Papers

We study the regularity of the free boundary for a multi-phase obstacle-type problem associated to quadrature domains, which arises from the minimization of the energy $$J(U,F):= \int_{B_1} (\frac 1 2 |\nabla U|^2 + F \cdot U)dx$$ among all competitors $U=(u^1,\ldots, u^N)$ with non-negative components $u^i \geq 0$ with disjoint supports, and constant forcing terms $f^i>0.$ We focus on the geometry of the free boundary at points where at least three distinct components meet (triple junctions).

The Radon transforms on Grassmannians define two families of geometric operators on convex bodies when applied to their projection functions. On the one hand, these are Minkowski's projection body maps and on the other hand these operators coincide on origin-symmetric bodies with the mean section body maps introduced by Goodey and Weil in 1992. Over the past two decades various sharp isoperimetric inequalities for projection bodies were shown to also hold for mean section bodies. However, an analogue of Lutwak's log-concavity inequalities for the intrinsic volumes of projection bodies remained open for mean section bodies. We establish new Aleksandrov–Fenchel inequalities for averages of certain mixed volumes and apply them to show that also all the intrinsic volumes of mean section bodies are indeed log-concave.

Motivated by a recent work of G. Knese [Canad. Math. Bull., 2026], we introduce the operator-valued Schur-Agler class in infinitely many variables, consisting of functions whose restrictions to every finite-dimensional polydisk lie in the operator-valued Schur-Agler class. We first establish an operator-valued analogue of Montel's theorem and, using it as a key ingredient, establish an Agler decomposition and a realization formula for functions in this class. The series in the Agler decomposition converges in the strong operator topology, and we show that this convergence cannot, in general, be strengthened to norm convergence. The realization formula yields a holomorphic extension of every function in the operator-valued Schur-Agler class to $\mathrm{Ball}(\ell^\infty)$, the open unit ball of $\ell^\infty$. Finally, as an application, we prove an Agler-Pick interpolation theorem in infinitely many variables.

We study the task allocation problem, where our goal is to develop an online algorithm that efficiently distributes tasks among workers based on their computational performance. This task is challenging since computation times are random and non-stationary in practice. Recent works have introduced different online algorithms to address this problem. One of them is Adaptive Task Allocation (ATA), a bandit-style algorithm. However, previous theoretical guarantees leave substantial room for improvement, and existing methods are designed only for stationary computation times. Using new proof techniques, our work establishes an improved analysis of ATA with better gap-dependent and gap-independent regret guarantees, improving upon all previous approaches. We also introduce a new method, Sliding Window Adaptive Task Allocation (SW-ATA), designed for more practical non-stationary environments. Finally, we prove new lower bounds in both the stationary and non-stationary settings.

We investigate the influence of decay rates on the dynamics of three-dimensional gene regulatory networks modeled by nonlinear differential equations with logistic sigmoidal regulatory functions. Particular attention is paid to the existence, location, and stability of equilibrium points in systems involving numerous interacting parameters. An explicit analytical representation of a central equilibrium is obtained under suitable relations between the regulatory weights, thresholds, and decay coefficients, allowing its location to be determined without solving the corresponding nonlinear algebraic system numerically. General formulas for the characteristic polynomial of the linearized system are derived, and spectral conditions for instability are established for several classes of regulatory interactions. We demonstrate how variations in decay rates can change the number and stability of equilibria, transform oscillatory dynamics, and influence the emergence or suppression of complicated attractors. The possibility of chaotic behavior is investigated through eigenvalue configurations, Lyapunov exponents, and the Kaplan-Yorke dimension. Numerical examples illustrate the sensitivity of the dynamics to decay parameters, including the suppression of previously observed chaotic behavior. The results provide an analytical framework for investigating equilibrium configurations, stability transitions, and complex dynamics in parametrized gene regulatory networks.

We study spectral measures of one-dimensional quasiperiodic Schrödinger operators. For subcritical Schrödinger operator with Diophantine frequency, we prove sharp measurable-set bounds, answering a question of Avila and Jitomirskaya \cite{AJ}; we prove optimal Fourier decay and the sharp decay order of the time-averaged quantum return probability, confirming a conjecture of Aloisio, Carvalho, and de Oliveira \cite{ACO} and a prediction of Zhong and Mosseri \cite{ZhongMosseri}. For the subcritical almost Mathieu operator, we establish summable square-root envelopes for the spectral density, with an optimal density-one formulation of the lower bound, thereby answering a question of Avila \cite{Aqi}. These results follow from a general theorem for reflectionless measures under a summable square-root Craig-type condition. We establish an exact residue expansion of the squared Borel transform, which connects local density blow-up with long-time averaged return probabilities: the endpoint residues determine both the square-root edge profiles and the leading coefficient of the time-averaged return probability.

We study interior Hessian estimates for $k$-convex solutions of Hessian quotient equations $\frac{\sigma_k(D^2u)}{\sigma_\ell(D^2u)}=1$ with $2\leq k\leq n$, $0 \leq \ell < k$, and $k-\ell \in\{1,2\}$. We first adopt the unified quantitative separation-propagation method of Li–Wu [LW26b] to derive a doubling inequality, and then use the compactness argument of Shankar–Yuan [SY25] to obtain the estimates. This approach also applies to a class of general Hessian equations.

Let $\mathcal O$ be the Lubin–Tate deformation ring of a one-dimensional commutative formal group of finite height $n\geq1$ over a perfect field $k$ of characteristic $p$. We identify Rezk's Witt filtration $\{I_d\}$ with the divisibility filtration of integral trace pairings on subgroup algebras of fixed source lifts. This gives the geometric recursion \[ I_{d+1}=pI_d+J_{d,z}\qquad(d\geq0), \] where $J_{d,z}$ defines the targets of order-$p^d$ quotient isogenies from a $W(k)$-valued lift $z$ of the closed point. The corrected ideal is independent of $z$. We also compute the cokernel of the universal source trace pairing over $\mathcal O$: it is $\bigoplus_{j=1}^d(\mathcal O/p^j)^{r_j-r_{j-1}}$, where $r_j$ counts order-$p^j$ subgroups of $(\mathbf Q_p/\mathbf Z_p)^n$. The same duality extends the generic trace along two-step isogeny flags to an integral surjection, even when the flag-forgetting map is not flat. This computes composition in the subring of weighted trace operations. Applications include sharp projector denominators and congruence fibre products for full-torsion loci.

We propose a model for the one-dimensional isothermal gas flow through a valve in a pipe. The valve aims at keeping the outgoing flow in an assigned interval; moreover, its opening and closing depend on a further parameter. The model is realized by a system of two conservation laws with suitable conditions coupling the ingoing and the outgoing flow. We study the corresponding Riemann problem, where non-classical shocks show up. Although simple, the model allows to study two key oscillatory behaviors of the flow, which are due, respectively, to the chattering of the valve, especially seen in the numerical simulations, and to the stabilizing hysteresis induced by the valve operation. We characterize the states leading to chattering and show how they depend on the parameters of the model. Through several numerical simulations, we study the interplay between the two above phenomena.

We introduce river ribbon tableaux, where a river of non-trivial ribbons is placed between two banks of single boxes. Compared to other ribbon peelings, this reduces the geometric differences across diagrams of different shapes, and the maximal number of ribbon configurations. This allows us to prove a uniform character bound for symmetric groups which, specialized to balanced diagrams, proves the conjecture of Moore and Russell. As an application, we prove a conjecture of Dousse and Féray on the number of standard tableaux of skew shapes.

Let $P_{n,k}$ count permutations of $[n]=\{1,\ldots,n\}$ whose longest increasing subsequence has length $k$. Chen conjectured that $(P_{n,k})_{k=1}^n$ is log-concave for every positive integer $n$. In this paper, we prove that Chen's conjecture is true when $n\ge 10^{10^{65}}$. The proof combines Gessel's determinantal formula with quantitative coefficient extraction, Tracy–Widom density estimates, and exact inequalities for the extreme lengths.

In many applications, covariates are naturally organized into groups, yet conventional group-selection methods often retain all variables within selected groups, failing to achieve within-group sparsity. To address this limitation, we propose an adaptive two-stage sparse-group selection procedure for high-dimensional quantile regression. The proposed method integrates convolution-smoothed quantile loss with $\ell_{2,0}$ and $\ell_0$ penalties to identify important groups and relevant variables within them, achieving sparsity at both levels. We establish theoretical guarantees for sure screening and exact support recovery at both stages, as well as consistent model-size selection using information criteria, with explicit bounds on failure probabilities. The proposed procedure is computationally efficient by restricting optimization to selected groups and active variables. Simulation studies and a breast cancer gene-expression analysis demonstrate its competitive performance in variable selection, estimation accuracy, and computational efficiency, while identifying biologically relevant genes and delivering accurate quantile predictions.

Smoothed decentralized Bregman proximal-gradient (SDBPG) methods combine relative-smoothness geometry with edge-dual consensus. We extend SDBPG to node-specific convex nonsmooth regularizers and unbiased stochastic local gradients while retaining its Bregman primal-dual smoothing mechanism. The analysis isolates temporal subgradient motion and aggregation mismatch as two defect sequences. For trajectories in a forward-invariant compact subset of the kernel domain and admissible parameters, finite accumulated defects yield an $O(T^{-1})$ bound on the squared best-iterate proximal residual over $T$ iterations. The stochastic bound separates accumulated oracle variance from expected defects and attains the same order when the variance bounds and expected defects are summable. Common active-structure identification, together with local subgradient conditions, gives sufficient finite-defect conditions for $\ell_1$ and group $\ell_{2,1}$ regularization. Experiments with common regularizers and heterogeneous sensing operators relate defect decay to observed support stabilization and compare reconstruction and communication with decentralized proximal and gradient-tracking methods. A separate phase-retrieval study evaluates the effect of a quartic Bregman kernel. Code and experiment scripts are available at https://github.com/shengze-xu/composite-stochastic-sdbpg .

We give a counterexample to the implication from generic local to generic global identifiability for linear networks with a fixed topology, independently unknown edge coefficients, and selected excited and measured nodes. A six-node network with nine unknown coefficients has exactly two compatible parameter vectors generically. Adding six excited and measured nodes preserves this ambiguity while making the added coefficients directly recoverable. The resulting twelve-node structure has forty-two independent unknown coefficients and zero diagonal. For every positive bound on the largest eigenvalue magnitude, a nonempty open set of parameters has two distinct compatible networks satisfying that bound. This holds for both real and complex coefficients. Other open regions have exactly one admissible solution. The proof uses a rational transformation exchanging the two solutions and a common limiting matrix whose fourth power vanishes. Exact rational-number certificates verify an explicit pair. The construction also extends algebraically when the unknown coordinates are whole rational transfer functions. Genericity in prescribed numerator and denominator coefficient families, and the requirement that every transfer function vanish at infinite frequency, remain separate questions.

Let $f$ be a monic polynomial of degree $n$ with integer coefficients. We use Fourier analysis to prove a new upper bound on the number of $n\times n$ matrices with characteristic polynomial $f$ and integer entries in $[-T,T]$, as $T\to \infty$. The implied constant in our bound only depends on $n$ and is otherwise uniform in $f$. Our bound improves on recent work of Habegger, Ostafe and Shparlinski for $n\geq 4$.

In \cite{Dan-Guo}, the authors provided a characterization of all sequences of doubly commuting pure contractions that have the sequence $\textbf{M}_z,$ the sequence of the multiplication by the coordinate functions on the vector-valued Hardy space $H^2_\mathcal E(\mathbb {D}_2^{(\infty)})$ over the Hilbert's multidisk $\mathbb D_2^{(\infty)},$ as their minimal isometric dilation for some Hilbert space $\mathcal E$. In this article, we remove the hypothesis of doubly commutativity and characterize all sequences of commuting contractions that are unitarily equivalent to $(P_{\mathcal Q}M_{z_i}|_{\mathcal Q})_{i\geq 1}$ for some quotient module $\mathcal Q$ of $H^2_\mathcal E(\mathbb D_2^{(\infty)})$. We further characterize those sequences for which $\mathcal Q$ is a Beurling quotient module. This generalizes Theorem 3.2 from \cite{BKD} to a sequence of commuting contractions.

We classify Frobenius liftable normal isolated complete intersection singularities of dimension at least three over an algebraically closed field of positive characteristic. In dimension three, they are precisely the regular points and the ordinary double points; in higher dimensions, they are regular. We also characterize Frobenius liftability of the complete local hypersurfaces defined by $xy-g(z,t)$ without assuming that the singularity is isolated. Such a hypersurface is Frobenius liftable if and only if, after a formal change of coordinates, $g$ is a unit times $H^pz^at^b$ with $0\le a,b<p$.

We construct $T^2$-invariant Einstein metrics with positive Ricci curvature on #_m(S^2 x S^2) for every integer $m>1$. These provide infinite family of Einstein metrics on closed $4$-manifolds with $\lambda>0$, which are in addition not Hermitian. We further establish the uniqueness of the $T^2$-invariant Einstein metrics on $S^4$, CP^2, S^2 x S^2, and CP^2#\bar{CP}^2.

We give a directed comparison beginning with the classical zero strip \(\lvert\Ima z\rvert\le1\) for the de Bruijn–Newman heat deformation. Under explicitly stated complete model-to-function source, localisation and global contact premises, translated-probe estimates imply \[ \Lambda\le\frac{148611789321711}{10^{15}} =0.148611789321711<0.148612. \] The construction uses Raymond's finite source data and 57 fixed translated kernels, independently checked by rational Bernstein certificates. An explicit finite Weyl estimate improves the positive first-head tail and its centred logarithmic moment. A centred cone transfer retains the original physical error allowances, exceptional debit and additive cone reserve, and charges the change in existing phase-surplus credits. Directed tail enclosures are regenerated at 192 and 256 bits; a separate exact-rational auditor accepts all 19,023 comparison steps. The controlled decrease from the complete two-source comparison is \(0.000021760452379\). We also prove a compatible matched logarithmic tail transfer and identify the finite requirements for stronger exponent pairs. A supplementary section records Raymond's ceiling below \(0.116398\) under the distinct \(7/8\) zero-free hypothesis.

We study the local $\mathrm v$-numbers of binomial edge ideals from a combinatorial perspective. For each minimal prime $P_{\mathcal S}$ of $J_G$, indexed by a cutset $\mathcal S$ of $G$, we consider two combinatorial invariants, $\mathrm v^-_{\mathcal S}(G)$ and $\mathrm v^+_{\mathcal S}(G)$, defined in terms of transversals of suitable hypergraphs, and show that the local $\mathrm v$-number $\mathrm v_{P_{\mathcal S}}(J_G)$ satisfies $\mathrm v^-_{\mathcal S}(G)\leq \mathrm v_{P_{\mathcal S}}(J_G)\leq \mathrm v^+_{\mathcal S}(G)$. Under a suitable hypothesis on the connected components, we provide explicit graph-theoretic descriptions of these invariants and analyze when they coincide, giving a combinatorial way to determine the local $\mathrm v$-numbers. As an application, we derive a sufficient condition for this coincidence, which applies to a broad class of graphs. Furthermore, we identify a constant that controls the gap between $\mathrm v^-_{\mathcal S}(G)$ and $\mathrm v^+_{\mathcal S}(G)$, thereby showing that the local $\mathrm{v}$-number lies within a bounded interval. Applications to cycles and trees illustrate both the equality criterion and the resulting bounds.

This work proposes a biomechanical registration framework designed to combine functional optical-mapping data with structural high-resolution MRI of cardiac tissue. The two acquisitions are performed in different mechanical configurations: optical mapping is obtained after stretching the sample, whereas MRI is acquired in a fixed resting state. To estimate the deformation between these configurations, the tissue is modeled as a nearly incompressible transversely isotropic hyperelastic material, and the traction amplitude is identified from optical-map marker positions through an inverse problem. Two-and threedimensional simulations show physically admissible deformations and provide a first proof of concept for the proposed strategy.

In the literature on multiple testing methods that control the familywise error rate, a fundamental concept is "coherence". A procedure is coherent if it is "logically consistent" (Hochberg and Tamhane, 1987) in the sense that if a hypothesis is rejected, then all hypotheses that imply it are also rejected. If a method is not coherent, then it can be made coherent without increasing the familywise error rate. Recently, the concept of coherence was generalized to a much broader class of multiple testing procedures, namely all methods that provide simultaneous confidence bounds for false discovery proportions. Such procedures were defined to be coherent if they cannot be further improved through a process called "interpolation". However, we show that such a coherent procedure can sometimes be further improved by exploiting remaining logical inconsistencies. This leads to what we call a "perfectly coherent" method. Thus, interpolation does not necessarily lead to perfect coherence. However, we provide conditions under which it does. Fortunately, many methods based on interpolation turn out to be perfectly coherent.

A critical set in a Latin square is a set of entries that determines the square uniquely and is minimal with this property. We prove that every partial Latin square of order $n$ with fewer than $\lfloor n^2/4\rfloor$ filled cells has an even number of completions. Consequently every critical set in every Latin square of order $n$ has at least $\lfloor n^2/4\rfloor$ entries. Since the cyclic Latin square has a critical set of exactly this size, the minimum size of a critical set over all Latin squares of order $n$ is $\lfloor n^2/4\rfloor$. This confirms a conjecture attributed to Nelder (1979) and made independently by Mahmoodian and by Bate and van Rees. Hatami and Qian had shown that this minimum is at least $n^2/10^4$ for large $n$. The proof is a counting argument modulo two that uses only expansion of products, cancellation in pairs and the pigeonhole principle, and it is illustrated with partial Latin squares of orders three to seven. An appendix discloses the substantial use of AI in developing the proof and writing the exposition.

We prove that every tangent flow at the first singular time of a smooth, closed, embedded mean curvature flow has smooth support, provided that the total curvature satisfies $\sup_{t\in[0,T)}\int_{M_t}|A|^n\,d\mu_t<\infty$. This extends Ilmanen's smoothness result for tangent flows to higher dimensions.

We extend Burnside's $p^a$-Theorem on conjugacy classes of prime-power size by proving that no non-abelian finite simple group possesses a conjugacy class of size $2p^n$ for any prime $p$ and $n\geq 0$. Based on this, we partially extend a well-known result of L. Kazarin by showing that, given a finite group $G$, the subgroup generated by a conjugacy class $x^G$ of size $2p^n$ where $x$ has odd order, is a solvable normal subgroup of $G$. If instead the subgroup $\langle x^G \rangle$ generated by such a class $x^G$ is non-solvable (which forces $x$ to have even order), then $p= 5$ and $\langle x^G \rangle/S(\langle x^G \rangle)$ is isomorphic to the symmetric group $S_5$, where $S(\langle x^G \rangle)$ denotes the solvable radical of $\langle x^G \rangle$.

We study the $(K,N,S,M,K_c)$ distributed linearly separable computation (DLSC) setting, where a master employs $N$ workers with a per-worker computation load of $M$ to recover $K_c$ deterministic linear combinations of $K$ messages in the presence of up to $S$ stragglers. We propose a DLSC framework based on nested cyclic codes that achieves $M=\frac{K}{N}(S+K_c)$. For sufficiently large $K$, we establish the matching converse $M\geq\frac{K}{N}(S+K_c)$, proving the asymptotic computation load optimality of the proposed framework. By instantiating cyclic locally recoverable codes (LRCs) as base codes in the framework, the master can recover the desired $K_c$ linear combinations by contacting fewer than $N-S$ workers under many straggler scenarios. This essentially softens the straggler threshold without increasing the computation load or communication cost. Finally, we extend the construction to reduce the per-worker communication load by a factor of $m$, while achieving an asymptotically optimal computation load of $M=\frac{K}{N}(S+mK_c)$.

This work is a step towards a general understanding of fluctuations for systems of particles described at large scale by fractional diffusion. First, we derive formally a Large Deviation Principle for the empirical measure of mean-field interacting particles with a Lévy-type noise. In the case of independent particles, this unveils a generalized gradient flow structure at large scales. Second, we derive and solve numerically a stochastic Partial Differential Equation describing the fluctuating hydrodynamics of this system. The results compare well with direct numerical integration of the particles' dynamics.

We investigate the effect of initially enlarging a Brownian filtration by its terminal value on backward stochastic differential equations (BSDEs) and reflected BSDEs (RBSDEs). We establish the Brownian innovation representation and the predictable representation property in the enlarged filtration, highlighting the singular information drift arising from the semimartingale decomposition of the original Brownian motion. We derive transfer identities under suitable integrability conditions and provide explicit counterexamples showing that the resulting singular equations may fail to admit unique square-integrable solutions, including in the reflected case. We further develop a conditional parametrization of regular BSDEs and establish a conditional comparison principle for Snell envelopes, illustrating both the strict value of additional information and the failure of general pathwise domination. An exact projection identity is also obtained for zero-driver terminal payoffs. Finally, we discuss possible extensions to irregular obstacles and more general information structures.

We prove the uniqueness of global solutions to the two dimensional barotropic compressible Navier–Stokes equations within the regularity class introduced by Danchin and Mucha [14], for general $C^1$ pressure laws. A crucial assumption is that both solutions have bounded divergence, locally integrable in time, which ensures that their densities remain bounded without excluding vacuum or requiring spatial regularity of the density.

We consider the two-dimensional isentropic compressible Euler equations for a polytropic gas, $p(\rho)=k\rho^{\gamma}$. The $R$–vacuum state–$R$ configuration—two rarefaction waves bounding a vacuum—occurs when the two constant states move apart with sufficiently large relative velocity and is essentially the only vacuum-forming Riemann pattern in the absence of external forces. Local-in-time rarefaction fronts have been constructed only away from vacuum, where the sound speed is bounded away from zero; their construction through the vacuum edge, and hence the structural stability of the $R$–vacuum state–$R$ configuration, has remained open. We establish this stability for $\gamma\in(1,1.3]\cup\{\frac{5}{3},\frac{7}{5}\}$: for sufficiently small perturbations, the Riemann problem admits a continuous entropy solution on a fixed time interval, with the domain partitioned by four characteristic surfaces and the solution in the three central regions belonging to $W^{2,\infty}$ in the self-similar coordinates $(t,x_{1}/t,x_{2})$, even though it may contain logarithmic singularities of the form $c^{\alpha}(\ln c)^{\beta}$ at the vacuum edge. The proof develops a geometric formulation directly on the first-order system, together with a direct analysis of the initial slice at the vacuum edge, where the formal high-order expansion fails.