Samples are learned sequentially, and each update must preserve the endpoint mappings at previously learned samples while protecting private data. This gives each agent three roles: (i) a learner that updates the model parameters, (ii) a teacher whose sample is learned at the current iteration, and (iii) a protected agent whose sample has already been learned. We build on Tuning without Forgetting (TwF) method to preserve previously learned mappings and show that TwF provides an indistinguishability guarantee for the learner whenever the set of protected agents contains another sample with the same label. For the teacher, we formulate a minimax optimal control problem that models the differential privacy noise as a worst-case disturbance to prevent performance loss while maintaining the same level of privacy for the gradient. For the protected agents, we compute the projections locally and aggregate them using a private push-sum gossip protocol. We prove geometric convergence of the decentralized gossip algorithm and of the distributed projection for TwF.

We determine the optimal exponential scale in Erdős's robust polynomial interpolation problem and identify the leading exponential coefficient for Chebyshev–Lobatto grids. Given $C>0$ and $0\le\rho<1$, put $H=C/(1-\rho)$. We prove that every configuration of $n\ge n_0$ nodes in $[-1,1]$ admits sign data for which every polynomial of degree at most $(1+\epsilon)n$ and uniform norm at most $C$ has error greater than $\rho$ at more than $\epsilon n$ nodes. One may choose $\epsilon\ge e^{-A(1+H)}$ and $n_0\le e^{A(1+H)}$, where $A$ is an absolute constant, and a matching upper bound shows that the exponential order is optimal. For the same obstruction restricted to Chebyshev–Lobatto grids, the optimal parameter $\epsilon_{\mathrm{Ch}}(C,\rho)$ satisfies $\log(1/\epsilon_{\mathrm{Ch}}(C,\rho))=\frac{\pi}{2}H+O(\log(H+1))$ as $H\to\infty$. The proofs combine finite interpolation estimates in Bernstein spaces with periodic sign constructions and averaging. We also establish obstructions for random signs and stability under angular perturbations of the nodes.

This is an expository article describing the works of Yu Deng, Zaher Hani and Xiao Ma on the long-time derivation of the fluid equations from the statistics of Newtonian particle collisions.

We consider the positive principal eigenvalue of an elliptic problem with a Robin boundary condition and bang–bang indefinite weight $\kappa\mathbf 1_D-\mathbf 1_{\Omega\setminus D}$, and ask where a favourable region $D$ of prescribed small volume $|D|=\delta$ should be located. Put $\varepsilon=\delta^{1/N}$ and $\tau_\delta=\alpha_\delta\varepsilon$. We prove that there is a finite threshold $\tau_*=\tau_*(N,\kappa)$, independent of the ambient domain, such that $\delta^{2/N}\Lambda_\delta(\alpha_\delta)\to\Lambda_{\mathbb H}(\tau)$ whenever $\tau_\delta\to\tau<\infty$. If $\tau<\tau_*$, optimal small regions concentrate at the boundary. If $\tau>\tau_*$, their concentration centres move to distances much larger than $\varepsilon$ from the boundary, and after recentring and rescaling the favourable sets converge in measure to the whole-space optimal ball. At $\tau=\tau_*$, the half-space problem admits a compact boundary optimiser as well as minimising sequences escaping to infinity. In the finer regime $\tau_\delta=\tau_*+\sigma\varepsilon+o(\varepsilon)$, we determine the first-order competition between the boundary and interior configurations. Tangential symmetry of compact threshold optimisers reduces the geometric correction to mean curvature. In particular, every fixed finite Robin coefficient is asymptotically in the boundary regime. Numerical experiments illustrate the boundary–interior transition, the three cases in the first-order selection law, and the curvature-dependent boundary location; for $N=2$ and $\kappa=1$ they suggest a transition near $\tau_*\approx3.2$.

We construct a nonconstant weakly harmonic map from a three-dimensional ball into the two-sphere with constant trace. This gives an affirmative answer to a question raised by Rivière in 1995.

In the conventional theory of quantum computation, universality is discussed in terms of properties of unitary gate sets. In many experimental setups, however, we instead have access to a parametrized set of Hamiltonians, which can be exponentiated for any desired time. Accordingly, we formulate a continuous analogue of universality, where adding one expensive Hamiltonian control to a cheap set of local control operators allows for universal computation. We find several concrete sets of 1- or 2-local Pauli generators $S$ and $n$-qubit “magic" Hamiltonians $H$ that together generate the full Lie algebra $\mathfrak{su}(2^n)$, with a special focus on the Ising Hamiltonian $J$. We propose a method to directly implement any unitary by exponentiating these generator sets, and give bounds on the needed time of application of the magic Hamiltonian $H$. We find that the number of applications of $H$ needed to approximate any unitary up to accuracy $\epsilon$ scales exponentially in the number of qubits as expected from the standard application of the Solovay-Kitaev theorem. We also find phase transitions in the size of the generated Lie algebras as the parameters of the Ising Hamiltonian are varied. We find an algorithmic application of the Chow/Rashevskii technique to implement unitaries corresponding to iterated commutators.

Let $(S_n)_{n\geq 1}$ be the one-dimensional Elephant Random Walk (ERW) in the superdiffusive regime, with memory parameter $p\in(3/4,1)$ and first-step bias $\mathbb{P}(S_1=1)=q$. Set $a=2p-1\in(1/2,1)$. If $f_{q,a}$ denotes the density of the superdiffusive limit $L_{q,a}$, we prove the uniform local limit theorem $\lim_{n\to\infty}\sup_{j\in\mathbb{Z},\,j\equiv n\;(\mathrm{mod}\;2)}\left|\frac{n^a}{2}\mathbb{P}(S_n=j)-f_{q,a}(j/n^a)\right|=0$. The proof uses the exact recurrence for the probability mass function of $S_n$, yielding uniform $O(n^{-a})$ and $O(n^{-2a})$ bounds for the p.m.f. and its first discrete differences, respectively. We then disprove a conjecture of Guérin, Laulin, Raschel and Simon concerning eventual log-concavity. Writing $L_a=L_{1,a}$ and $f_a=f_{1,a}$, let $a_{\rm ev}$ be the eventual-log-concavity threshold and let $a_\star$ be the upper failure threshold for log-concavity of $f_a$. Their results imply $a_{\rm ev}\geq(\sqrt{5}-1)/2$, and we prove $(\sqrt{5}-1)/2\leq a_{\rm ev}\leq\min\{0.80399,a_\star\}\leq a_\star<0.918$. The bound $a_\star<0.918$ follows from the first three exact moments of $L_a$ and the sharp skewness inequality for centered log-concave distributions, while $a_{\rm ev}\leq0.80399$ follows from a certified finite-order analysis of the exact p.m.f. recurrence. Direct numerical iterations provide evidence that $a_{\rm ev}\approx0.80399$, equivalently $p_{\rm ev}\approx0.902$.

Hamiltonian and Lagrangian mechanics are powerful frameworks for analyzing physical systems. Previous work has extended these formalisms to ecological systems, such as the predator-prey Lotka-Volterra (LV) system. In this Article, we derive a canonical Lagrangian for the two-dimensional LV model directly from its Hamiltonian representation. We find that the two-dimensional LV system admits a standard canonical Lagrangian formulation with one degree of freedom and a non-quadratic kinetic structure. This formulation admits a mechanical interpretation in terms of a particle moving in a potential well, where the non-standard kinetic structure produces a position-dependent damping term that can instead act as "revving." The derivation provides a direct connection between predator-prey dynamics and a canonical formulation of mechanical dynamics. As a verification of the construction, we apply Noether's procedure to the explicitly time-independent derived Lagrangian and reveal that the well-known Hamiltonian of the LV system is the corresponding conserved quantity. We also uncover a subtle redundancy associated with the choice of canonical momentum and its identification with the original population variables.

A structure-preserving symmetric Galerkin boundary element method (SGBEM) is developed for the anisotropic multi-phase Mullins-Sekerka problem in $\mathbb R^2$. The bulk harmonic problems are eliminated through a discrete Dirichlet-to-Neumann operator assembled from region-wise SGBEM discretization, leading to a boundary-only formulation coupled to the evolution of a polygonal curve network with triple junctions. The Galerkin approximation of the resulting boundary problem is shown to be uniquely solvable, and the fully discrete scheme is unconditionally stable with respect to the time step as long as the evolving polygonal network remains in an admissible class. The accuracy of the proposed method is demonstrated by a convergence test under coupled space–time refinement against a similarity-rescaled exact solution consisting of three concentric circles.

In this paper, we introduce and study conformal permutation matrices. In particular, for a permutation $\pi_s\in S_n$, we define the corresponding conformal permutation matrix $H_{\pi_s}$ and show that its $d-$th power, where $d=n/\gcd(s,n)$, is a diagonal block matrix whose blocks are products of certain matrix blocks $h_i$. We further show that these diagonal blocks share a common subset of eigenvalues and prove that $H_{\pi_s}$ is block-diagonalizable. We also introduce the notion of conformal circulant matrix and investigate its eigenvalues. Finally, we establish a necessary and sufficient condition for a rectangular matrix to be conformal circulant.