Papers

Let $f\colon X\to X$ be a polarised endomorphism of a complex projective manifold, and let $D\subset X$ be a reduced totally invariant hypersurface. We prove that every irreducible component of the singular locus of $D$ has codimension one in $D$. In particular, if $D$ is normal, then it is smooth. As an application we prove the linearity of totally invariant divisors for endomorphisms of $\mathbb P^4$.

Extreme learning machines (ELMs) are a class of machine learning algorithms that randomly intialize and subsequently fix the hidden layer parameters, and solve for the last layer coefficients using a least squares method. ELMs have been successfully applied to the solution of various partial differential equations, but the cost of solving large least squares problems limits their use in large scale problems or when high accuracy is desired. To address this limitation, domain decomposition methods (DDMs) have been developed for ELMs, reducing training times via parallel computation. Yet these methods, even those incorporating a coarse space, have struggled with scalability. This paper introduces a novel nonoverlapping DDM for ELMs that treats and enforces continuity and flux conditions equally. Additive Schwarz preconditioners and GenEO (Generalized Eigenvalues problems in the Overlaps) coarse spaces are constructed to accelerate the method. Numerical experiments show the new method outperforms previous DDMs for ELMs in both iteration count and accuracy, especially for the Stokes and wave equations. Furthermore, weak scalability has been observed in up to 4,096 subdomains when employing GenEO coarse spaces.

Let $t(n)$ denote the minimum number of transversals in a Latin square of odd order $n$. Improving upon a recent bound of Dai, Divoux and Kelly, we prove that for every $n$ such that $n \equiv 3 \pmod 6$, \[ t(n) \leq \left( \left(1+o(1)\right) \frac{2n}{3e^2}\right)^n . \] Our proof is based on a family of $3 \times 3$ block Latin squares whose transversals are constrained to either lie entirely in the diagonal blocks or avoid them altogether.

Network science is increasingly focused on how node dynamics influence emergent phenomena such as oscillations, waves, chimeras, and turbulence. In oscillatory systems modeled by networks of coupled ordinary differential equations (ODEs), a common approach is to reduce the system to phase variables. When understanding how network delays shape emergent properties, the delays are often absorbed in the reduced description as a phase shift. The reduced system is a set of ODEs that loses some information about the full delay differential equation (DDE) system. We adopt a less restrictive approach and consider limit cycle oscillator networks with multiple delays that can be reduced to a DDE system. This captures the effects of delayed couplings, including coexistence and multistability of states. We analyze relative equilibria in the form of phase-locked states with tools previously developed for more general DDE settings. This allows us to explore patterning in networks with space-dependent delays, including in neuroscience, using symmetric bifurcation theory, linear stability analysis, and numerical simulations and continuation. In brain dynamics, time delays are determined by the speed of a communicating signal (action potential) along a fiber (axon). Importantly, these are now known to be state-dependent since the myelin (white matter) that insulates axons is plastic and can change in response to neuronal activity. A simple phenomenological model of this process (in the phase reduced description) is introduced and analyzed by extending techniques developed for fixed delays. Our analysis suggests that white matter plasticity can drive networks to more coherent behavior.

Within the framework of classical contact geometry, the first author introduced the notion of contact symplectic structure –the data of an exact symplectic fibration with a contact base– and proved, under mild conditions, that the total space of such a fibration inherits a contact structure compatible with the fibration map, thereby establishing the contact Thurston theorem. This paper provides a derived contact version of that result by incorporating our prior work on the derived symplectic Thurston theorem. In this paper, we prove, under certain conditions, that if a morphism $\pi: X \rightarrow S$ of derived stacks has a shifted exact symplectic fibration structure and the target stack $S$ admits a shifted contact structure, then one can construct a shifted contact structure on the source stack $X$, compatible with $\pi$ in a sense similar to the smooth case. Our framework relies on the theory of relative shifted structures; hence our result, called the derived contact Thurston theorem, in fact establishes a relative-to-absolute type construction. As an application, we present examples of our relative-to-absolute construction formalism in the derived contact setting, including conormal stacks, quotient mapping stacks, and affine exact symplectic fibrations.

Let $T$ be a finite group and let $kB_T$ denote the shifted Burnside biset functor over a commutative ring $k$. We study the essential algebra $\widehat{kB_T}(G)$ of $kB_T$ at an arbitrary finite group $G$. We first establish criteria determining when a covering subgroup of $G\times G\times T$ factors, with respect to the star product, through a group of order strictly smaller than $|G|$; the first criterion places no hypothesis on $T$. As an application, we describe $\widehat{kB_T}(G)$ completely whenever no nontrivial quotient of $G$ is isomorphic to a section of $T$, extending the coprime case treated by Romero. When $T$ is abelian and $|T|$ is invertible in $k$, we prove that $\widehat{kB_T}(G)$ decomposes as a direct sum of matrix algebras over the group algebras of the groups $Out_T(A)$, where $A$ runs over the linkage classes of reduced subgroups of $G\times T$, and we parametrize the simple modules of $\widehat{kB_T}(G)$.

We are interested in computing approximate dual bounds for nonconvex optimal control problems using McCormick inequalities. To handle the numerical burden concerned with the tightening of these inequalities, we locally average the bilinear term in the PDE constraint. We extend recent theory in this area to elliptic PDEs on multidimensional domains that involve bilinear terms. As an additional result, we also generalize the necessary assumptions to allow for more flexibility in the choice of the discretization for the total variation penalty term from the underlying control problem. Furthermore, we prove the existence of valid bounds on the state variable, which are crucial to enable the optimization-based bound-tightening (OBBT) procedure to tighten the McCormick inequalities. Since theoretical bounds on the state variable are rather conservative, OBBT is both crucial and the main computational bottleneck for deriving a sharp approximate dual bound. We show how the sequence of LPs arising during OBBT may sensibly be warmstarted, allowing us to successfully scale the numerical experiments to finer discretizations than what was possible before.

We prove an asymptotic Chvátal-type codegree criterion for tight Hamiltonian cycles in $k$-uniform hypergraphs for every fixed $k\ge3$. The criterion allows small degrees to be compensated by large degrees in the link graphs of $(k-2)$-tuples. It strengthens the asymptotic Dirac-type theorem by Rödl, Ruciński and Szemerédi and implies a Pósa-type criterion conjectured by Schülke.

We consider two higher-order generalizations of the Godbersen conjecture for mixed volumes of convex bodies, which were first proposed by Schneider in 2000. First, we establish the conjectures in several special cases, in particular in low dimensions. Second, we prove that certain consequences of the conjectures hold. More precisely, we define a higher-order version of the unbalanced difference body and prove related weighted inequalities, generalizing previous results of Artstein-Avidan and Putterman. Finally, we introduce higher-order analogs of unbalanced joins of convex bodies considered by Artstein-Avidan, Einhorn, Florentin, and Ostrover, prove the corresponding volume bounds, and propose conjectures that interpolate between the higher-order Godbersen conjectures and a conjecture due to Fáry and Rédei.

In this paper, we study sequential lossy compression under a causal conditional perception criterion comparing source and reconstruction distributions given the same reconstruction history. For first-order Markov sources, we formulate the finite-horizon nonanticipative rate-distortion-perception function (NRDPF) with stagewise constraints and establish one-shot lower and upper bounds on the minimum variable-length sum rate using a strengthened strong functional-representation lemma (SFRL) and common randomness. For time-varying scalar Gauss–Markov sources under pointwise mean-squared error (MSE) and conditional squared Wasserstein-$2$ fidelity, we prove Gaussian optimality, derive a log-variance characterization, and obtain a closed-form solution that recovers the classical Gaussian nonanticipative rate-distortion function (NRDF) when perception is unconstrained and the classical Gaussian RDPF when the source is stationary and memoryless.

Polynomial modular reduction is central to binary finite-field arithmetic and repeated Frobenius powering. Top-down shift/XOR folding can use few operations when the nonleading support is sparse, but a tap near the leading term creates a long feedback chain. We formulate this recurrence as inversion of a nilpotent shift operator and factor its inverse by characteristic-two Frobenius powers. The resulting Frobenius-factorized reduction (FFR) applies to every monic binary modulus without materializing a reciprocal or dense reduction matrix, and its shifts can be generated online without a persistent modulus-specific schedule. For degree $m$ and nonempty nonleading support of size $s$, with nearest-tap distance $\Delta_{\min}$, FFR uses exactly $\lceil\log_2(m/\Delta_{\min})\rceil$ sequential feedback stages and has scheduled work $O(ms(1+\log(m/s)))$. A portable-C evaluation on 1,096 supports through degree $131072$ identifies distinct FFR, López–Dahab, and gf2x-backed Barrett regions. On four certified irreducible moduli, FFR makes complete Rabin irreducibility testing $1.35$–$8.04$ times faster than NTL and $1.60$–$6.81$ times faster than the matched Barrett implementation.

This note presents a proof that the Riesz projection is contractive from $L^q$ to $L^{4(1-1/q)}$ for the case $1<q<\infty$. This fills in the previously unresolved parameter range of a conjecture of Brevig, Ortega-Cerdà, Seip, and Zhao.

In this article, we investigate refined and generalized versions of the Bohr and Bohr–Rogosinski inequalities for a normalized subclass $\mathcal{P}_{\mathcal{H}}^{0}(\alpha)$ ($0 \le \alpha < 1$) of univalent close-to-convex harmonic mappings $f = h + \overline{g}$ defined on the open unit disk $\mathbb{D} \subset \mathbb{C}$. By incorporating non-negative monotone increasing functions associated with the planar area integral $S_r/\pi$ of the image domain $f(\mathbb{D}_r)$, we establish new sharp Bohr-type inequalities expressed in terms of the Euclidean distance $d(f(0), \partial f(\mathbb{D}))$. Furthermore, we establish sharp Bohr–Rogosinski-type inequalities involving powers of the modulus of the mapping $|f(z)|^p$ ($p \ge 1$). All associated radii are proven to be sharp, and extremal functions realizing the equality cases are explicitly identified. As applications, our results generalize and unify several well-known classical and recent theorems in geometric function theory.

For a given number $N$, we consider the problem of computing two integers $1\leq r,f < N$ such that the set $$\mathcal{X}(N,r,f) = \{(a+b)-(f+\frac{Nr+1}{f}): ab=Nr\}$$ consists only of positive integers. Computing a solution to the problem is equivalent to finding a pair $(r,f)$ satisfying $l(Nr) < f \leq l(Nr+1)$, where $l(x)$ is the largest divisor of $x$ bounded by $\sqrt{x}$. This requires factoring both $Nr$ and $Nr+1$. We present a simple randomized algorithm that - avoiding factoring - computes pairs $(r,f)$. We give an exact formula for the total number of possible pairs $(r,f)$, and with the aid of empirical data we estimate that the ratio $$\frac{\phi(N)-2}{|\mathfrak{F}(N)|}$$ is approximately about $c*\log \log N$. Here, $\mathfrak{F}(N)$ is the set of unique $r$ appearing among all possible pairs $(r,f)$, $\phi(.)$ is the Euler's Totient function, and $c$ is a constant equal to 2 for prime $N$ and oscillates much for composite $N$. As a separate and independent case, we study the same problem of computing $(r,f)$ with $r>N$. We present a procedure to find such an $r$, which requires finding the least prime in an arithmetic progression.

We prove four explicit continued fraction representations for the Lerch transcendent $\Phi(z, s, M+1)$, where $\Re M>0$ and $(z,s)\in\{(-1,1),(-1,2),(1,2),(1,3)\}$. All of the continued fractions have unit partial numerators, while partial denominators depend on the parameter $M$. The proofs combine equivalent transformations of continued fractions, generalized hypergeometric functions, three-term recurrences, and asymptotic analysis of minimal solutions. For $z=1$, the corresponding representations give continued fractions for the values of the Hurwitz zeta function $\zeta(2, M+1)$ and $\zeta(3, M+1)$.

Dunshee and Ellingham recently showed that seven natural properties of a cellularly embedded graph form a Fano-plane framework. We establish an analogous framework for binary delta-matroids. For a binary delta-matroid $D$ and $\tau$, let $Z_3(D,\tau)$ denote its associated binary tight $3$-matroid. The six outer points are represented by evenness or bipartiteness of $D$ and its global vertex-flip transforms. For the seventh point, we call $D$ $Z_3$-bipartite when every circuit of $Z_3(D,\tau)$ has even cardinality. We show that the satisfied properties are precisely the nonzero vectors of a subspace of $\Ftwo^3$. For ribbon-graphic delta-matroids, $Z_3$-bipartiteness is equivalent to bipartiteness of the medial graph, so the construction recovers the Fano-plane framework for embedded graphs.

We develop the mathematical analysis of the Time-Dependent Density Functional Theory introduced by Manfredi [J. Plasma Phys., 86.2 (2020), 825860201] in the setting of Vlasov equations with a time-dependent exterior potential. In this context, we establish the invertibility of the potential-density mapping for the Vlasov-Poisson system on $\mathbb{T}^d \times \mathbb{R}^d$ when the exterior potential is (real-)analytic in the space variable. We also recover a Runge-Gross type Theorem for the Vlasov-Poisson equation for smooth (real-)analytic in time exterior potential. The surjectivity of the potential-density mapping is an inverse problem for the Vlasov-Poisson equation: we find the electric field that generates a given density profile on a given time interval.

We characterize equality in the mass-capacity inequality for asymptotically flat manifolds with arbitrary ends in every dimension \(n\ge 3\). If \(u\) is the capacity minimizer, then the equality forces \((M,u^{4/(n-2)}g)\) to be isometric to \((\mathbb R^n\setminus S, g_{\mathrm{Euc}})\) for a compact set \(S\) satisfying \(\operatorname{dim}_{\mathcal H}S\le (n-2)/2\). A key ingredient is that conformal changes by positive harmonic functions transform nonnegative Ricci curvature into nonnegative Bakry–Émery Ricci curvature of effective dimension \(4-n\).

For $0<\lambda<1/2$, let $K_{\lambda}$ be the attractor of the iterated function system $\{\lambda x, \lambda x+1-\lambda\}$, and put $C_\lambda=K_{\lambda}\times K_{\lambda}$. We study the intersection \[ E_\lambda=C_\lambda\cap S^1 \] and the Minkowski sum \[ A_{\lambda}=C_\lambda+S^1, \] where $S^1$ is the unit circle. For the intersection problem, we prove that $E_\lambda$ has the cardinality of the continuum for $(\sqrt{3}-1)/2<\lambda<1/2$, and $\dim_{\rm H} E_\lambda>0$ for $\sqrt{2}-1<\lambda<1/2$. We also establish quantitative lower bounds for $\dim_{\rm H} E_\lambda$ for $\lambda$ near $1/2$; in particular, $\dim_{\rm H} E_\lambda$ approaches $1$ as $\lambda\uparrow1/2$. For the Minkowski sum problem, we prove that $A_{\lambda}$ has nonempty interior throughout the previously open range $1/4<\lambda<1/3$, answering a question of Simon and Taylor. Together with earlier results of Simon and Taylor, our theorem yields the complete classification: $A_{\lambda}$ has nonempty interior in $\mathbb{R}^2$ if and only if $1/4<\lambda<1/2$. More generally, we prove that $C_\lambda+\Gamma$ has nonempty interior for every regular $C^1$ closed curve $\Gamma$ whenever $1/4<\lambda<1/2$.

We study the design of feedback recommendation policies for networked multi-topic opinion dynamics. The design of recommendations is formulated as an infinite-horizon linear-quadratic optimal control problem that favors suggestions aligned with each agent's current opinion, thereby using opinion alignment as a proxy for engagement. At the same time, the performance index penalizes polarization, deviation from the uncontrolled equilibrium of the opinion dynamics, and from the current agents' opinion, recommendation effort, and favors coherence among neighbors' recommendations. An agent-wise representation exposes the interconnection structure of the social network and enables structured controller and certificate synthesis. Under the assumption that the stage cost of the performance index is strictly positive definite, the affine optimal control problem separates into a strictly convex steady-state optimization, which determines the optimal equilibrium and the affine controller offset, and an LQR problem, which yields the static state-feedback gain. The resulting centralized Riccati controller provides a performance benchmark, while structured gains are obtained from dissipativity-based and $\mathcal H_2$-based LMI surrogate formulations. Possibly overlapping certificate clusters yield scalable local sufficient conditions for the structured gain design, while a separate steady-state cluster decomposition provides an exact consensus reformulation of the steady-state optimization. Numerical results on a social network illustrate the closed-loop behavior and the trade-off between performance and controller locality.

The $a$-number of a function field $F$ is the dimension of the kernel of the Cartier operator acting on the holomorphic differentials of $F$. We compute the action of the Cartier operator on the holomorphic differentials on the Hermitian function field $\mathcal{H}$, function fields of Fermat type $\mathcal{F}_{n,m}$ and the Hurwitz function field $\mathcal{H}_n$. From this, we obtain various results about the $a$-number of these function fields, including a generalisation of the results on the Fermat function field from Kodama and Washio.

We study distributed one-dimensional mean estimation under a 1-bit communication constraint. Each agent observes one sample, drawn independently from an unknown distribution, and returns a single bit in response to a query $Q: \mathbb{R}\to\{0,1\}$ chosen by a central learner. The distribution has mean in $[-\lambda,\lambda]$ and $k$-th central moment at most $\sigma^k$, for a fixed $k>1$. The order-optimal two-stage protocol of Lau and Scarlett uses responses from the first batch to choose the second-batch queries, motivating the question of whether this single round of interaction is necessary. We answer this negatively: for every $k>1$, a non-adaptive protocol attains the adaptive 1-bit minimax rate (and concurrent works reached the same conclusion via different strategies). We further determine the minimax sample complexity among non-adaptive 1-bit estimators when every one-set $Q^{-1}(1)$ is restricted to a union of at most $s$ intervals. Relative to unrestricted non-adaptive 1-bit querying, this constraint adds a term of order $(\lambda\sigma/(s\varepsilon^2))\log(1/\delta)$, giving the full tradeoff between sample complexity and interval complexity to within $k$-dependent constant factors. As a corollary, we identify, order-wise, the minimum interval budget needed to retain the unrestricted 1-bit minimax sample rate.

Permutation pattern avoidance on Ferrers boards has become a central topic in enumerative combinatorics with important connections to matchings, set partitions, and other combinatorial structures as it allows one to build families of Wilf-equivalent patterns. While shape-Wilf-equivalence classes have been completely determined for individual patterns and pairs of patterns of length three, the corresponding classification for larger pattern sets has remained open. In this paper, we provide a complete classification of the shape-Wilf-equivalence classes of triples of patterns of length three. Our proofs use a bijective encoding of pattern avoiding transversals to establish all equivalence classes. As an application, we enumerate matchings avoiding triples of patterns of length three for all, extending previous results of Bloom and Elizalde. These enumerative results identify additional families of combinatorial objects counted by the Fuss-Catalan numbers and by other integer sequences appearing in the OEIS.

Class disentanglement–the separation of a representation's class-conditional point clouds along depth and over training–is measured by descriptive curves: the sentence such a study wants to write, layer l+1 is more disentangled than layer l, is an eyeball judgement with no null. We supply the inferential layer for a topological measurement of class overlap, the Intersection Euler Characteristic Profile: the Euler characteristic of the overlap of the clouds' ball unions as a function of scale, from one Alpha-complex sweep with no boundary-matrix reduction. Every number carries a test–exact permutation tests in both directions, a guarded separation certificate the invariant requires, and a paired sign-flip test for comparative claims. Building that test taught a lesson outliving this invariant: its statistic must be scale-free. On the raw profile mass, which has units of feature length, 12,375 paired tests return 5,633 significant steps of which every one at the first epoch points the wrong way, certifying feature-norm dynamics as disentanglement; the dimensionless statistic returns 2,707, with 2,026 decreases. Across 111 networks and 52,650 measurements, disentanglement is depth-graded and early, and interaction quotients rank class pairs by confusability (rho=0.83), on par with cheap separability statistics. In a 96-model factorial, augmentation is the one training choice that separates classes relative to chance; weight decay compresses the overlap without separating. Only a k-fold statistic can pose the structural question: the joint entanglement of a class triple sits below its strongest pair in 97% of triple-layer cells and 99.5% of deep cells, at median ratios far below a measured null floor, in vision encoders and frozen language models–a regularity, not a law. The unnormalized mass predicts test accuracy (R^2=0.94), the quotient does not, and neither beats a linear probe.

We define a construction of Legendrians inside contact manifolds that arise by doubling an exact Lagrangian filling in the page of an open book decomposition. This can be seen as a generalization of a previous construction by Courte and Ekholm to arbitrary open books. These Legendrians, called Legendrian doubles, are shown to admit regular flexible exact Lagrangian fillings, and they are thus classified up to Legendrian isotopy by classical data. Finally, we show that the Legendrian suspension construction, as defined by Arikan and the author in previous work,-this is a Legendrian contained inside a page of an open book that is obtained by using Seidel's suspension of Lefschetz fibrations- is a Legendrian double.

This paper establishes a formal framework for the cocyclic development of Hadamard matrices over loops based on a novel cohomology theory with associativity obstructions. In this context, we formalize the notion of a loop-cocyclic Hadamard matrix, so that both the classical cocyclic Hadamard matrices over groups and the recently introduced pseudococyclic Hadamard matrices over loops are naturally embedded within this broader algebraic architecture. To validate the computational viability of this framework, we prove the existence of four new Hadamard equivalence classes of order 24 and eight new classes of order 28 that are strictly non-cocyclic over any finite group, arising uniquely as loop-cocyclic developments over Moufang and right Bol loops.

We formulate a conjecture predicting the existence of exotic isomorphisms between Igusa stacks associated to different Shimura data whose underlying groups are pure inner forms of each other, and prove it in many cases of interest. In combination with the spectral action of Fargues–Scholze, this allows us to relate the cohomology groups of the two Shimura varieties, giving a geometric incarnation of the Jacquet–Langlands correspondence. An important ingredient in our proofs is the work of Xiao–Zhu on exotic Hecke correspondences between the perfect special fibers of different Shimura varieties, which we reinterpret in terms of exotic isomorphisms of perfect Igusa stacks.

As a natural generalization of the classical hyper-Kloosterman family studied by D. Haessig and S. Sperber, we study the $k$-th symmetric power $L$-functions attached to a weighted hyper-Kloosterman family $$Kl_{n,m}(t;x_{1},\cdot\cdot\cdot,x_{n})=x_{1}^{m}+x_{2}\cdot\cdot\cdot+x_{n}+\frac{t}{x_{1}x_{2}\cdot\cdot\cdot x_{n}}.$$ Under suitable conditions, we determine the bounds of degrees of these $L$-functions and prove that their $q$-adic Newton polygons admit uniform lower bounds.

We study variance-optimal stock hedging and the convergence of approximate strategies in the rough Hawkes–Heston model. Starting from the model's affine conditional transform and the affine Volterra jump framework, we obtain semi-explicit hedges for European calls and a representation of the minimum quadratic error through the Galtchouk–Kunita–Watanabe projection. Our main approximation result keeps the original stock, variance driver, and information flow fixed while regularizing the kernel used to evaluate the hedge. To handle singular memory and common marked jumps, we construct the approximate holdings from histories available before trading and preserve the conditional transform's random modulus envelope. Riccati–Volterra stability and weighted truncation then yield convergence in the original stock's trading norm on compact Fourier intervals. For calls, a joint choice of kernel regularization and Fourier cutoff gives convergence of the initial capitals and strategies, uniform-in-time square-mean convergence of continuous-time gains, and convergence of the terminal mean-square error to the variance-optimal value. A numerical experiment with shifted fractional kernels illustrates the construction on common original-market paths.

Let $G$ be a simple graph with maximum degree $\Delta\ge 3$, and let $P(G,k)$ denote its chromatic polynomial. For each positive integer $k$, the list-color function $P_{\ell}(G,k)$ is the minimum number of $L$-colorings of $G$ over all $k$-assignments $L$. In this paper, we prove that $P_{\ell}(G,k)=P(G,k)$ for every integer $k\ge 23.41\Delta$. This gives a threshold for equality that is linear in the maximum degree and independent of the number of vertices or edges. It improves the known sufficient condition $k\ge |E(G)|-1$ for graphs with sufficiently many edges relative to their maximum degree.