Papers

Han and Xiong recently extended the Gaussian binomial coefficient $\genfrac{[}{]}{0pt}{}{r+k}{k}_{q}$ to positive rational $r$ and conjectured that its integer trace, the integer-exponent part of the resulting power series, is coefficientwise largest at $r=1/2$. We prove a support-dominance theorem comparing rational parameters under an explicit divisibility condition. It settles the conjecture for every $r\geq 1/2$ and reduces the full conjecture to the unit fractions $r=\frac{1}{2m}$, only finitely many of which are nontrivial for each fixed $k$. A computer computation then verifies the conjecture for every positive rational $r$ and every $k\leq 200$. The theoretical results were autonomously produced and verified in Lean by AxiomProver.

We prove a uniform stratified generic vanishing theorem for perverse sheaves on $\mathbb{G}_m^d$ over finite fields, with constants depending only on the dimension and on the complexity à la Sawin. As a corollary, we prove an equidistribution theorem for $\mathbb{G}_m^d$ extending Katz's theorem for $\mathbb{G}_m$. The corollary applies to sequences of perverse sheaves of bounded complexity with common tannakian monodromy group, over finite fields of varying characteristic. As the cardinalities of the fields tend to infinity, the Frobenius conjugacy classes attached to the multiplicative characters equidistribute in the space of conjugacy classes of a maximal compact subgroup.

We observe that the motivic analogue of Suslin reciprocity (and similar degree-zero statements) is a formal consequence of localization (plus purity/some six functor formalism). In particular, the statement of Suslin reciprocity for smooth schemes over fields due to Kriz is a corollary of localization for higher Chow groups, over any base; we write down the framework yielding such relations with coefficients for schemes smooth over any base in $\A^1$-invariant motivic cohomology. As an application, we refine some relations between cup products of modular units to be integral in coefficients and in the base: first, we imitate the (rational-coefficients, complex-analytic) Busuioc–Park–Patashnick–Stevens argument for full-level-$N$ elliptic schemes, extending the result to integral bases and coefficients using the elementary reciprocity statement. We then refine the construction and resulting relations to the setting of motivic sheaves; in particular, this gives analogous relations at non-full level structure, as well as over any smooth global quotient stack.

In this paper we study a quadratic dynamical system that is a perturbation of a system of Markov processes. We give necessary and sufficient conditions for such a perturbation to be stochastic, show the existence of fixed points, and present a few examples and possible applications. Finally, we derive explicit contraction conditions that guarantee uniqueness of the fixed point and geometric convergence to it from every initial state.

We develop a new phase-field (diffuse interface) approximation of multiphase curvature motion with triple junction drag - an important sharp interface model for the evolution of microstructure in polycrystalline materials during heat treatment. This sharp interface model arises as gradient flow for the total length of the interfacial network with respect to a certain metric. Accordingly, we derive our diffuse interface approximation - a coupled system of partial differential equations that is a variant of the Allen-Cahn system - from a variational perspective, in the style of minimizing movements: starting from a discrete in time approximation that entails a convex optimization problem to advance from one time step to the next. In the process, we propose a simple integral expression that counts the number of junctions using the order parameter that appears to be new even for the standard multiphase Allen-Cahn system. The convergence of the resulting flow to the desired sharp interface limit is then verified via the method of matched asymptotic expansions. Numerical convergence studies against both known exact solutions as well as highly accurate benchmark solutions obtained via front tracking provide clear further evidence for this convergence. Moreover, the method retains the most desirable feature of diffuse interface methods: Automatic handling of topological changes in the network of interfaces.

The question of why males invest more into competition than offspring care is an age-old problem in evolutionary biology. On one hand, paternal care could increase the fraction of offspring surviving to maturity. On the other hand, competition could increase the likelihood of more paternities and thus the relative number of offspring produced. While drivers of these behaviours are often intertwined with a wide range of other constraints, here we present a simple dynamic model to investigate the benefits of these two alternative fitness-enhancing pathways. Using this framework we evaluate the sensitivity of equilibrium dynamics to changes in payoffs for male allocation to mating versus parenting. Even with strong effects of care on offspring survivorship, small competitive benefits can outweigh benefits from care. We consider an application of the model that includes men's competition for hunting reputations where big game supplies a benefit to all, and find a frequency-dependent parameter region within which, depending on initial population proportions, either strategy may outperform the other. Results demonstrate that allocation to competition gives males greater fitness than offspring care for a range of circumstances that are dependent on life-history parameters and, for the large-game hunting application, frequency dependent. The greater the collective benefit, the more individuals can be selected to supply it.

A missing integer in a partition $\lambda$ is a positive integer less than the largest part of $\lambda$ that does not appear as a part in $\lambda$. In the recent paper On the number of missing integers in partitions, Bhoria, Eyyunni, and Santra examined the number of partitions (overpartitions) of $n$ with a fixed number of missing integers and established generating functions, identities, and congruences for them. In this article, we provide combinatorial proofs for several of their results.

We present a robust physical-space approach to establish time decay and global asymptotics of solutions to variable-coefficient Schrödinger equations in $(3+1)$-dimensions. As an immediate nonlinear application, we obtain new small data global existence and asymptotics results for quasilinear Schrödinger equations with cubic, Hamiltonian nonlinearity, variable coefficients in their linear part, and possibly outside obstacles, even in the presence of trapped bicharacteristics (provided that they are suitably unstable). Our approach relies on three primary ingredients. First is the concept of a good commutator, which extends Klainerman's classical commuting vector field method, and develops upon earlier approaches of Cuccagna–Georgiev–Visciglia and Rodnianski–Tao in the variable-coefficient case, and of Ifrim–Tataru, Ifrim–Koch–Tataru in the nonlinear case. Second, we apply Ifrim and Tataru's testing-by-wave-packets method, which is key for obtaining global asymptotics and handling sharp decay assumptions on the coefficients. Finally, we introduce a systematic technique for analyzing the control provided by the good commutator, termed two-scale elliptic analysis. Together, these techniques significantly broaden the applicability of physical-space methods to a wider class of variable-coefficient nonlinear problems.

We study the approximation of the Fourier transform of a function from finitely many samples. Departing from the equispaced setting, we sample the function at deterministic non-equispaced nodes obtained by transforming equispaced points on $[0,1]$ through the inverse cumulative distribution function of a probability density. This change of variables compactifies the real line, so that no truncation of the space domain is necessary. Combining an exact aliasing identity for the midpoint rule with stationary-phase estimates for the transformed oscillatory integrals, we derive deterministic error bounds for every frequency and in $L_p$. For functions with polynomial decay in space and frequency, an explicitly optimized density recovers the equispaced convergence rate, while an additional variance parameter substantially reduces the pre-asymptotic error constants. For (sub-)exponentially decaying functions a polynomially decaying density yields (sub-)exponential rates. Numerical experiments confirm the theory and demonstrate the reduced pre-asymptotic error compared with optimally scaled equispaced sampling.

In a previous paper we defined real Heegaard Floer homology, an invariant of three-manifolds equipped with involutions. Here we prove that real Heegaard Floer homology is natural, and that it admits an action of the equivariant mapping class group. We also establish naturality of real link Floer homology and real sutured Floer homology, and give a definition of involutive real Heegaard Floer homology.

We study convex approximations of mixed-integer recourse functions in two-stage stochastic programming. For first-stage decisions, the relevant approximation error is the signed expected residual between the convex approximation and the integer-recourse value function. On bounded uncertainty boxes, we identify residual conditions that yield deterministic bounds for this expectation error. The central condition is coordinate slice-centering, under which integration by parts gives an anisotropic total-variation bound and a mixed-derivative bound whose all-coordinate case is expressed through Vitali variation of the density. Exact centering, however, may be incompatible with convexity, even for totally unimodular ceiling recourse. We therefore use commuting coordinate projectors to decompose an arbitrary residual into a centered component and an explicit slice-mean defect. The resulting defect-adjusted bounds provide selectable error certificates and uniform first-stage decision-quality guarantees. We further establish sharp constants, extend the mixed-derivative estimate to nonsmooth densities, and develop tensor-product and geometric extensions of the framework, alongside a max-affine convex fitting and audit methodology.

The spectrum of the Hodge Laplacian on differential manifolds encodes rich topological and geometric information and thus provides a powerful tool for analyzing data on manifolds. However, the classical unweighted formulation is restricted in its ability to study data with varying local features. To address this limitation, we propose a weighted Hodge Laplacian framework for manifolds with boundary, both in theory and in computation, by incorporating a weight function on the manifold. Under appropriate boundary conditions, we formulate the corresponding weighted de Rham-Hodge theory, in which the kernel of the weighted Hodge Laplacian coincides with the weighted harmonic space, and remains isomorphic to the de Rham cohomology of the underlying manifold. The harmonic spectrum of the weighted Hodge Laplacian captures the global topological information, while its non-harmonic spectrum encodes the local geometric property induced by the weight. The proposed framework therefore enables the study of topological and geometric features of data on manifolds across varying weights, and in addition, allows local structure to be highlighted by choosing weights that emphasize regions of interest. We demonstrate the effectiveness of the proposed method through proof-of-principle experiments in protein flexibility analysis, and the results show its promise.

Density deconvolution is an ill-posed inverse problem because recovering the latent density amplifies high-frequency variation in the observed data. We propose a two-stage likelihood procedure. The first estimator maximizes the convolution likelihood over a regularized Gaussian-mixture sieve; the second treats this fit as an empirical prior and averages the fitted conditional latent densities. Under fixed-model misspecification, the estimators generally have different population targets. For bounded local multiplicative perturbations of the latent density, the population posterior map is locally contractive in chi-square divergence under identifiable convolution. Under correct finite-dimensional specification, however, reconstruction adds a nonnegative first-order variance component. For growing sieves, we separate observed-domain estimation, inverse stability, and posterior empirical variation. We derive explicit rates under Gaussian error and recover the classical ordinary-smooth exponent, up to logarithmic factors, with a constructive verification for Laplace error. Simulations and a Framingham blood-pressure application provide empirical performance evidences.

We quantify viscous energy loss for two-dimensional Delort vortex sheets. Let $u^\nu$ be Leray-Hopf solutions on $\mathbb{T}^2$ with uniformly bounded kinetic energy and total vorticity variation, and write $\omega_0^\nu=\mu_0^\nu+f_0^\nu$, where $\mu_0^\nu\geq0$ and $f_0^\nu$ is bounded in $L^p$, $p>1$. For every fixed $0<\delta<T$, $\nu\int_\delta^T\|\omega^\nu(t)\|_2^2\,\mathrm{d}t\lesssim_{\delta,T}\frac{1}{|\log\nu|}$. This improves the $O(|\log\nu|^{-1/2})$ bound of De Rosa and Marcotullio under the same assumptions and thereby disproves their Conjecture 1.6 (arXiv:2602.15670, v1). The proof uses a sharp $L^2$-$H^1$-$H^{-1}$ interpolation inequality for nonnegative densities, retaining the total interaction energy of the positive vorticity rather than only its largest local mass. The bound also remains effective when the observation time grows with the Reynolds number. If the initial velocities are relatively compact in $L^2$, the loss still vanishes whenever $\log T_\nu=o(|\log\nu|)$; in particular, any prescribed energy loss must wait at least until $\nu^{-a}$ for some $a>0$. Previous estimates covered only $T_\nu=o(\exp(|\log\nu|^\kappa))$, $\kappa<1/2$, so this gives a polynomial lower bound on the energetic lifetime of the inviscid vortex-sheet model. If instead $f_0^\nu$ is bounded in $L(\log L)^\alpha$, the rate is $O(|\log\nu|^{-q_\alpha})$, $q_\alpha=\min\{2\alpha,1\}$, and the loss vanishes when $\log T_\nu=o(|\log\nu|^{q_\alpha})$. On $\mathbb{R}^2$, exact radial solutions attain these exponents for $0<\alpha\leq1/2$. At the endpoint, a bounded-energy $L^p$ family attains the rate $1/|\log\nu|$, while every fixed radial datum dissipates $o(1/|\log\nu|)$ and can lose a fixed amount of energy only on the diffusive scale $1/\nu$.

Self-consistent Bogoliubov-de Gennes calculations have long shown that solving the Poisson equation inside a superconducting vortex turns a one-signed charge depletion into a modulation that alternates in sign with period $\pi/k_F$. We give an elementary account of that result. Taking the Caroli-de Gennes-Matricon bound states in a step-like gap, we show that the normalization of the bound-state spinor is nearly independent of angular momentum, which collapses the mode sum into closed form. Inside the core the vortex winding removes one Bessel channel from a completeness sum, so the density vanishes on the vortex line and carries Friedel-like oscillations of wavevector $2k_F$; outside it the sum gives a $1/r$ envelope decaying over a coherence length, with a residual ripple. The bound-state charge does not integrate to zero, so neutrality obliges the extended states to compensate it exactly. That compensation is complete at long wavelength but fails at the diameter of the Fermi circle, and what survives is a sign-alternating $2k_F$ modulation reduced only by $4k_F^2/(4k_F^2+k_{TF}^2)$, a factor lying between one-half and three-quarters for any metal. The oscillation is therefore not a delicate effect but a consequence of neutrality and the inefficiency of screening at large momentum transfer: in the screened total the smooth terms cancel and only the ripple is left.

The Jacobian conjecture, open since 1939, asks whether every polynomial map of $\C^n$ whose Jacobian determinant is a nonzero constant must have a polynomial inverse. It was refuted in dimension three by Alpöge on July 19, 2026, with an infinite family by Gallagher (July 20) and a geometric explanation by Speyer (July 23): the counterexample sweeps the tangent lines of a plane curve. We give a self-contained account of this tangent-sweep mechanism and generalize it from plane curves to direction fields on hypersurfaces. The resulting construction, including stabilization of the three-dimensional family, produces counterexamples in every dimension greater than two and, in each dimension, of arbitrarily large geometric degree (the number of preimages of a typical point). We work it out in five explicit maps: one three-dimensional of degree four, two four-dimensional of degrees five and ten, and two five-dimensional of degrees six and twelve. We show that our explicit examples in dimensions $n\ge4$ are not polynomially left-right equivalent to the counterexamples of Alpöge and Gallagher when these are embedded in $\C^n$ by adjoining identity coordinates. The counterexamples provide explicit examples of étale morphisms $\C^n\to\C^n$ that are not proper: they are everywhere unramified, and fail to be injective only through points escaping to infinity. All identities were verified in exact rational arithmetic. Part~\ref{part:equivalence} proves the higher-dimensional inequivalence results, and Appendix A determines exact fiber structures for the three-dimensional examples. A complete classification of the fibers of $F_4,F_5,F_6,F_7$ remains open.

We study inverse potential mean-field games (MFGs), in which an unknown spatial inverse-cost (mobility) map is inferred from observed population densities. We solve the forward MFG with a preconditioned primal-dual hybrid gradient (PDHG) method and develop Jacobian-free backpropagation (JFB-$r$), which records only the final $r$ iterations from a detached warm start while retaining the full forward solve. To analyze this truncated differentiation method, we show that the exact-proximal dual-extrapolated PDHG map is a metric resolvent of the maximal monotone KKT operator. This resolvent view shows that JFB-$r$ exactly differentiates a finite-trajectory surrogate and, under a locally fixed active set at an exact equilibrium detach point, converges to the implicit gradient as the tracked depth increases. Across several inverse-MFG settings, numerical experiments show that JFB-r at moderate tracked depths can achieve recovery accuracy comparable to full unrolling while reducing memory and runtime.

A graph G with m edges has an antimagic orientation if its edges can be oriented and bijectively labelled 1, ..., m so that the oriented vertex sums, the labels on in-arcs minus the labels on out-arcs, are pairwise distinct. Hefetz, Mütze and Schwartz conjectured that every connected graph admits an antimagic orientation. We prove that this conjecture holds for every graph with a dominating clique.

It is well known that the inclusive cross section in QED is infrared finite. In the standard diagram techniques this result follows from cancellation of infrared divergences. We suggest a new approach that should lead to IR -safe computational technique for QED with massive fermions.

We study Fisher-information-based item-selection rules for multidimensional computerized adaptive testing (MCAT) with intentional and nuisance abilities. Although such rules are widely used, rigorous asymptotic theory for the coupled dynamics of adaptive selection and latent-trait estimation remains limited. We consider a weighted A-optimality criterion that prioritizes intentional abilities while controlling accuracy in nuisance dimensions. For the multidimensional two-parameter logistic model, we first establish asymptotic optimality under an item-type reuse regime: paired with the maximum likelihood estimator, the weighted A-optimal rule attains the minimum asymptotic weighted mean squared error. We then prove consistency and asymptotic normality of the maximum likelihood estimator and express its asymptotic weighted mean squared error through a limiting Fisher information matrix. We also extend the analysis to the operational setting in which each item is administered at most once. These results justify normal approximations and information-based standard errors in MCAT. A simulation study examines finite-sample behavior at practical test lengths and shows favorable weighted mean squared error, with item-selection patterns consistent with the predicted limiting behavior.

We study global regularity of solutions to Dirichlet or Neumann elliptic problems in spherical sectors $S_{D,R}$ of radius $R>0$ in $\mathbb{R}^N, N\ge 2$, where $D$ is the bounded domain on the unit sphere $\mathbb{S}^{N-1}$ which spans the spherical sector. One of the main results shows that boundedness of the gradient of the solutions of Poisson equations holds whenever $\lambda_1(D)\ge N-1$, where $\lambda_1(D)$ is the first nontrivial eigenvalue of the Laplace Beltrami operator $-\Delta_{\mathbb{S}^{N-1}}$ on the domain $D$ with Dirichlet or Neumann boundary conditions on $\partial D$. As an example of Maz'ya shows, the condition on the eigenvalue is sharp. For general spherical sectors and for $p$-Laplacian equations, $p>1$ we prove weighted global lipschitzianity of the solutions, as well as second order regularity.

This paper provides an abstract framework for studying inertial manifolds associated with a class of nonlocal parabolic problems. In particular, by suitably modifying the nonlocal term outside the absorbing ball and changing the scale of time, we derive a corresponding spectral gap condition. As applications, we establish the existence of inertial manifolds for two classes of two-dimensional modified nonlocal parabolic equations on a square domain, whose diffusion coefficients depend on the $L^2$-norm of the solution and of its gradient, respectively.

In this paper we develop an approach to the full quaternionic moment problem. We define a hierarchy $ \mathbb{H}^{k}[q^{*}, q]\subset \mathbb{H}^{k+1}[q^{*}, q]$, $k\in \mathbb{N}_{0}\cup \{\infty \}$, of two-sided $\mathbb{H}$-linear spaces of quaternionic polynomials which are invariant under conjugation of quaternions and determine the hermitian parts of these spaces explicitly. Using a generalization of Choquet's theorem on adapted spaces to quaternions we provide necessary and sufficient solvability criteria for the quaternionic moment problem of each space $ \mathbb{H}^{k}[q^{*}, q]$. The hermitian part of $ \mathbb{H}^{\infty }[q^{*}, q]$ is the real polynomial algebra $\mathbb{R}[x_{0}, x_{1}, x_{2}, x_{3}]$. This enables us to apply real algebraic geometry (Positivstellensätze) to the quaternionic moment problem on $ \mathbb{H}^{\infty }[q^{*}, q]$.

Given a Steinhaus random multiplicative function $f$, a $1$-periodic function of bounded variation $g$, and an irrational number $\alpha$, we study the distribution of $\sum_{n=1}^N f(n) g(\alpha n)$. We determine a necessary and sufficient condition for these sums, normalized by their standard deviation, to converge to the standard complex Gaussian distribution. On the other hand, we show that if we restrict the summands to have all their prime factors $>z$, with $z$ tending to infinity arbitrarily slowly, then a central limit theorem always holds for such sums.

There is a finite dimensional $\mathbb{C}$-algebra with $\mathrm{gldim}(A)=\infty$ and $\mathrm{HH}_{n}(A)=0$ for $n\geq 1$.

We construct a geometric model for the derived category of the weighted projective line $\mathbb{CP}^1_{\omega}$ of type $(2,2,2,2)$ using a graded sphere with four binaries. More precisely, we give a bijection between the set of indecomposable rigid objects and the set of certain arcs, such that the dimensions of $\operatorname{Hom}$-spaces are computed by oriented intersection numbers. As applications, we provide a geometric realization of the indecomposable rigid objects in the cluster category $\mathcal{C}(\mathbb{CP}^1_{\omega})$ via tagged arcs on the four-punctured sphere, with $\operatorname{Hom}$-space dimensions given by tagged intersection numbers. When the weighted points of $\mathbb{CP}^1_{\omega}$ are $(0,1,\infty,\frac{1}{2})$, both correspondences are compatible with the natural actions of the automorphism groups and the corresponding mapping class groups.

We study the complexity of approximating the finite-dimensional vector space embedding $\ell_p^m \hookrightarrow \ell_q^m$ for $2 \leq p < q \leq \infty$ based on non-adaptive randomized algorithms that use up to $n$ arbitrary linear functionals as information on a problem instance $x \in \mathbb{R}^m$, where $n \ll m$. We prove lower bounds on the non-adaptive randomized approximation error with a joint dependence on $(n,m)$ matching previously known upper bounds.

Combinatorial discrepancy theory is a subject with roots in combinatorics, geometry, and number theory, and with numerous applications to mathematics and computer science. At its core, discrepancy theory is about dividing a collection of objects into two parts that are as balanced as possible. For example, given a collection of subsets of a finite universe, we may wish to color the elements with two colors so that each set is approximately evenly split. Other problems in discrepancy are more geometric in flavor, and ask for example, to assign signs to a collection of vectors, so that the sum of the signed vectors is as small as possible. Classical results, such as the Beck-Fiala theorem and Spencer's "six deviations" result, show that it is often possible to attain remarkably small discrepancy, often far smaller than what naive random colorings achieve. In recent years, discrepancy theory has undergone a transformation, driven by new algorithmic techniques and a rich interplay between probability, optimization, and convex geometry. These developments have led not only to new constructive proofs of foundational theorems, but also to several new results and research directions. This monograph aims to provide an accessible and unified introduction to these modern developments, with a focus on the core algorithmic and convex geometric ideas that have driven them. For several results, we provide new simpler analyses, while highlighting the intuition behind the proofs.

The method of Convex Integration has revealed a number of rather disturbing facts concerning well-posedness of the Euler system of gas dynamics. In particular, there is a dense set of "wild" initial data, for which the problem admits infinitely many physically admissible (entropy) weak solutions. We identify the class of initial data enjoying the following properties: (a) they give rise to a family of weak solutions with increasing entropy profiles; (b) the solutions are "discrete", meaning they attain only a finite number of constant states; (c) the solutions reach a prescribed terminal entropy profile when time goes to infinity.

We prove bijectively a recursive excision formula for maps of arbitrary topology. This yields the topological recursion formulae governing their enumeration, already at the level of formal generating series and without any analyticity assumption. We extend the result to maps carrying self-avoiding loop models and to stuffed maps, i.e. a variant of maps allowing faces of arbitrary topology. Our results give a precise combinatorial meaning to all terms of the (blobbed) topological recursion known for (stuffed) maps, including the recursion kernel, which had so far been missing. The construction relies on the simple idea of iterating Tutte's algorithm until the topology changes. Equivalently, it can be interpreted as a pair-of-pants decomposition driven by a path issuing from the root of the first boundary, providing an analogue for maps of the Mirzakhani-McShane identity for hyperbolic surfaces.