We study a dynamic assortment problem on a two-sided service platform with incomplete information and heterogeneous customers in a discrete-time setting. In each period, a customer arrives seeking service, and the platform chooses an assortment of sellers to display. The customer then proposes a transaction to at most one seller in the assortment according to a multinomial logit choice model. After a fixed number of periods, sellers review the proposals they have received and each chooses at most one customer according to another multinomial logit choice model, after which the cycle repeats. A key challenge is that the platform does not know the choice-model parameters of either customers or sellers in advance. To our knowledge, this is the first study of a dynamic assortment problem in which both sides' choice parameters are unknown. We develop a data-driven algorithm that learns these parameters while optimizing the platform's objective over time. We evaluate performance using regret, which measures revenue loss relative to a clairvoyant benchmark that knows all parameters and customer arrivals in advance. We show that the algorithm's worst-case regret grows polylogarithmically over time, and we derive a matching lower bound, establishing its rate optimality.

The Graph Burning Problem (GBP) is a NP-Hard combinatorial optimization problem that models the propagation of influence or contagion in a network. The propagation is represented through the metaphor of a fire spreading through the vertices of a graph. A burning process takes place in a series of discrete time-steps. At each time step, the burning process is characterized by a propagation (where burned nodes spread the fire to their neighbors), and an ignition (where one additional unburned node is chosen to become burned). The minimum number of steps required to burn all vertices of a graph defines its burning number. Literature provides integer linear programming formulations to solve the problem, but with no surprise, such approaches struggle to converge as the graph size increases. Therefore, reducing the search space explored by these formulations becomes a key point to improve performances. In this work, we study the similarities of the Graph Burning Problem with the well-known Dominating Set Problem. We propose a new formulation based on this study and apply dominance rules and symmetry-breaking techniques to reduce the search space and consequently speed up resolution time. We also introduce a perturbation of the proposed objective function, as well as a pruning rule for the perturbed model in order to further accelerate its resolution.

Let $\Sigma$ be a closed embedded minimal hypersurface in $\mathbb{S}^{n+1}$. We establish an improved lower bound for the first non-zero eigenvalue of the induced Laplace-Beltrami operator on $\Sigma$. It is slightly better than the bound of Duncan-Sire-Spruck.

In this paper, we develop a quiver approach to coquasi-Hopf algebras with the dual Chevalley property. We introduce a modified generalized path coalgebra $\Bbbk(\mathrm{Q},\mathcal{S})$ associated with a given quiver $\mathrm{Q}$ and a collection of simple coalgebras $\mathcal{S}=\{C_i\mid i\in \mathrm{Q}_0\}$ indexed by the vertices of $\mathrm{Q}$, such that its link quiver coincides with $\mathrm{Q}$. We prove that such a coalgebra admits a graded coquasi-Hopf algebra structure with the dual Chevalley property if and only if $\mathrm{Q}$ is a generalized Hopf quiver and $\bigoplus_{i\in \mathrm{Q}_0}C_i$ forms a cosemisimple coquasi-Hopf algebra. Moreover, we provide a classification of these coquasi-Hopf algebra structures. We then study the link-indecomposable components of a coquasi-Hopf algebra with the dual Chevalley property, and give the generalized dual Gabriel's theorem for such coquasi-Hopf algebras. As an application, we apply the quiver method to classify finite integral tensor categories with the Chevalley property of finite representation type. We also give structural characterizations of coradically graded coquasi-Hopf algebras of tame corepresentation type. Furthermore, we investigate finite braided integral tensor categories with the Chevalley property via the quiver approach.

Let $G$ be a finite group and $p$ a prime. It is interesting to determine when two ordinary irreducible representations of $G$ have the same $p$-modular reduction; this is the same as saying that the corresponding rows of the decomposition matrix are equal, or that the characters of the two representations agree on $p$-regular conjugacy classes. In fact we consider the more general problem of asking when two rows of the decomposition matrix are proportional. In the case where $G$ is a double cover of the alternating or symmetric group, this problem has been solved except when $p=3$. Here we resolve the missing case for spin characters (i.e. characters which are not lifted from the covered group), which completely solves the problem for the double cover of the symmetric group. There are surprising parallels to our solution to the corresponding problem for $p=2$.

We classify when a spin and a non-spin irreducible character of the alternating group have proportional 2-modular reductions. Equivalently, we classify when such a pair of characters are proportional on elements of odd order.

Building on preliminary results about lower sections of real-valued functions on a metric space, we provide a constructive counterpart of Ekeland's theorem on the approximate optimisation of real-valued functions.

Covariance estimation yields a fundamental second-order statistic underlying representation learning, dimension reduction, and dependence modeling. While covariance has been well understood in Euclidean spaces, it is ill-defined for random objects residing on nonlinear Riemannian manifolds, which increasingly arise in modern machine learning applications involving shapes, symmetric positive definite (SPD) matrices, etc. This paper introduces an intrinsic Riemannian cross-covariance for manifold-valued random objects. Our approach defines covariance and correlation by transporting local variations to a common tangent space via parallel transport, yielding a second-order descriptor that is independent of arbitrary coordinate choices. We establish that the proposed covariance inherits desirable properties of its Euclidean counterparts and characterize its asymptotic behavior. Numerical studies on spheres and SPD manifolds, together with real-data experiments on heart valve shapes in Kendall's shape space, demonstrate the effectiveness of our estimators and verify the stated properties. Our results position the Riemannian covariance as a fundamental tool for second-order learning and analysis in non-Euclidean representation spaces.

This paper studies the complexity of a dual dynamic programming (DDP) method for solving a class of convex optimization problems with linear coupling constraints. Existing complexity results based on DDP depend on the dimensions of the state vectors and, in particular, grow exponentially with dimension. The goal of this paper is to establish a complexity bound that is independent of the dimension. Our approach first studies an unconstrained strongly convex problem and develops a flexible framework for DDP, called FDDP, for solving the associated dynamic programming equations, and establishes an iteration-complexity bound for it that is independent of the dimension. A dimension-independent complexity bound for the original linearly constrained problem is then obtained by applying FDDP to a corresponding unconstrained strongly convex problem obtained via smoothing and penalization. Inspired by the literature on bundle methods, FDDP updates lower approximations of cost-to-go functions through a generic procedure that includes both the classical multi-cut DDP and a new two-cut DDP variant as special cases. The two-cut variant maintains only two affine cuts per stage at each iteration, making it more memory-efficient while retaining the same theoretical guarantees. Finally, numerical experiments illustrate the practical behavior of multi-cut and two-cut DDP, including their dependence on problem parameters and their performance relative to a direct quadratic constraint reformulation.

The category of noncommutative quadratic quadric hypersurfaces, ${\tt Quad}\text{-}{\tt QHS}$, consists of pairs $(A, f)$, where $A$ is a quadratic algebra and $f \in A$ is a nonzero degree $2$ element. We associate to such $(A, f)$ a pair $(\bar{A}^!, f^!)$, and show that this association makes ${\tt Quad}\text{-}{\tt QHS}$ into a category with duality. We construct a faithful functor from the category of graded modules over $\bar{A}^!$ to the homotopy category of curved DG modules over a canonical curved DG algebra $(A \otimes \bar{A}^!, d, f \otimes f^!)$. If $A$ satisfies the left strong rank condition and $f \in A$ is not a right zero divisor, we show that the restriction of our functor to a natural full subcategory of the category of graded modules over $\bar{A}^!$ is valued in a stable category of noncommutative matrix factorizations of $f$. When $A$ is Koszul of finite global dimension and $f \in A$ is normal and regular, we prove that the even Clifford algebra, $\bar{A}^![(f^!)^{-1}]_0$, is isomorphic to a canonical PBW-deformation of a Zhang twist of the $2$-Veronese subalgebra of the Koszul dual $A^!$. Finally, we study several classes of Artin-Schelter regular algebras to illustrate our results.

Thurston gave a simple way to construct all triangulations of the sphere for which 5 or 6 triangles meet at each vertex, using the Eisenstein integers $\mathbb{E}$. While such triangulations can be defined purely combinatorially, Thurston noticed that given such a triangulation, one can make all the triangles into flat equilateral triangles with the same edge length, and this gives the 2-sphere a flat Riemannian metric except at 12 cone points with angle deficit $\pi/3$. He showed that up to rescaling, all such Riemannian metrics arise from his procedure. He studied the moduli space $\mathcal{M}$ of all such metrics modulo rescaling, and showed that $\mathcal{M}$ is open and dense in an orbifold $\overline{\mathcal{M}} = \mathbb{PC}^{10}_+/\Gamma$. Here $\mathbb{C}^{10}_+ = \{ v \in \mathbb{C}^{10} \vert \; Q(v) > 0\}$ for some quadratic form $Q$ of signature $(1,9)$ on $\mathbb{C}^{10}$, $\mathbb{PC}^{10}_+$ is its projectivization, and $\Gamma$ is a certain discrete group of linear transformations of $\mathbb{C}^{10}$ preserving both $Q$ and the lattice $\mathbb{E}^{10} \subset \mathbb{C}^{10}$. He also showed that $\overline{\mathcal{M}}$ is the moduli space of flat Riemannian metrics on the sphere with at most $12$ cone points and angle deficits that are positive integer multiples of $\pi/3$. Here we briefly outline the basic ideas behind this work, and illustrate them with examples.

We show that for a generic real or complex-valued compactly supported potential, the corresponding Schroedinger operator achieves maximal resonance density, in the sense that its integrated resonance counting function achieves the optimal asymptotic upper bound. For odd dimensions this follows from results of Dinh-Vu once we adapt an argument of Christiansen Hislop. The proof for even dimensions constitutes the bulk of the paper, and we prove several new results on resonances which have analogues in the odd dimensional case. This includes a sharp upper bound on the integrated resonance counting function for any compactly support potential, a proof that the characteristic function of a ball has resonance counting function which achieves the optimal upper bound, and an even-dimensional analogue of the result of Dinh-Vu on asymptotics of the resonance counting functions for complements of pluripolar subsets of analytic families of potentials. We use the characterization of resonances as zeros of certain Fredholm determinant functions related to the scattering matrix, allowing us to apply techniques and results from the theories of one and several complex variables. Our proof that the characteristic function of a ball has counting function achieving the optimal upper bound uses the uniform asymptotics of Bessel functions and follows ideas of Zworski, Christiansen-Hislop, and Dinh-Vu.

Ballantine, Beck, Merca, and Sagan conjectured four identities, collectively Conjecture 19, relating the image of the map pre_k on integer partitions to four OEIS sequences. We prove parts (i) and (iii) unconditionally, prove part (iv) unconditionally using the injectivity of pre_2 on partitions of n (Conjecture 1 of the same paper, proved by Li in arXiv:2508.00971 ), and show that this injectivity is in fact equivalent to part (iv). For part (ii) we prove the partition-theoretic half unconditionally and reduce the remaining content to a 2006 conjecture of Dean Hickerson on the OEIS concerning Huffman coding. We also correct a sign error in the published statement of part (iii): the correct identity is chi(ImP_3(n)) = A213213(n) - 1, not 1 + A213213(n) as stated.

An entire slice regular function $f:\mathbb H\to\mathbb H$ can omit four prescribed quaternionic values only in the affine-dependent case. More precisely, four affinely independent omitted values force $f$ to be constant, while the converse follows from the plane-omission theorem of Bisi--Winkelmann. The proof passes to the real-symmetric stem function. For each omitted value a quadratic zero-divisor criterion gives a zero-free entire function $Q_j$, and the component normal to the affine span is governed by a square-discriminant identity. Finite-order data are excluded by Hadamard factorization and a rigidity argument on the real axis. In the general case, logarithmic Bloch--Ochiai places the $Q$-curve in a translated algebraic torus. The Laurent-square case reduces to the finite-order contradiction, and the nonsquare case is excluded by an even-ramification argument together with the level-one truncated Second Main Theorem of Noguchi--Winkelmann--Yamanoi.

Let $f(n,k)$ be the minimum size of a collection of bicliques such that (i) every edge of the complete graph $K_n$ is covered by at least one and at most $k$ bicliques in the collection, and (ii) for each edge $\{u,v\}$, the number of bicliques in which $u$ appears in the first class and $v$ in the second class differs by at most one from the number of bicliques in which $u$ appears in the second class and $v$ in the first class. For $k=1$, $f(n,k)$ reduces to the biclique partition number of $K_n$, and the Graham-Pollak theorem gives $f(n,1)=n-1$. For $k=2$, $f(n,k)$ is the ordered biclique partition number of $K_n$, for which it is known that $c_1 n^{1/2} \le f(n,2) \le c_2 n^{1/2+o(1)}$ for some positive constants $c_1$ and $c_2$. In this note, we give almost tight bounds for $f(n,k)$ for fixed $k \ge 2$: \[ (1+o(1))c_1(k)\cdot n^{\frac{1}{\lceil k/2\rceil+1}} \le f(n,k) \le (1+o(1))c_2(k)\cdot n^{\frac{1}{\lfloor k/2\rfloor+1}+o(1)}, \] where $c_1(k)$ and $c_2(k)$ are positive constants.

Reduced-precision arithmetic offers significant opportunities to improve performance, memory usage, and energy efficiency in numerical applications, provided that numerical accuracy is preserved. This work investigates automated precision tuning through customized floating-point formats with user-defined exponent and significand sizes, enabling the emulation of emerging low-precision formats and the exploration of non-standard precision configurations within a unified mixed-precision framework. The proposed methodology, implemented in the PROMISE precision autotuning tool, combines numerical validation with a systematic search to generate program variants that satisfy user-defined accuracy requirements. To address the computational cost of this exploration, a containerized benchmarking framework supports parallel execution across multiple algorithms and parameter configurations. The approach is evaluated on a suite of numerical programs, including linear solvers and applications from the Rodinia benchmark. Results show that a substantial proportion of variables can be safely reduced to lower precision while preserving accuracy, indicating that standard double precision is often over-provisioned. These findings highlight the potential of automated precision tuning to derive efficient mixed-precision configurations tailored to application-specific accuracy requirements.

The 1970 spectral extension of Mantel's theorem, proved by Nosal, states that every graph with $m$ edges and spectral radius $\rho_1>\sqrt{m}$ contains at least one triangle. Its quantitative refinement by Ning and Zhai later established that any graph $G$ with $m$ edges and spectral radius $\rho_1\geq\sqrt{m}$ contains at least $\lfloor\frac{\sqrt{m}-1}{2}\rfloor$ triangles, unless $G$ is a complete bipartite graph. In this paper, we further investigate the minimum number of triangles guaranteed under the strengthened spectral condition $\rho_1\geq\sqrt{m}+c$, where $c$ is a positive constant. We prove that for any constant $c\in (0,\frac{1}{2}]$ and all sufficiently large $m$, if $s=s(m)$ is a real-valued function satisfying $\lim_{m\to\infty} \frac{s}{m}=c$, then every $m$-edge graph $G$ with spectral radius $\rho_1$ satisfying $\rho_1^2\geq m-1+\frac{2s}{\rho_1-1}$ contains at least $s$ triangles. Moreover, we characterize the extremal graph achieving the minimal number of triangles. In particular, when $s=\frac{m-1}2$, our result settles a conjecture proposed by Li, Feng, and Peng.

We show that every derivation of a simple and rational vertex operator algebra of CFT type is an inner derivation.

We introduce the calibration energy for oriented immersions into Euclidean space, quantifying the deviation from calibrated geometry. A key property is that this energy may remain finite for infinite-volume immersions, while a null-Lagrangian structure ensures that it has the same first variation as the volume functional. We establish an exact dissipation identity for the calibration energy along oriented, proper mean curvature flows in arbitrary dimensions and codimensions, under a mild local-volume bound. This provides a new, finite variational framework for mean curvature flow beyond the finite-volume setting. Our result yields several applications, including rigidity for solitons and convergence for two-dimensional immortal solutions. In particular, every proper self-expander with finite constant-coefficient calibration energy must be a plane in all dimensions and codimensions.

We use recent results, among which modular inequalities for curves, to determine the Alexander polynomials for some classes of pencil-type conic-line arrangements. For these classes of curves we prove that the Alexander polynomial is (at least partially) combinatorial. To this end, we exemplify new techniques that are suitable for broader use, lending themselves to more general classes of curves.

We study a class of Sturm--Liouville problems of the form $$ -(p\,y')' + q\,y = \lambda\, p\, y, $$ on a finite interval with complex-valued coefficients, where $p$ is piecewise smooth and $q$ is bounded. We prove that all eigenvalues in the open left half-plane are contained in a bounded set, which implies that only finitely many eigenvalues lie in this region. A canonical instance of this class arises in transverse-magnetic (TM) diffraction by metallic lamellar gratings, a benchmark problem in computational photonics that has been central to the development of Fourier modal methods. These methods exhibit long-standing convergence difficulties in this setting, associated with the loss of definiteness of the underlying operator and the emergence of spurious modes. Our result yields a rigorous criterion for identifying such non-physical modes in low-loss metallic gratings. Numerical examples illustrate the practical utility of the criterion.

In this paper, we study the class \Sigma_{H}^{k}(p) of sense-preserving univalent harmonic mappings in the unit disk \mathbb{D} that possess a simple pole at p\in[0,1) and admit a k-quasiconformal extension to the extended complex plane for k\in[0,1). In 2024, Bhowmik and Satpati established an area theorem and derived a sufficient condition for the k-quasiconformal extension of harmonic mappings belonging to \Sigma_{H}^{k}(p) without logarithmic terms. Motivated by their work, we investigate the corresponding problem when a logarithmic singularity is present. Our main contributions are two-fold: we first prove a generalized area theorem for all mappings in \Sigma_{H}^{k}(p); we then obtain a sufficient condition for sense-preserving univalent harmonic mappings in \mathbb{D} to admit explicit k-quasiconformal extensions. These results extend the aforementioned work to the setting where logarithmic singularities are allowed.

In this paper, we study the noncommutative fractional symmetric Sobolev spaces on noncommutative torus. We prove noncommutative distributional fractional Sobolev inequality and as its application, we obtain Sobolev embeddings. In order to obtain these results, we first prove a noncommutative version of the famous O'Neil inequality for the convolution. As a first application of our main results, we obtain a Cwikel-Solomyak-type estimate. As an another application, we show a $L_2$-time decay for the mild solution of the Cauchy problem for the diffusion equation in this noncommutative setting. When $\theta=0,$ our results recover many known results on Sobolev embedding on the torus.

While the uncrowding algorithm on set-valued tableaux has long been instrumental in proving the Schur positivity of stable symmetric Grothendieck polynomials, lattice models have emerged as a modern framework for investigating symmetric functions, in particular symmetric Grothendieck polynomials. In this work, we synthesize these combinatorial and lattice-theoretic approaches by defining both the Robinson--Schensted--Knuth (RSK) correspondence and the uncrowding operation directly on a 5-vertex model of Motegi and Sakai and its subsequent reinterpretation by Buciumas, Scrimshaw, and Weber. Our lattice-based RSK formulation yields a powerful new result: the direct construction of the associated crystal structure on the states of the 5-vertex model.

Let $\mathbb Z_n$ be the cyclic group equipped with the uniform probability measure $\pi$, and let $A_{\psi_n}$ be the Laplacian with word length \[ \psi_n(k) = \min(k,n-k). \] We prove the sharp log-Sobolev inequality \[ \text{Ent}_{\pi}(f^2) \le 2\pi(f A_{\psi_n} f), \qquad f:\mathbb Z_n \to [0,\infty), \] for every $n \ge 4$. The proof is inspired by the recent work of Frank and Ivanisvili~\cite{FrankIvanisvili2026} on a sharp log-Sobolev inequality for nearest-neighbor simple random walk. We use their cubic-majorant reduction, which turns the problem into a 3rd moment estimate; the new point is a blockwise 3rd moment estimate adapted to the word-length multiplier. The same 3rd moment argument also recovers the log-Sobolev inequality for Poisson-semigroup on the circle, first proved by Weissler~\cite{Weissler1980}. The same sharp inequalities were also obtained recently by Yao~\cite{Yao2026} by a different method.

We reinvestigate the theory of spectral sequences by studying the $(\infty,1)$-category of filtered spectra through the cofiber-of-$\tau$ formalism of Burklund-Isaksen-Pstragowski-Wang-Xu. In this framework, we define and analyze hidden extensions along arbitrary maps of filtered spectra, establishing computational principles that extend the generalized Leibniz rule and the generalized Mahowald trick of Lin-Wang-Xu, as well as Burklund's Leibniz rule for total differentials, from the Adams spectral sequence to this broader setup. Our formulation uses a more refined, layered notion of extension, which slightly sharpens these statements even for the Adams spectral sequence. As an application, we study equivariant slice spectral sequences and obtain new families of "exotic transfer" differentials in the $C_4$-slice spectral sequences for the Hill-Hopkins-Ravenel theories $\mathrm{BP}^{((C_4))}\langle m\rangle$ for every $m \ge 1$.

We study local pure coordination games on finite social networks, continuing the framework of Hutchcroft, Rospuskova, and Tamuz. They showed that low inefficiency in local coordination forces the underlying graph to be amenable, with a square-root loss in the amenability parameter. We improve this loss in the binary unbiased setting. Using Shapley values of a mutual-information game associated with the players' local outputs, we prove that if the average disagreement is at most $\varepsilon$, then the graph is $(O(\varepsilon\log(1/\varepsilon)),r)$-amenable. This gives a sharper quantitative converse between local coordination and graph amenability.

We consider the periodic quintic nonlinear Schrödinger and prove small-mass global well-posedness in $H^s(\mathbb{T})$ for $s>0$. The proof relies on a new derivative-loss-free $L^6_{t,x}$ Strichartz estimate which is established using the high-low method, an asymmetric superlevel set estimate and a new refined broad-narrow argument. Although our $L^6_{t,x}$ Strichartz estimate is not sharp, being valid on slightly shorter time scales than the optimal logarithmic scale, combining it with the $I$-method enables the extension of local solutions to arbitrary times.

We present a constructive version of the Brøndsted-Rockafellar theorem in general geodesic metric spaces. Applications include a constructive form of the Caristi theorem and quantitative relations between metric slope error bounds and the global growth of functionals.

Fractional Gaussian noise and $\FARIMA(0,d,0)$ have the same long-memory pole $|\theta|^{-2d}$ and hence the same leading PACF law $\alpha(n)\sim d/n$. We show that this agreement breaks at the first non-universal order. For $0<d<1/2$, the pure fGn PACF satisfies $$ \alpha_{\fGn}(n)=\frac d n+\frac{C_{\fGn}(d)}{n^2}+o(n^{-2}), \qquad C_{\fGn}(d)<d^2, $$ The proof uses the Bingham--Inoue--Kasahara representation, a phase-coefficient expansion for fGn, and a Hankel-operator perturbation argument. Thus the fGn spectral envelope is invisible at first order but visible in second-order finite prediction, explaining why short-memory order selection can differ when fGn data are fitted by FARIMA-type models.