Papers

Let $\iota: R\to A$ be a Frobenius extension and let $T$ be a Wakamatsu tilting left $R$-module. We give sufficient conditions for the induced module $A\otimes_R T$ to remain Wakamatsu tilting and establish an ascent–descent result for split centrally projective Frobenius extensions. We also characterize the tilting case by the vanishing of $\mathrm{Ext}_R^i(T,A\otimes_R T)$ for all $i>0$. Moreover, if $A\otimes_R T\in\mathrm{add}_R(T)$, then the natural map $S=\mathrm{End}_R(T)\to B=\mathrm{End}_A(A\otimes_R T)$ of endomorphism rings is a Frobenius extension and $T\otimes_S B\cong A\otimes_R T$ as $R$-$B$-bimodules. Applications to Brenner–Butler–Miyashita equivalences and some specific classes of Frobenius extensions are also discussed.

Consider the matrix polynomial $\mathbf f_n(z)=I_qz^n+A_1z^{\,n-1}+\cdots+A_n,$ where $A_j\in\mathbb C^{q\times q}$. Write $\mathbf f_n(z)=\mathbf h_n(z^2)+z\,\mathbf g_n(z^2).$ A matrix polynomial $\mathbf f_n$ is called a Hurwitz-type matrix polynomial if, for $n=2m$, $\mathbf g_n(z)\mathbf h_n(z)^{-1}$, and for $n=2m+1$, $\mathbf h_n(z)\bigl(z\mathbf g_n(z)\bigr)^{-1}$, admit finite continued fraction expansions with positive definite coefficients. We derive explicit formulas for the coefficients of Hurwitz-type matrix polynomials in terms of orthogonal matrix polynomials, Markov parameters, and Schur complements. We introduce the associated block Hurwitz matrix and establish determinant identities relating it to the corresponding block Hankel matrices. Finally, we settle a conjecture concerning the positivity of the determinants of the coefficient matrices of Hurwitz-type matrix polynomials. We prove that this property holds for degrees at most three, but fails in degree four by means of an explicit counterexample.

Relative-frequency trajectories are often used to infer selective advantage in competing biological populations. A common empirical approach is to fit a linear function to the logit-transformed frequency of an invading type and interpret the slope as the relative advantage. Here we test how this estimator is affected when the competing types are observed after type-specific delays. We use SARS-CoV-2 variant replacement in the United Kingdom as empirical motivation and study the mechanism with simple two-type models. In an ideal exponential replacement model, fixed or randomly distributed observation delays change the intercept of the observed log-odds trajectory but not its slope, provided that delayed counts are aggregated before frequencies are formed and boundary effects are absent. In nonlinear SIR-type models, where the relative growth is time-dependent, delays can compare the competing variants at different dynamical phases and substantially bias the fitted logit slope. This bias can occur even when the delayed replacement trajectory remains nearly linear on the logit scale. Thus, a good logistic fit to the observed data is not sufficient to guarantee that the inferred logit slope approximates well the relative growth advantage. Observation delays should therefore be accounted for when logit slopes are used as proxies for relative fitness in nonlinear evolutionary competition.

Anti-Latin squares were introduced in connection with non-linear secure network coding, and the extremal problem for large mutually orthogonal families is motivated by that setting. We study the maximum size $N_A(d)$ of a family of mutually orthogonal anti-Latin squares of order $d$. We prove that $N_L(d)+1\le N_A(d)\le N_L(d)+2$ for every $d\ge 3$, where $N_L(d)$ denotes the classical maximum size of a family of mutually orthogonal Latin squares of order $d$, and we show that in fact $N_A(3)=N_L(3)+1$ whereas $N_A(d)=N_L(d)+2$ for every $d\ge 4$. The upper bound is obtained by passing through balanced matrices, while the lower bound is given by a deterministic permutation argument. For all $d\ge 8$, and also for the exceptional order $d=6$, the upper bound is shown to be attainable by a general probabilistic construction. On the structural side, we show that a saturated family of size $d+1$ induces an affine plane of order $d$, and that the saturated case is characterized by the existence of an anti-coordinate grid decomposition; after transporting this condition to the fixed cell set $[d]^2$, it becomes a direction-completeness condition on the corresponding row-blocks and column-blocks. The remaining small orders are treated separately: $d=3$ is handled by direct analysis and classification of orthogonal triples, $d=4$ by an explicit saturated construction and an analysis of its finite-geometric structure, and $d=5$ and $d=7$ by explicit saturated examples arising from the random-grid framework. Thus $N_A(d)$ is determined in terms of $N_L(d)$ for every $d\ge3$, and its numerical value is obtained explicitly for every $3\le d\le9$.

Suppose a set of integers $A\subseteq\{1,\dots,N\}$ has no solutions to $a-a'=n\lfloor\sqrt[3]2n\rfloor,$ for distinct $a,a'\in A,$ and $n\in \mathbb{N}.$ We show that $|A|\ll N^{1-c}$ for some absolute constant $c>0.$ To do this, we prove quantitative bounds on the van der Corput property for certain sets of bracket quadratics. This comes as a consequence of establishing exponential sum estimates for these sets, utilising a theorem of Green and Tao on the quantitative equidistribution of polynomial orbits on nilmanifolds, closely following the approach of Neale who went on to prove a Waring-type result. We also extend our result to differences avoiding a family of bracket polynomials (also known as generalised polynomials).

For every $n \geq 1$ there is a distinguished $\mathfrak{sl}_2(\mathbb{R})$ action on the Poincaré disk, arising as the infinitesimal isometries of a hyperbolic metric with conical singularity of order $n-1$ at the origin. For $n=1$ this is the standard infinitesimal Möbius action, and for $n>1$ these vector fields have singularities at the origin and are incomplete, preventing integration to a global Lie group action. However, they naturally define an action Lie algebroid $\mathcal{A}_n=\mathfrak{sl}_2(\mathbb{R})\ltimes \Dbarstar $ over the punctured disk. We construct an explicit Lie groupoid $\mathcal{G}_n$ integrating $\mathcal{A}_n$ and compare it to the \v Severa–Weinstein groupoid. Although $\mathcal{G}_n$ is not an action groupoid, its restriction to the boundary recovers an $n$-fold Möbius action on the boundary circle.

We introduce a distribution-free goodness-of-fit test, termed the omega-1 test, which naturally complements the Kolmogorov–Smirnov test and Cramér–von Mises test and can be viewed as their (piecewise) linear analog. Defined as an $\mathrm{L}^{1}$-functional of the empirical process, the test statistic improves on balancing sensitivity to localized and diffuse alternatives and gives a robust and interpretable measure of distributional discrepancy, apart from close connections to the Wasserstein 1-distance. For finite samples, we derive a finite-dimensional computational form for the statistic under general conditions, which leads to various explicit formulas for its null distribution. Under mild continuity assumptions, the limiting statistic is distribution-free, with explicit distribution formulas. In composite settings, the statistic is also compatible with the Khmaladze transformation, enabling asymptotically distribution-free testing. The limiting transformed statistic also has an explicit distribution that escapes reliance on intractable compensator processes or purely numerical evaluation. Simulation results indicate rapid convergence of the finite-sample distributions to their limiting counterparts and support the practical applicability of the test.

We discuss the dichotomy for fourth order Q curvature on Riemannian manifolds with positive Yamabe invariant and dimension at least 5.

Token prediction is a central pre-training objective for modern language models. Despite its empirical success, why token prediction learns broadly useful representations remains incompletely understood. We develop a statistical framework connecting token prediction with representation geometry, encoder approximation, and downstream performance. Under a softmax prediction head, we show that accurate token prediction organizes token embeddings according to similarities between the distributions of contexts in which different token types appear, as measured by Hellinger distance, with explicit errors governed by prediction accuracy and token frequency. Meanwhile, the contextual representation provides a low-dimensional coordinate for the conditional distribution of the target token relative to these embeddings. We further introduce a self-consistency principle showing that repeated applications of a shared representation block can progressively refine the contextual representation without introducing additional block parameters. Among representations with the same prediction accuracy, this recurrent construction favors those that can be stably reconstructed from their contexts. Finally, we establish downstream guarantees for token generation, token community recovery, and classification by a linear probe, showing how prediction accuracy and recovered geometry translate into performance beyond the pre-training objective. Together, these results explain how the simple objective of predicting tokens can recover semantic geometry and produce broadly useful representations. A controlled simulation illustrates the theoretical mechanisms.

We study the high-energy spectral effect of Robin boundary perturbations on complete Laplace eigenspaces and its consequences for eigenfunction non-localization. Highly degenerate Neumann levels generally split under Robin perturbation, but this splitting need not produce a simple spectrum: distinct modal classes may still contribute to the same Robin eigenvalue. We prove that, for every fixed positive Robin parameter, the number of modal classes contributing to any one eigenvalue is uniformly bounded over the spectrum on the equilateral triangle and on every rectangle with rational squared aspect ratio. On the equilateral triangle, for all sufficiently small Robin parameters, the Neumann complete-eigenspace observation estimate persists with a constant uniform both in the eigenvalue and in the boundary parameter. For an arbitrary fixed positive Robin parameter, a high-frequency shell-splitting analysis gives a spectrum-wide bound on the number of modal classes contributing to any one Robin eigenvalue. In the nonsquare rectangular case, the corresponding splitting is governed by a strongly convex profile on weighted quadratic shells. Since each modal class has uniformly boundedplane-wave complexity, the spectral complexity bound implies observation on every measurable set of positive measure through a multidimensional Turán–Nazarov inequality, without a frequency-separation assumption. Thus, spectral simplicity is not required for complete-eigenspace non-localization: uniformly bounded spectral coincidence complexity is sufficient.

Sparse representations are often expected to make models smaller and also reduce inference cost. For Fourier Neural Operators (FNOs), these objectives are not equivalent or do not always align: removing parts of the learned operator can leave the underlying transforms and dense computations unchanged, while changing the grid on which the model is evaluated can introduce overhead of its own. We therefore distinguish sparsity in the representation, in the stored parameters, in the theoretical operation count, and in measured runtime, and present an empirical study of several routes toward sparse FNOs that tests each transition between them separately. Coarsening the execution grid reduces the theoretical cost without reducing measured latency, and adding a correction term recovers accuracy at the cost of making the model slower. Even an 83% parameter reduction remains slower than the dense baseline under ordinary execution. These results motivate a stricter definition of useful sparsity: the deployed operator must preserve solution accuracy and map its reduced support to a genuinely cheaper execution path.

We study matrix representations of the Jacobsthal sequence generated by binary 3x3 matrices with determinants 0, 2, and -2, using linear algebraic methods analogous to Fibonacci-type constructions. Explicit formulas for matrix powers are obtained, yielding identities for Jacobsthal numbers, including convolution formulas, trace relations, determinant expressions, and Cassini-type identities. We further derive congruence relations and recurrence formulas, and analyze arithmetic properties such as partial sums and the Sidon-type structure of the sequence. Finally, we prove that exactly three conjugacy classes of binary 3x3 matrices generate the Jacobsthal sequence, providing a unified algebraic framework that connects matrix theory with second-order linear recurrences.

We develop an immersed interface finite difference method for a one-dimensional nonlinear parabolic interface problem with jump condition \[ [u]_\alpha=\lambda u^+u^-. \] The method combines a Crank–Nicolson immersed interface discretization with an \(s\)-parameter reduction of the nonlinear interface condition, thereby reducing the nonlinear coupling to a scalar quadratic equation. We also discuss two different viewpoints for combining Newton iteration with immersed interface discretization, namely the IIM–Newton and Newton–IIM formulations. Numerical experiments are presented to illustrate the behavior and accuracy of the method.

In this article, we introduce a refinement of the method of Heap and Soundararajan to obtain lower bounds for the $2k$-th moment of a broad class of $L$-functions for all real $k\ge 0$. In particular, our method circumvents the need to estimate the twisted moments of $L$-functions.

Let \(q\in\mathbb N\), let \(G=(0,\infty)^q\), and let $ S:G\times E\longrightarrow E, (r,x)\longmapsto S_rx $ be a jointly measurable left action on an arbitrary measurable space \((E,\mathcal E)\). For \(\alpha=(\alpha_1,\ldots,\alpha_q)\in(0,\infty)^q\) set $ \chi_\alpha(r)=\prod_{i=1}^q r_i^{\alpha_i}. $ We study nonzero \(\sigma\)-finite measures \(\nu\) satisfying $ \nu(S_rA)=\chi_\alpha(r)^{-1}\nu(A), r\in G, A\in\mathcal E.$ Motivated by the scalar case \(q=1\) studied in [1] we derive equivalent conditions for the existence of an \(E\)-valued random element \(Z\) such that $ \nu(A) = \mathbb{E}\{\int_G\mathbb I_A(S_rZ)\prod_{i=1}^q \alpha_i r_i^{-\alpha_i-1}dr_i\}, A\in\mathcal E. $ We also characterise when two random elements generate the same homogeneous measure, using multihomogeneous moments and, after fixing an admissible product gauge, weighted transverse measures. When the corresponding weighted transverse measure is finite, tilting and gauge normalisation produce a canonical representer, unique in law on the prescribed gauge shell. Finally, we characterise stationarity under an action commuting with \(S\) and construct positive semidefinite tail-overlap kernels directly from \(\nu\).

As a generalization of Gabe and Neagu's inclusions of real rank zero, we introduce a notion of Property (SP) for inclusions of C*-algebras. We begin the systematic study of SP-inclusions, establishing a topological description in the commutative setting and permanence under many constructions. We also extend known results about the permanence of Property (SP) for C*-algebras under classical symmetries to the setting of quantum symmetries; we show that Property (SP) for (simple, separable) C*-algebras is preserved under irreducible C*-discrete inclusions. Finally, we resolve (in the negative) a question raised by Gabe and Neagu about inclusions coming from Furstenburg boundaries. Namely, we show the existence of a countable, discrete, non-amenable group $G$ such that $C_r^*(G) \subseteq C(\partial_F G) \rtimes_r G$ is not an SP-inclusion (and therefore not an inclusion of real rank zero).

I give an elementary categorical lift of the three-variable Negami polynomial and of the boundary state structure underlying the splitting formula.

For $P=X_1^2+\cdots + X_n^2$, let $T_t^Pf(x)$ denote the solution to the linear Schrödinger equation at time $t$. In 1980, Carleson asked for the minimal regularity of an initial data function $f\in H^s(\mathbb{R}^n)$ that guarantees pointwise convergence of $T_t^Pf(x)$ to $f(x)$ as $t\rightarrow 0$. This was resolved by Bourgain, who constructed counterexamples for the Schrödinger maximal operator to show that $s\geq n/(2(n+1))$ is necessary, and Du and Zhang, who proved that $s> n/(2(n+1))$ is sufficient. Rogers, Vargas, and Vega studied the analogous question for the non-elliptic Schrödinger maximal operator, where $P = X_1^2-X_2^2 \pm X_3^2\pm \cdots \pm X_n^2$, and proved that, for all $n\geq 2$, $s\geq 1/2$ is necessary and $s>1/2$ is sufficient. In this paper, we construct counterexamples for generalizations of the non-elliptic case and prove a necessary condition of $s\geq 1/2$ for an infinite class of polynomial symbols $P$.

We explore Fischer decompositions from the perspective of inframonogenic functions in Clifford analysis, a non-commutative version of harmonic functions in Euclidean spaces. We study decomposition formulas for polynomial spaces involving inframonogenic and harmonic components, providing fresh insights into their algebraic structures and further important applications.

We construct Fortin operators for discontinuous Petrov-Galerkin (DPG) discretizations of the advection equation with a piecewise constant divergence-free advection vector $\beta$, on simplicial meshes of any spatial dimension $N$ and for any polynomial degree $k \ge 1$ of the trial space. A minimal test space is built on each element from facet and interior bubbles and later augmented. The Fortin operator is shown to be bounded, uniformly over shape-regular mesh families, in the natural $\beta$-weighted broken test graph norm built on $L_q$ for every $1 < q < \infty$, where $q$ is the exponent conjugate to the trial exponent $p$. A non-characteristic facet condition is assumed when $k \ge 2$, while the lowest-order case requires no such condition and admits characteristic facets. As applications we prove that the practical fully discrete residual minimization method is quasioptimal in the DPG energy norm for every $1 < p < \infty$, with a quasioptimality constant governed solely by the Fortin operator, that its computable residual is a globally reliable and efficient a posteriori error estimator, and that augmenting the test space and changing the test norm improves the results in the lowest-order case.

Given a finite set $A$ of natural numbers whose greatest common divisor is one, the Frobenius number $g(A)$ is the largest integer that is not a non-negative integer combination of the numbers in $A$. In a 2016 preprint, Matsubara states that given $A$ and $k$, deciding if $g(A) \geq k$ is $\Sigma_2^P$-complete. A decade has passed since without peer-reviewed publication of this result. At the same time, the community has found it difficult to verify this result. In this note, we give a write-up of the completeness proof based on Matsubara (2016).

We study the geography of taut foliations on compact $3$-manifolds with toroidal boundary components. The central object of our work is the set of boundary multislopes realized by foliations transverse to a fixed flow on such a manifold. We prove that these sets exhibit remarkable structural properties (rationality, rigidity, and convexity) which motivate the name ziggurats. Our main tool, of independent interest, is a two-way correspondence between foliations and contact structures on $3$-manifolds with boundary: we generalize the Eliashberg-Thurston theorem, which produces pairs of positive and negative contact structures from foliations, and a construction of the first author, which builds foliations from such contact pairs. Using both directions of this correspondence, we bring contact-geometric methods to bear on the architecture of ziggurats.

We determine the long-time behavior of the canonical Ricci flow on a closed three-manifold of Thurston type $\mathbb{R}^3$, Nil, or Sol, starting from an arbitrary initial metric. In the Euclidean case, the metric $g(t)$ converges exponentially fast to a flat metric. The Gromov–Hausdorff limit of $(M,t^{-1}g(t))$ is a point in the Euclidean and Nil cases, and a metric circle or a compact interval of positive length in the Sol case. In all three cases, the blowdown flows lifted to the universal cover, $s^{-1}\widetilde g(s\tau)$, converge, in the pointed Cheeger–Hamilton sense, to an explicit homogeneous expanding Ricci soliton solution.

Finite difference methods for interface problems often exhibit large local truncation errors near the interface, particularly when discontinuities in coefficients or solution derivatives are present. Nevertheless, many such schemes achieve second-order accuracy, a phenomenon not fully explained by standard pointwise consistency arguments. In this work, we provide a precise explanation of this behavior through the structure of the discrete Green's function associated with the underlying conservative difference operator. An explicit representation of the Green's function reveals a two-plateau structure in its weighted increments. By expressing the numerical error in terms of this Green kernel, we show that the dominant interface truncation errors, although individually of order \(O(1)\), possess a cancellation structure that reduces their effective contribution to the global error. The exact Green's-function analysis leads to a class of conservative interface flux–balance schemes for which the cancellation mechanism yields second-order accuracy in the maximum norm. The weighted harmonic discretization is included as a particular member, and its cancellation property is established directly. For Cartesian grids with a flat interface parallel to a coordinate direction, the same conservative cancellation mechanism persists along grid lines crossing the interface. Numerical experiments in one and two dimensions directly illustrate the predicted cancellation behavior and the resulting second-order accuracy. These results demonstrate that second-order accuracy in interface problems can arise from the interaction between localized truncation errors and the global structure of the discrete operator, rather than from pointwise consistency alone.

Integrals of spectral densities are frequently used to summarize spectral characteristics of linear processes. This work studies self-normalization for estimators of such integrals based on sequential periodograms and establishes weak convergence of the corresponding processes. For linear functionals of the spectral density, self-normalization yields pivotal limiting distributions that are free of unknown spectral quantities. For non-linear functionals, however, additional components with distinct covariance structures may arise in the limiting process. We demonstrate this phenomenon for the integrated squared spectral density.

Enforcing stochastic state constraints over the full prediction horizon in Model Predictive Control (MPC) can be computationally demanding. Here we study stochastic MPC without terminal ingredients in which chance constraints are enforced only over a shorter constraint horizon. Using stochastic relaxed dynamic programming, we derive an explicit upper bound on the average expected closed-loop cost that depends on both prediction and constraint horizons. For linear quadratic problems with affine chance constraints and bounded uniform disturbances, we provide a deterministic reformulation via coordinate transformation and constraint tightening. Simulations illustrate the trade-off between computational effort and performance.

We study polynomial absolute and relative joint first-order differential projective invariants for configurations of $n$ points in the plane. The main approach is based on passing to a homogeneous vector–covector representation, which reduces the problem to the classical invariant theory of the group $SL(3,\mathbb C)$. The algebra of polynomial absolute invariants is described and shown to be generated by cyclic invariants. For polynomial relative invariants of weight $-1$, a graded description is obtained, their finite generation as a module over the algebra of absolute invariants is proved, and an explicit finite generating system is constructed. For $n=3$, the corresponding module is shown to be free of rank $1$. For $n=4$, a minimal homogeneous generating system consisting of $39$ elements is constructed. The obtained results complement the rational theory of joint projective differential invariants by its polynomial counterpart and provide an algebraic foundation for the further construction of projectively invariant integral characteristics.

We prove, under some assumptions, the uniqueness of universal quantum dimension formulae. We first show that the quantum dimensions of classical algebras have a specific form that can be represented universally, in Vogel's sense, as the product/ratio of q-dimension factors with arguments $a\alpha+b\beta+c\gamma$ where $c=0, 1, 2$ and $a, b$ are integers. On this basis, we derive simple uniqueness criteria, according to which all known universal quantum dimensions are unique.

We study avoidance of patterns of length $3$ with three distinct letters in canon permutations. We reduce the problem to studying pattern avoidance in lattice words and show that there are $6$ such pattern avoiding classes. This shows that there are $12$ classes for the original Canon permutation pattern avoidance problem. We also give descent refinements for these classes and classify the patterns for which the descent enumeration gives palindromic and $\gamma$-positive polynomials. When the polynomials are $\gamma$-positive, we explain the $\gamma$-positivity through a group action analogous to Foata-Strehl valley hopping. Additionally, we study the avoidance of patterns in the relabelling orbit of $1213, 12112, 1231$ after a conjecture about their cardinalities by Laudone and give bijective proofs for the results.

This survey explores the rich interplay between (taut) foliations and (tight) contact structures in dimension three, highlighting recent work of the author. We outline a new method for constructing taut foliations from suitable pairs of contact structures, and discuss some applications and future research directions. In particular, we give a brief account of work in progress with Jonathan Zung on ziggurats for taut foliations transverse to pseudo-Anosov flows, and we propose a (mostly speculative) contact perspective on the $L$-space conjecture.