Papers

The classical Blaschke-product approach provides an elegant description of triangles inscribed in a circle and circumscribed about an ellipse. Motivated by the Siebeck–Marden theorem, we derive a symmetric parametrization for cyclic $p$-gons circumscribed about a Siebeck–Marden curve of class $p-1$, recovering the Blaschke product parametrization as the triangular case. We then use this parametrization to establish several geometric invariance results, including a characterization of when the sum of the squares of all sides and diagonals is invariant. We revisit Cayley's criterion for 3- and 4-Poncelet pairs and obtain a complete classification of the associated central conics solely in terms of their foci. In particular, we show that the formulas for the lengths of the major axis obtained under the assumption on the foci of the conic lying inside the circumcircle remain valid when one focus or both foci lie outside the circumcircle, thereby extending the classical theory from ellipses to all admissible central conics. Several geometric properties of the associated cyclic quadrilaterals are also obtained.

Kirkland and Šmigoc constructed a family of stochastic matrices realising the Type III boundary polynomials in the Karpelevič region and conjectured that, conversely, every stochastic realisation of such a polynomial must come from their construction. We prove this conjecture for the full nonzero parameter range $0<\alpha\le1$, for genuine Type III reduced Ito polynomials of order $n$, $f_\alpha(x)=x^y(x^q-(1-\alpha))^d-\alpha^d$, where $n=qd+y$. For $0<\alpha<1$, the proof first reduces every realisation to a two-shift cyclic normal form using the Dmitriev–Dynkin boundary theorem. The remaining argument is finite and combinatorial: Coates' coefficient formula and the equality case of a weighted Turán theorem force the $q$-cycles associated with the backward edges to split into $d$ complete multipartite classes of equal total weight. A circular-arc telescoping argument then converts this additive equality into the product condition required by Kirkland and Šmigoc. The endpoint $\alpha=1$ is treated separately. We also explain why the closed endpoint $\alpha=0$ is degenerate: the literal extension to this endpoint fails, because reducible realisations with closed $q$-cycles and transient states need not contain the global $n$-cycle.

Given a fractional-order linear equation $L^\alpha u = f$, we define an appropriate symmetric bilinear form so that the fractional operator $L^\alpha$ is symmetric with respect to that bilinear form. Using the bilinear form, we then define a functional of the fractional calculus of variations proving that the solutions of the given fractional-order equation are critical points of the fractional variational functional. In the case of fractional integral equations, the provided bilinear form is non-degenerate, and all critical points are solutions of the given equation. In the case of fractional differential equations, a relation with the least-squares method is obtained.

To help evaluate the mathematical skills of current AI systems, we present a set of formulas for fundamental mathematical constants. These problems are attractive for AI evaluation because they are concrete and can be checked numerically to arbitrary precision, yet proving them may require non-obvious mathematics. Mathematical constants such as $\pi$, $e$, Catalan's constant, and special values of the Riemann zeta function have fascinated mathematicians for centuries. The search for formulas evaluating mathematical constants has produced some of the most beautiful mathematics in the field, especially in cases that yield irrationality proofs or fast convergence rates. Ramanujan's legacy is emblematic of this tradition. The list we provide contains two types of problems: formulas whose proofs are known to the authors but will remain encrypted for a short initial period; and formulas that are not yet proven. We are curious to see the achievements of AI in both cases.

This paper offers a novel analytical method for obtaining new non-commuting solutions for the Yang-Baxter-like matrix equation $AXA=XAX$, when the coefficient matrix $A$ satisfies the square-cyclic condition $A^n=tA^2$. All the consistency criteria are proved and the obtained results are demonstrated on worked examples. The findings provided by this article simultaneously generalize several existing special cases regarding this topic. Consequently, by employing the obtained results, the previously unstudied inhomogeneous Yang-Baxter matrix equation $AXA=XAX+B$ is solved when $A$ is either invertible, or satisfies $A^2=0$.

Andrews and El Bachraoui investigated two-color partitions in which even parts may occur only in blue and obtained several identities relating parity refinements of these partitions to overpartitions and minimal excludants. Some of their proofs are analytic and rely on $q$-series manipulations. In this paper, we construct explicit weight-preserving bijections and parity-reversing involutions that provide combinatorial proofs of these identities. We also give a new combinatorial proof of related overpartition refinements involving $\overline{p}(n)$ and $\overline{p}_o(n)$.

Motivated by recent developments in complex difference equations and Nevanlinna theory in several complex variables, we investigate finite-order transcendental entire solutions of the coupled Fermat-type difference system: \beas \begin{cases} f_1^{n_1}(z)+ f_2^{m_1} \left(z+c \right) = 1,\\ f_2^{n_2}(z) + f_1^{m_2} \left(z+c\right) = 1, \end{cases} \eeas where $z,c=(c_1,c_2,\ldots,c_n) \in \mathbb{C}^n$ for various choices of $n_i,m_i$, $i=1,2$. where $n_i,m_i\in\mathbb N$ and $n_i+m_i\ge2$ $(i=1,2)$. Extending the classical investigations of Gross–Yang, Liu, Liu–Cao–Cao and more recently, Xu et al. in one and two complex variables, to a general coupled system in $\mathbb C^n$ we establish a complete characterization of all finite-order transcendental entire solutions. We have determined that the solution structure is completely determined by the relative sizes of the exponents.

Modern multivariate regression problems involve several related outcomes whose regression effects are not only nonlinear, heterogeneous, and outcome-specific, but also where the residual dependence among outcomes is scientifically meaningful. Existing multivariate Bayesian tree-based methods typically address only part of this problem: some impose substantial sharing of tree architecture across outcomes, which is overly restrictive when responses depend on distinct predictors or effect modifiers, while others accommodate residual dependence but retain simpler mean structures. This paper develops multiVCBART, a multivariate varying-coefficient Bayesian additive regression tree framework that jointly models flexible outcome-specific coefficient surfaces and a sparse residual precision matrix. Each entry of the coefficient matrix $B(x)$ is represented by an independent BART ensemble, allowing predictor effects to vary nonlinearly with modifiers $x$ across outcomes, while a Graphical Horseshoe prior on the precision matrix $\Omega$ captures parsimonious residual conditional dependence. To permit efficient computation, we introduce a sampler that reduces the multivariate Gaussian likelihood to a sequence of scalar pseudo-response updates, decoupling the tree backfitting from the Graphical Horseshoe step. Theoretically, we establish the first posterior contraction rates for a multivariate BART model with jointly estimated residual dependence, proving near-minimax adaptation to underlying smoothness and structural sparsity. Empirically, multiVCBART outperforms existing multivariate tree models and Bayesian SUR competitors on sparse, high-dimensional datasets. Finally, in a re-analysis of the Genomics of Drug Sensitivity in Cancer dataset, our method identifies distinct biomarker signals and recovers a coherent residual pharmacologic network.

We study the asymptotic properties of the posterior on the latent space for infinite mixtures driven by a Dirichlet process, both in terms of mixing measure and clustering behaviour. In the well-specified regime, where the data are generated by a finite mixture of location densities, we show that the posterior is adaptive to the true number of components $K$: indeed the cumulative mass assigned to weights of the stick-breaking representation beyond the $K$-th one vanishes as $n^{-1/2}$, up to terms growing slower than any polynomial. This also implies a nearly optimal posterior contraction rate for the mixing measure in Wasserstein distance. A remarkable phase transition underlies this result: approximating the mixing measure to any precision finer than $n^{-1/2}$ requires a number of components growing logarithmically with the sample size. We show that this has a profound impact on the clustering behaviour: the number of clusters grows logarithmically, as in the prior case, but the proportion of observations outside the $K$ largest clusters vanishes polynomially fast. Finally, we turn these results into posterior guarantees for truncation-based approximations: while any truncation with at least $K$ elements recovers the optimal contraction rates for both density and mixing measure, $\mathcal{O}(\log n)$ components are both necessary and sufficient to reproduce the clustering of the exact posterior.

We give examples of varieties $X$ defined over a non-algebraically closed field $k$ with nontrivial unramified cohomology, in the case when the field $k$ is of bounded cohomological dimension, or the variety $X$ is a conic bundle over a rational surface and $k$ is an arbitrary field of characteristic different from $2$.

Casanovas and Potier proved that algebraic quantification preserves stability of formulas. They also gave a nonsimple example, answering a question of Laskowski, showing that the algebraicity hypothesis cannot simply be replaced by NFCP, and asked whether a similar example exists in a simple theory. We give such an example in elementary form. The edge-set structure of the random bipartite graph has two definable equivalence relations, both stable and NFCP, whose relational composition has the order property. The resulting theory is simple and $\aleph_0$-categorical. We also prove a formal sharpness observation: every formula in every first-order theory is an existential composition of two stable NFCP formulas algebraic in the two outer variables. Consequently, closure of stable NFCP formulas under this composition characterizes stability of the whole theory.

We propose a derivative-free matrix conjugate-subgradient method for unconstrained nonsmooth optimization of locally Lipschitz functions. The method constructs discrete gradients using only function values and forms a finite sampled model of the Goldstein subdifferential. A minimal-norm element of the convex hull of the sampled discrete gradients is then computed and used both as a stationarity measure and as the reference vector for generating descent-oriented directions. To improve robustness beyond the basic steepest-descent direction, we introduce a matrix memory correction together with coefficient damping, diagonal scaling, bounded-angle correction, and matrix-stability safeguards. A two-point line-search procedure with enrichment is used to obtain either a serious step or an improved local model. Under suitable consistency assumptions on the discrete-gradient approximation and line-search sampling, the method generates directions satisfying a safeguarded descent property and computes approximate Goldstein stationary points. Numerical experiments on nonsmooth test problems with dimensions up to 1000 show that both proposed variants are robust for lower and medium accuracy requirements, while the matrix conjugate-subgradient variant remains the most reliable under the strictest tolerance.

Koopman theory promises linear structure in nonlinear dynamics, but numerical Koopman spectra are easy to compute and hard to trust. A finite EDMD matrix always has eigenvalues; the problem is that many of them may have nothing to do with the infinite-dimensional operator. In this paper we make spectral reliability the objective of dictionary learning. We train neural-network dictionaries not merely to predict the next snapshot, but to minimize Residual Dynamic Mode Decomposition residuals: operator-level a posteriori errors that test whether computed eigenvalues and modes are genuine Koopman spectral objects. To keep the learned observables from collapsing into an unstable coordinate system, the loss also penalizes the condition number of the lifted data matrix. Thus the method couples two requirements that should not be separated: small Koopman residuals and a well-conditioned representation. The result is a learned dictionary that is expressive, numerically stable, and spectrally disciplined. Across conservative and dissipative benchmark systems, the method sharply reduces spectral pollution, improves residual pseudospectral inclusion, and lowers forecast error relative to standard fixed dictionaries. On sea-surface temperature data, it gives cleaner Koopman diagnostics and substantially better one-step forecasts from noisy observations with no governing equations. The message is simple: neural Koopman learning should be judged not by prediction alone, but by whether its spectral claims can be certified. Residuals provide the certificate; conditioning makes it computable.

We resolve Erdős Problem 731 under the explicit dyadic-regularity formalization of "reasonable." Let $A(n)$ be the least positive integer not dividing $\binom{2n}{n}$. On dyadic intervals $X\le n<2X$, put $L=\log(2X)$ and $F_X=\sqrt{2}(\log 2)^{1/4}L^{1/4}\exp\sqrt{(\log 2)L}$. Uniformly for $1\le z\le Z(X)=o(L^{1/4})$, we prove $\mathbb{P}_X(A(n)\le F_X\exp(-z))\asymp \exp(-2z)$ and $\mathbb{P}_X(A(n)>F_X\exp(z))\ll \exp(-2z)$. Consequently $\log A(n)=\sqrt{(\log 2)\log n}+\frac{1}{4}\log\log n+O_{\mathrm{dens}}(1)$. We also prove dyadic nonconcentration: no scalar center on a large dyadic block, and hence no dyadically regular deterministic scale $f$, can satisfy $A(n)/f(n)\to 1$ in natural density. The proof retains the exact least-common-multiple divisibility condition and replaces heuristic cross-base independence by a moving-base restricted-digit variance theorem. The resolution proved here has been formally verified in Lean.

In this article we study the polynomial identity (PI) property of skew PBW extensions. We show that every bijective skew PBW extension over a prime PI-algebra has nontrivial center. This fact allows us to determine, from the known description of the center in several classes of examples, whether such extensions satisfy a polynomial identity. Furthermore, building on results of Brown and Zhang \cite{BrownZhang2022}, we investigate the PI property of certain $\K$-algebras over fields of positive characteristic.

We study a parabolic obstacle problem for surfaces evolving by anisotropic mean curvature flow subject to an obstacle constraint. Given a convex obstacle and initial data, we seek an evolving surface minimizing an anisotropic energy functional while remaining above the obstacle; as a special case, this framework includes the anisotropic Stefan problem, where the free boundary represents a phase transition interface with direction-dependent surface tension. The central tool is the Cahn–Hoffman transform $S(x) = A^{-1/2}x$, which maps the Wulff ellipsoid $\{x : x^T A^{-1}x \leq 1\}$ to the Euclidean unit ball and converts the anisotropic problem into an equivalent isotropic one with a generalized Robin-type condition on the free boundary. We prove optimal regularity of the solution ($C^{1,\alpha}$ in space and $C^{0,\alpha/2}$ in time up to the free boundary) and $C^{1,\alpha}$-regularity of the evolving free boundary at non-degenerate points. The parabolic Hausdorff dimension of the space-time singular set is shown to be at most $n - 1$.

Let $X/K$ be a smooth projective variety defined over a number field and $f:X\to X$ be a morphism defined over $K$. Assuming there exists a point in $X(K)$ whose $f$-orbit is Zariski dense in $X$ and up to replacing $K$ by a finite extension, Pasten and Silverman studied the distribution of grand $(f,K)$-orbits and proved that many sets of representatives of grand $(f,K)$-orbits on various classes of varieties are Zariski dense. In particular, they showed that if $X$ is a geometrically simple abelian variety, then all such sets of representatives are Zariski dense. We demonstrate the existence of a dense set of representatives for maps on all abelian varieties.

We study an obstacle problem for surfaces minimizing an anisotropic surface energy of ellipsoidal type. Given a convex obstacle and a boundary datum, we seek a surface that minimizes the anisotropic area functional while remaining above the obstacle. The central novelty is the systematic use of the Cahn-Hoffman transform to convert the anisotropic problem into an equivalent isotropic problem with a generalized Robin boundary condition. We prove optimal regularity of the solution ($C^{1,1}$ up to the free boundary) and $C^{1,\alpha}$-regularity of the free boundary itself under a non-degeneracy condition. The singular set of the free boundary is shown to have Hausdorff dimension at most $n-1$, and a logarithmic epiperimetric inequality yields its $(n-1)$-rectifiability. The approach combines Caffarelli's classical theory of obstacle problems with the geometric theory of anisotropic mean curvature and the Alexandrov reflection principle adapted to the anisotropic setting.

We prove a compactness theorem for the space of closed embedded minimal surfaces with area bounded from above and injectivity radius bounded from below in a closed Riemannian $3$-manifold. This result is a variant of the Choi–Schoen compactness theorem in which the genus bound is replaced by a lower bound on the injectivity radius of the surface.

Let \(S_b\) be the class of birational morphisms between smooth varieties over a field \(F\), and let \(L_n=S_b^{-1}d_{\leq n}\Sm(F)\). Kahn and Sujatha asked whether the natural functor \(L_n\to S_b^{-1}\Sm(F)\) is fully faithful. We prove that it is fully faithful exactly for \(n=0\). More strongly, for every \(n\geq1\) and every \(N\geq n+1\), the transition functor \(L_n\to L_N\) has an infinite fibre on an endomorphism set. The proof identifies a sharp dimension threshold: if \(\dim X+r\leq n\), then \(X\times\mathbb A^r\to X\) is invertible in \(L_n\) precisely when \(\dim X+r\leq n-1\). We also give proper and projective analogues.

Standard statistical methods are often inadequate for modeling the joint dependence between linear and circular variables, and existing methods for modeling this dependence are designed only for continuous variables. However, circular data are frequently observed on a finite set of equally spaced directions, either due to rounding prior to reporting or because of the experimental design employed for data collection. To address this gap, we propose a flexible, analytically tractable model for jointly representing a discrete circular and a continuous linear variable. The construction combines a wrapped symmetric geometric distribution, a Weibull distribution, and a trigonometric linking function. This formulation yields closed-form expressions for the joint, marginal, and conditional distributions. The choice of the Weibull distribution facilitates direct sample generation using the inverse transform technique. Additionally, it provides explicit expressions for conditional moments, enabling a flexible circular-linear regression framework. We detail the theoretical interpretation of the model parameters, mathematically establishing the monotonicity of the conditional mean and variance with respect to the dependence parameters. The performance of the estimators is demonstrated through extensive simulations, and the utility of the model is illustrated by analyzing two empirical environmental datasets.

We determine the limiting distribution of partial sums of a Steinhaus random multiplicative function $\sum_{x\le n \le x+y} f(n)$ over short intervals $[x, x+y]$, where $y \rightarrow \infty$ but $y=o(x)$. We show that with appropriate normalization, the limiting distribution is Gaussian for all such $y$. A key new feature of our result is that the normalization factor is different from the standard deviation $\sqrt{y}$ when $y$ is very close to $x$. In contrast, when $y \asymp x$ there is no normalization for which the limiting distribution is a non-degenerate Gaussian.

A finite group $G$ is said to be semi-rational if the set of generators of each cyclic subgroup of $G$ is contained in at most two $G$-conjugacy classes. This is equivalent to the following condition: for every column of the character table of $G$, the values appearing in the column are contained in a quadratic extension of the field of rational numbers (possibly a different one for each column). When the analogous condition holds for the rows, that is, when the field of values of every irreducible character is contained in a quadratic extension of the rationals, we say that the group is character-quadratic (these groups are often called quadratic rational in the literature). We obtain several results concerning the structure of the Gruenberg-Kegel graph of a finite solvable group that is either character-quadratic or semi-rational. More precisely, we first provide a complete classification of such graphs in the disconnected case. Also, we prove that if the graph has at most three vertices and the group is nontrivial, then it belongs to an explicit list of $20$ graphs (in the semi-rational case, this result is proved under the additional assumption that the order of the group is not divisible by $17$), and all of them are realizable except perhaps one. Finally, we show that if the graph has four vertices, then it must have at least four edges.

We consider the largest interpoint distance $M_n=\max_{1\le i<j\le n}\|X_i-X_j\|$ among independent random points $X_1,\ldots,X_n$, uniformly distributed on a $d$-dimensional ellipsoid. We assume that the largest semi-axis has length 1 and multiplicity $k\ge 2$, whereas the remaining semi-axes are strictly smaller. In this situation, the diameter is attained on a manifold of dimension $k-1$, and the extremal points are no longer isolated. We establish a weak limit law for the diameter deficit $2-M_n$. Writing $q=d-k$ and $\alpha=q+(k+3)/2$, we show that $n^{2/\alpha}(2-M_n)$ converges in distribution to a Weibull random variable. The proof is based on a local analysis near the diameter manifold, a sharp asymptotic formula for the two-point tail probability, and a Chen–Stein Poisson approximation for rare nearly diametral pairs.

We study compatible Lie algebras from algebraic and representation-theoretic points of view, obtaining counterexamples to some fundamental theorems from classical Lie algebra theory, namely the theorems of Lie, Weyl and Levi. We also classify the two-dimensional compatible Lie algebras up to isomorphism and explore their representation theory, presenting families of indecomposable non-semisimple representations, showing that the solvable two-dimensional compatible Lie algebras have wild representation type, and classifying all irreducible finite-dimensional line representations. Finally, we prove a Clebsch-Gordan decomposition for tensor products of finite-dimensional irreducible line representations.

We propose a semi-Lagrangian adaptive-rank (SLAR) method that combines the large time-step capability of semi-Lagrangian schemes with the efficiency of adaptive-rank tensor representations while simultaneously enforcing local conservation laws for mass, momentum, and energy. The method builds on the high-dimensional SLAR framework introduced in our previous work and achieves high-order accuracy in both space and time. To address the loss of conservation in long-time simulations, we extend the implicit local macroscopic conservative (LoMaC) correction technique for the BGK equation to the high-dimensional Vlasov–Poisson (VP) system. The implicit macroscopic system is discretized using backward differentiation formulas and solved with a Jacobian-free Newton-Krylov method. This approach enables a consistent coupling with semi-Lagrangian methods which are capable of taking large time steps. A novel component of the proposed method is a unified adaptive-weight projection technique that eliminates the ad hoc parameter tuning required by previous LoMaC approaches. These weights capture problem-dependent velocity space structures and are constructed from the low-rank velocity bases of the solution. The local semi-Lagrangian method used in this work reconstructs the solution at the feet of the characteristics using efficient tensor contractions. To the best of our knowledge, this is the first successful implementation of an implicit LoMaC method for the VP system up to the 2D–2V setting. Numerical experiments on several classical benchmark problems demonstrate the accuracy and efficiency of the proposed method, as well as its ability to preserve conservation laws in VP simulations.

The Bethe-Salpeter equation, which has many applications in both theoretical and applied physics, is generally solved via a matrix eigenvalue problem with a rich algebraic structure. The numerical solution of such structured eigenproblem calls for specific algorithms that are able to preserve the structure throughout the computation. Several structure-preserving methods have already been proposed in the literature. In this paper, we develop a polynomial filter strategy that is able to extract approximations of eigenvalues located inside a specified interval. For this, we have devised a structure-preserving Chebyshev polynomial series, along with a specialized subspace iteration method that preserves the Bethe-Salpeter structure at every step of the algorithm. All necessary details required for a robust implementation are incorporated, and the performance is illustrated with matrices arising from real applications.

We discuss the problem of characterizing equidistant binary codes of a given length $n$ having largest possible distance and the maximum number of codewords. Such characterizations have been studied by several authors over the years and they involve symmetric BIBDs with certain parameters. In this primarily expository paper, we investigate the history of this problem and give a unified presentation of the main results. Perhaps surprisingly, researchers on this problem were unaware of early relevant work by Marrero and Butson from 1973. Also, it turns out that published results on characterizations of equidistant binary codes have missed one of the possible subcases when $n \equiv 2 \bmod 4$.

We study centralizers of dilations in the quasi-isometry group of the positive real line. We introduce an asymptotic invariant defined via coarsely dense sequences at infinity and establish a rigidity theorem for quasi-isometries that coarsely commute with a dilation. As an application, we identify the subgroup of the centralizer consisting of elements with non-empty asymptotic invariant and prove that it is naturally isomorphic to the multiplicative group of positive real numbers.

We provide an intrinsic characterization of hyperelliptic stable curves of genus $g \geq 2$, independent of admissible covers or auxiliary moduli data. A stable curve is hyperelliptic if it admits an involution yielding a rational tree quotient, subject to a characteristic-dependent condition. By analyzing the action of this involution on the nodes and decomposing the curve based on its connectivity, we obtain an explicit structural description of hyperellipticity and prove that the hyperelliptic involution is unique. Furthermore, we explain the connection to the very ampleness of the dualizing sheaf. This framework applies in arbitrary characteristic, explicitly capturing the divergent geometric and combinatorial behavior in characteristic 2. We verify that this formulation precisely captures the geometric points of the moduli stack of hyperelliptic stable curves $\overline{\mathcal{H}}_g$, defined as the scheme-theoretic closure of the smooth hyperelliptic locus $\mathcal{H}_g$ within the moduli stack of stable curves $\overline{\mathcal{M}}_g$. Extending this definition to flat families yields an explicit modular description of $\overline{\mathcal{H}}_g$ over $\operatorname{Spec} \mathbb{Z}[1/2]$.