Papers

We study the exact enumeration of maximal cliques in graph classes defined by excluded clique minors and excluded clique immersions. For n-vertex K_t-minor-free graphs, we give an algorithm that lists all maximal cliques in n * 4^(2t/5+o(t)) time, significantly improving the previous n * 2^O(t log log t) bound of Eppstein, Löffler, and Strash. For n-vertex K_t-immersion-free graphs, we establish the first exact enumeration algorithm parameterized by immersion number, achieving a running time of n * 3^(t/3+o(t)). While both algorithms employ a common degeneracy-based root-assignment scheme, their analyses require distinct structural mechanisms. Crucially, rather than applying generic sparsity bounds, our algorithms deeply integrate the specific structural obstructions – local density thresholds for minors and minimum-degree branchings for immersions – directly into the enumeration logic. We also prove matching output-size lower bounds, up to sub-exponential factors in t, using specialized constructions. Consequently, the exponential bases 4^(2/5) and 3^(1/3) are asymptotically optimal.

In this study we have explored gradient $h$-almost Yamabe solitons on both compact and complete non-compact Riemannian manifolds. We have established several sufficient conditions for the triviality of such solitons with respect to integral inequalities involving the scalar curvature and the soliton function. In this regard, we have acquired scalar curvature estimation under certain $L^1$ and $L^2$ integrability conditions, and defined signal of the function $h$. Our results extend and refine former works on almost and $h$-almost Yamabe solitons, and characterize the geometric structures of generalized Yamabe solitons.

The paper is a synthesis of several works on the variational principles for application to the mechanics and the physics, inspired from original ideas of Brezis, Ekeland and Nayroles. On this basis, we developed an unified framework for dynamic dissipative systems that leads to a space-time variational principle of minimum constructed with tools of convex analysis and symplectic geometry. We stress the essential ideas and concepts. They are illustrated with various theoretical and numerical examples.

We study matching-removability under the degree/connectivity regime of Halin's theorem, which asserts that every $k$-connected graph $G$ with minimum degree $\delta(G)\ge k+1$ contains an edge $e$ such that $G-e$ remains $k$-connected. For $k,\ell\ge 1$, an $\ell$-matching is a matching of size $\ell$. A matching $M$ in a $k$-connected graph $G$ is {\it $k$-removable} if $G-M$ remains $k$-connected. We improve Halin's result by proving that every $k$-connected graph $G$ with $\delta(G)\ge k+1$ contains a $k$-removable $2$-matching, except when $k=1$ and $G$ is a cycle. For small $k$ we obtain stronger bounds: (i) $k=1$: a 1-removable $\min\{\lfloor n/2\rfloor,\delta(G)\}$-matching; (ii) $k=2$: a 2-removable $\lceil(\delta(G)+1)/2\rceil$-matching, with a unique tight exception when $\delta(G)$ is even and $G\cong K_{\delta(G)+1}$; and (iii) $k=3$: for $\delta(G)\ge 5$, a $3$-removable $\lceil(\delta(G)+1)/2\rceil$-matching. All these bounds are optimal with respect to removable matching size and minimum degree. We also show that for every $n\ge 2\delta$, there exists a $k$-connected $n$-vertex graph $G$ with minimum degree $\delta$ that does not contain a $k$-removable matching of size at least $\delta(G)+1$. Moreover, for $k\le 2$ there exists a $k$-removable $(\delta(G)-c)$-matching for some $c\le 3$, which is optimal up to the additive constant.

In this paper we consider a nonlinear equation $-\mathcal{L} u(x) = f(x, u(x))$ with a super-quadratic nonlinearity, $f$, and a nonlocal operator, $\mathcal{L}$, generated by a special class of radially symmetric $L^1$ convolution kernels with finite second moments. The assumptions on this operator are mild and allow for a variety of kernels used in biological and physical applications, including kernels with algebraic decay and sign changing kernels. Using the strong nonlinearities present in the equation, we prove the existence of nontrivial solutions using the classical Mountain Pass Theorem, a central result in minimax theory that equates solutions of our equation to critical points of a corresponding energy functional. This existence result holds with both homogeneous nonlocal Dirichlet and nonlocal Neumann boundary conditions. We supplement these theoretical results with numerical simulations for various nonlinearities with odd maximal degree in the unknown $u$. The numerical scheme exploits the resulting energy landscape which allows one to adapt a gradient descent algorithm.

This paper addresses the challenge of reconstructing full-field structural mode shapes from sparse sensor data. While Gaussian Process Regression (GPR) offers a robust non-parametric framework for spatial interpolation and uncertainty quantification, standard formulations often yield physically inconsistent mode-shape reconstructions under sparse sensing conditions. A Physics-Constrained Single-Output Gaussian Process (CONS-SOGP) framework is derived that utilizes independent modal kernels while coupling the optimization via a mass-orthogonality penalty. The paper presents derivations for the marginal likelihood, hyperparameter gradients, and penalty coupling. Numerical verification on a multi-degree-of-freedom structure demonstrates that the proposed method overcomes existing limitations in GP-based prediction, providing more accurate and reliable expanded mode shapes.

Fréchet regression generalizes linear regression to metric-space-valued responses by defining fitted values as minimizers of weighted Fréchet functionals. Since these weights may have mixed signs, the resulting objective is a signed barycenter problem rather than a standard convex barycenter problem. On Riemannian manifolds, this is further complicated by the lack of global geodesic convexity and possible nonsmoothness of squared distances near cut loci. We study signed Fréchet regression on complete manifolds with two-sided bounded sectional curvature. By restricting optimization to a strongly convex normal ball containing the response support, we use local smoothness, Hessian comparison, and Jacobi-field estimates to formulate the problem as a locally controlled Riemannian proximal DC problem. This leads to FRIDA (Fréchet Regression via Riemannian Iterative DC Algorithm), an exact and inexact proximal DC algorithm for computing regression fits. We prove existence and interiority of minimizers under explicit signed-weight conditions, establish curvature-dependent strong convexity of the proximal subproblems, and show descent and convergence of the iterates to stationary points. We also derive sublinear complexity estimates and, under real-analyticity, obtain full-sequence convergence with KL-type local rates. These results provide a rigorous optimization framework for signed Fréchet regression on manifolds with bounded curvature.

Studied here is the Zakharov–Kuznetsov equation with a linear transport term posed on a half-strip with nonhomogeneous boundary condition. Using Bourgain-type spaces adapted to the ZK dispersive structure, anisotropic smoothing and boundary trace estimates, we establish its local well-posedness in $L^2.$

We define a sequence of integer-valued invariants $\gamma^k(L)$ for a $3$-component link $L$. We prove that the resulting $\gamma$-invariants are invariant under concordance, and more generally under weak cobordism, and that they lift certain Milnor invariants of 3-component links. To establish this, we introduce an invariant $h(L)$, a $3$-component analogue of the Kojima–Yamasaki $\eta$-invariant, and show that it recovers the $\gamma$-invariants. As applications, we obtain a weak-cobordism classification when the distinguished component has trivial Alexander polynomial and characterize knots that bound continuously embedded disks in $B^4$ whose complements have fundamental group $\mathbb{Z}$.

Let $\{P_n\}_{n\ge 0}$ and $\{R_n\}_{n\ge 0}$ denote the Padovan and Perrin sequences, both satisfying the recurrence $U_{n+3} = U_{n+1} + U_n$, but with initial values $P_0 = P_1 = P_2 = 1$ and $R_0 = 3$, $R_1 = 0$, $R_2 = 2$, respectively. A Cullen number is a positive integer of the form $m\cdot 2^m + 1$ for some integer $m \ge 1$, while a Woodall number is a positive integer of the form $m\cdot 2^m - 1$ for some integer $m \ge 1$. In this paper, we determine all Woodall numbers in the Padovan sequence and all Cullen numbers in the Perrin sequence. Specifically, we prove that $1$ and $7$ are the only Woodall numbers in the Padovan sequence, and that $3$ is the only Cullen number in the Perrin sequence.

We determine sharp density thresholds for when discrete Pauli pairs with Gaussian decay must be classical Pauli pairs. More precisely, we identify when equality of the sampled moduli of two functions and their Fourier transforms forces equality of the global moduli in time and frequency, thereby answering a question of Ramos and Sousa. We also determine the sharp threshold for when discrete Pauli pairs must be weak Pauli pairs, where either the time-side or the frequency-side moduli agree identically. These can both be seen as phaseless versions of the results of Kulikov, Nazarov, and Sodin on Fourier uniqueness pairs.

In the torn paper channel, a transmitted codeword is broken at random locations into fragments that arrive at the decoder in an unordered manner. A central theoretical challenge within this model is global alignment – the task of determining each fragment's original position – in order to faithfully reconstruct the entire codeword. Prior work by Shomorony and Vahid introduced an interleaved-pilot scheme that successfully achieved a vanishing error probability. However, their alignment strategy relies heavily on global statistics, requiring fragments to exceed a minimum length and effectively discarding many shorter ones as erasures, which results in rates significantly below capacity. To address this gap, we propose an improved coding scheme that achieves a provable rate increase through a novel approach we call local alignment. This approach identifies global alignment bits within each fragment using only local information, allowing the decoder to determine the positions of fragments that are shorter than those used in previous work. Consequently, the decoder can extract information from a much larger fraction of the channel output than in previous work, yielding significantly higher rates. Furthermore, we extend our analysis to torn paper coding with lost pieces (TPC-LP), a generalized model that accounts for length-dependent fragment deletion. For a class of TPC-LP channels that delete all fragments below a logarithmic length threshold while allowing arbitrary length-dependent deletion probabilities for longer fragments, we show that the proposed local alignment strategy achieves an arbitrarily small additive gap to capacity as the threshold increases.

We construct an explicit isomorphism between the generalised Khovanov arc algebras of type D and the basic algebras of the anti-spherical Hecke category associated to the maximal parabolic subgroup $W (A_{n-1})$ of $W (Dn)$. This isomorphism maps generators to generators, thereby equipping the arc algebras with an Ext-quiver and relations presentation.

We consider the one-dimensional cubic nonlinear Schrödinger equation $$ \ii\partial_tu+\frac12\partial_{xx}u=\la|u|^2u,\,\lambda=\pm 1 $$ and solve the final-state (modified wave operator) problem for small asymptotic data. More precisely, given a small $W(\xi)$, we construct a solution $u$ such that \begin{equation*} u\rightarrow (2\pi)^{-1/2}(\ii t)^{-1/2}e^{\ii x^2/(2t)}\, W\!\Big(\frac{x}{t}\Big)\exp(-\ii\la|W(x/t)|^2\log t). \end{equation*} Crucially, we design a contraction map, so that we can run the analysis in the spirit of Kato–Pusateri \cite{KP} for $w$ with a forcing term depending only on the final data $W$. This scheme is easy to adapt to solving final state problems with a complete theory for the forward problems.

Standard neural network training relies on learning-rate schedules tied to a fixed horizon, leading to strong path dependence and costly re-tuning as data availability changes. Schedule-Free (SF) methods address this by removing explicit schedules, yet SF-AdamW, the current state-of-the-art anytime optimizer, consistently underperforms well-tuned AdamW baselines. We propose SF-NorMuon, a schedule-free spectral optimizer that closes this gap: with a single hyperparameter configuration, SF-NorMuon matches or exceeds tuned AdamW on 125M and 772M parameter language models across $1$–$8\times$ Chinchilla horizons, and trails horizon-aware, cosine-scheduled NorMuon by only $\sim 0.03$ nats, substantially narrowing the gap between anytime and scheduled spectral optimizers. On the theoretical side, we prove a stationarity guarantee for schedule-free spectral dynamics and identify weight decay at the fast iterate as essential for long-horizon stability. SF-NorMuon enables practitioners to obtain high-quality checkpoints at any point during training without committing to a horizon in advance. Fully matching horizon-aware spectral optimizers without a schedule remains an open problem, which we highlight as a direction for future work. By closing the performance gap with tuned AdamW baselines, SF-NorMuon makes horizon-free optimization more practical, taking a step towards truly open-ended, continual learning.

In this article, we develop an explicit categorical realization of sheafification based on colimits, products, and subobjects, emphasizing its behavior in algebraic and topological-algebraic settings. We prove that if $\mathcal{C}$ is a reflective subcategory of a category $\mathcal{A}$, then the presheaf category $\mathbf{Psh}(X,\mathcal{C})$ is reflective in $\mathbf{Psh}(X,\mathcal{A})$. We further investigate the interaction between reflections and sheafification, obtaining natural conditions under which these constructions are naturally isomorphic.

Let $p_1,...,p_L\in Z[x_1,...,x_d]$ be non-constant polynomials with zero constant term. The ergodic theoretical proofs of the polynomial and the IP-polynomial Szemeredi theorems as well as some of the ergodic-theoretical and combinatorial consequences of the Density Polynomial Hales-Jewett conjecture (DPHJ) naturally lead to the study of sets of large returns which are defined as $$ R_\epsilon^{p_1,...,p_L}(A):=\{n\in Z^d\,|\,\mu(A\cap T_1^{-p_1( n)}A\cap\cdots\cap T_L^{-p_L(n)}A)>\mu^{L+1}(A)-\epsilon\}, $$ where the $T_j$'s are commuting and invertible $\mu$-preserving transformations, $A$ is measurable, and $\epsilon>0$. We obtain new results dealing with the sets of the form $R_\epsilon^{p_1,...,p_L}(A)$. Among other things, we show that every set of the form $R_\epsilon^{p_1,...,p_L}(A)$ is syndetic if and only if $p_1,...,p_L$ are linearly independent, answering a question asked by Frantzikinakis-Kuca. Moreover, the linear independence of $p_1,...,p_L$ implies that every set of the form $R_\epsilon^{p_1,...,p_L}(A)$ has the A-IP$^*$ property (="almost" IP$^*$ property), which is stronger than syndeticity. The following is one of the new combinatorial results obtained in this paper. Suppose that $p_1,...,p_L$ are linearly independent. For any set $E\subseteq Z^D$ with upper Banach density $d^*(E)>0$, any non-zero $v_1,..., v_L\in Z^D$, and any $\epsilon>0$, the set $$ S_\epsilon^{p_1,...,p_L}(E):=\{ n\in Z^d\,|\,d^*(E\cap (E-p_1(n)v_1)\cap \cdots\cap (E-p_L(n)v_L))>(d^*(E))^{L+1}-\epsilon\} $$ is A-IP$^*$. Furthermore, we prove that when $D>L>1$, this result is sharp: the A-IP$^*$ property cannot be upgraded to IP$^*$. The techniques developed in this paper lead to some additional applications. For example, we show that an amplified form of the IP-polynomial Szemeredi theorem conjectured by Bergelson- McCutcheon follows from the DPHJ.

This paper focuses on the further development of the Lie bracket approximation approach for optimization and control via extremum seeking systems. Classical results in this area provide algorithms with exponential convergence rates for quadratic-like cost functions, and polynomial decay rates for cost functions of higher degrees. This paper proposes a novel design that ensures the motion of the extremum seeking system along directions associated with higher-order Lie brackets, thereby achieving exponential convergence for cost functions that are "flat-bottomed", i.e., polynomial-like but of degree greater than two and unlike literature assumptions, we do not require Hessian information or strictly non zero Hessian at the minimum. Numerical simulations are presented to demonstrate the effectiveness of the proposed designs and their exponential convergence on fourth-, sixth-, and even eighth-degree cost functions. We include a comparison that shows our design outperforming a Newton-based method.

Steinerberger introduced the Buffon discrepancy problem, asking how accurately a one-dimensional set of length $L$ in a convex body can match the Crofton-predicted line-intersection counts, and proved an $O\left(L^{1/3}\right)$ upper bound via a Steinhaus longimeter construction. Using the Aistleitner–Bilyk–Nikolov arbitrary-measure star-discrepancy theorem we demonstrate the existence of full-chord constructions with discrepancy $O\left((\log L)^{3/2}\right)$ for every fixed compact convex body with finite piecewise $C^2$ boundary. In the disk, we prove that every full-chord construction has discrepancy at least $\Omega\left(\log L\right)$, using Schmidt's two-dimensional rectangle discrepancy lower bound.

In 1976, Gallagher showed that, conditional on the Hardy–Littlewood conjectures, the number of primes below $x$ in a randomly chosen short interval of length $\lambda \log x$ asymptotically follows a Poisson distribution with mean $\lambda$. Correspondingly, the normalized gaps between consecutive primes follow an exponential distribution, provided that the scaling parameter $\lambda$ is fixed. We investigate the validity and limitations of the associated folklore Poisson Tail Conjecture as $\lambda$ is allowed to grow. For slowly growing $\lambda$, and conditional on a strong variant of the Hardy–Littlewood conjectures, we establish asymptotics showing that the local counting statistics agree with these predictions. Furthermore, we identify the scale of the phase transition and establish the breakdown of these distributions for larger $\lambda$. The proof relies on a novel combination of extremal interval sieve estimates and concentration inequalities from probability.

We study a concrete family of symmetric integral $Z$-matrices attached to weighted star trees. The arms are ordinary type-$A$ chains and the central diagonal entry is an arbitrary positive integer $k$ rather than being fixed to the Cartan value $2$. This gives a matrix-theoretic and graph-theoretic version of the so called Berger construction: it extends the simply laced affine Dynkin stars while remaining accessible through elementary linear algebra. For a star with arm lengths $r_1,\ldots,r_m$ we compute the determinant, the inertia, the positive-definite and affine regimes, and the primitive positive null vector in the affine case. The affine condition is exactly the unit-fraction equation \[ \sum_{i=1}^m \frac{1}{r_i+1}=m-k, \] so the classification of these affine weighted trees reduces to a finite Egyptian-fraction enumeration for each fixed pair $(m,k)$. The classical affine diagrams $D_4^{(1)}$, $E_6^{(1)}$, $E_7^{(1)}$ and $E_8^{(1)}$ appear as small subfamilies, while higher-arm cases give new integral positive-semidefinite star matrices with explicit Coxeter labels.

The $K$-homology groups of a $C^*$-algebra are receptacles for information from topology, operator algebra theory, and representation theory. For applications, one often wants to know if two $K$-homology classes are the same: the simplest way to deduce this is typically via the `primary' pairing between $K$-homology and the dual theory ($K$-theory). However, this pairing will typically miss some information: for example, it cannot detect torsion elements of $K$-homology. In this paper, we introduce a `secondary' pairing between subgroups of $K$-homology and $K$-theory that takes values in $\mathbb{Q}/\mathbb{Z}$. In good cases we show that this pairing will detect all the classes in $K$-homology that are missed by the primary pairing. We then relate our secondary pairing to the relative eta invariants of Atiyah-Patodi-Singer, and to the Thomsen exact sequence and zeta maps from $C^*$-algebra classification theory.

We provide a complete Sturm–Liouville spectral analysis of the Constant Elasticity of Variance (CEV) operator. By transforming the corresponding Fokker–Planck operator into a generalized Laguerre operator, we explicitly characterize its self-adjoint extensions, boundary conditions, spectra, and eigenfunctions across all elasticity regimes. We then relate these spectral features to arbitrage phenomena in the CEV model, showing how boundary behavior and positive harmonic functions encode the distinction between attainable-boundary arbitrage mechanisms and strict-local-martingale bubble regimes. The result is an explicit operator-theoretic perspective on the link between CEV dynamics, no-arbitrage, and spectral theory.

We derive explicit formulae for the subalgebra zeta functions of all higher Heisenberg Lie algebras over an arbitrary compact discrete valuation ring $\mathfrak{o}$. To this end, we develop Hecke-theoretic techniques for the enumeration, by two distinct invariants, of sublattices of an $\mathfrak{o}$-lattice of finite rank endowed with a non-degenerate symplectic form.

We generalize positivity results due to Popa-Wu and Popa-Schnell for Hodge modules and Higgs bundles on smooth projective varieties to the case of smooth proper DM stacks admitting projective coarse moduli spaces. This paper is the first in a series aiming to generalize results of Popa-Schnell and Wei-Wu on Viehweg hyperbolicity to the setting of DM stacks, and in particular, to certain KSBA moduli spaces.

We study $F$-graded systems of ideals in $R$, which are sequences of ideals giving rise to Cartier algebras on $R$. We identify how properties of these systems (or modifications of these systems) affect the singularity properties of the corresponding Cartier algebra. In particular, we show that in a Gorenstein and strongly $F$-regular local ring, strong $F$-regularity and $F$-splitting are the same for a special class of $F$-graded systems called $p$-families. Further, we make use of this and a new operation we introduce called $p$-stabilization to get a criterion that in a Gorenstein and strongly $F$-regular local ring, a system is strongly $F$-regular exactly when its $p$-stabilization is $F$-split. Finally, we associate a combinatorial object to systems built out of monomial ideals and show how this can help compute the $p$-stabilization.

Kuramoto's differential equation describes a synchronization process between several harmonic oscillators. It has been used to model biological phenomena such as the synchronization of heart cells, the circadian rhythm, or brain waves. It is also used in power system control. The simplest possible model assumes that all oscillators are identical and connected to each other with equal pairwise attraction. In this paper, we give a full geometric description of its global dynamics in terms of Morse theory and dynamical systems. Most of this description is stable in the sense that it is topologically preserved under small perturbations of the parameters.

In this paper, we show how to construct a special class of ruled hypersurfaces in the nonflat complex space forms $\mathbb{CP}^n$ and $\mathbb{C}H^n$. This is done by taking an arbitrary smooth curve in a totally geodesic (complex) one-dimensional submanifold and erecting an orthogonal ruling over each of its points. Concentrating on the $n=2$ case, we also examine the special situation in which the base curve has constant geodesic curvature. We show that, in this case, the construction yields precisely the real-analytic hypersurfaces of cohomogeneity one that satisfy a certain transversality condition.

Let $K$ be an algebraically closed field of characteristic zero, $A= K[x_1, \dots, x_n]$ the polynomial ring in $n$ variables, and let $W_n(K)$ be the Lie algebra of all $K$-derivations of $A.$ This Lie algebra also is the free $A$-module of rank $n$ over the ring $A,$ so every subalgebra of $W_n(K)$ has a rank $\leq n$ over $A.$ We prove that every maximal subalgebra of rank $\leq n$ of $W_n(K)$ is a simple Lie algebra. If a maximal subalgebra $L\subset W_n(K)$ has rank $n$ and is a submodule of $W_n(K)$ then $L$ is not simple. Moreover, $L$ is of the form $L=\{ D\in W_n(K) \ | \ D(I)\subseteq I\} $ for some ideal $I$ of the ring $A.$ It is also proved that, for a simple derivation $D$ on the ring $K[x, y]$, the subalgebra $K[x, y]D$ is a maximal subalgebra of $W_2(K).$

We determine which quasi-simple groups have a non-principal $2$-block that is stable under complex conjugation. As a corollary, we determine that the Mathieu group $M_{22}$ is the only simple group not possessing a nontrivial irreducible Brauer character of quadratic type, answering a recent question of Gow and Murray.