Papers

The valley Delta conjecture of Haglund, Remmel and Wilson predicts that the symmetric function $\Delta'_{e_{n-k-1}} e_n$ equals a generating function over labelled Dyck paths with $k$ decorated contractible valleys. Unlike the rise version, which is now a theorem, the valley version remains open; indeed it is not even known that its combinatorial side is symmetric. We prove that the coefficients of $t^0$ and $t^1$ in the valley generating function are symmetric functions for all $n\ge 1$ and $k\ge 0$; equivalently, the fixed-diagonal-multiset slices of area at most one are symmetric. The theorem follows from an adjacent exchange identity for scaffold classes, which we prove in a strictly stronger form refined by the numbers of undecorated rows carrying the labels $r,r+1$ between consecutive rows with other labels. The proof develops a transfer-operator calculus in a $q$-deformed two-variable algebra generated by two commuting half-twists. In this algebra the exchange reduces to two scalar symmetric-series identities for the operator $T=u\mathfrak{d}+v$. We also verify the refined identity computationally at area two over extensive finite ranges and state the resulting general conjecture.

By introducing high-order Besov-type variance functionals that generalize the canonical variance, we develop a unified framework for proving high-order Talagrand-type inequalities that relate high-order energies to Fourier weights. Applying this machinery, we establish high-order Poincaré-type, $L^p$–$L^q$, isoperimetric-type, Falik–Samorodnitsky and Eldan–Gross inequalities, all with explicit constants, in both the Boolean and Gaussian settings. Fundamentally, our semigroup-based framework relies primarily on hypercontractivity and high-order Bismut-type derivative estimates, and is broadly applicable.

We develop a regularity theory for fully nonlinear oblique transmission problems. Our main result establishes the optimal regularity of viscosity solutions for flat interfaces. For constant-coefficient equations, we prove that viscosity solutions are piecewise $C^{1,\alpha}$ up to the interface. The same regularity continues to hold for variable-coefficient problems, with sufficiently regular coefficients, in a more restrictive class of viscosity solutions. This is the first regularity result for variable-coefficient transmission conditions.

We introduce elliptic transmission problems involving fully nonlinear equations in both the interior bulk equations and the interface transmission condition. In contrast to all previous works, our transmission condition depends on both the normal and tangential derivatives of the solution from each side of the interface. Our main result establishes the existence and uniqueness of viscosity solutions for constant-coefficient oblique transmission problems with flat interfaces.

This paper, the second in a series, presents an efficient, self-contained algorithm for computing the modified Bessel function of the second kind, \(K_{\nu}(z)\) for complex argument and real orders, building on Part~I (for \(I_{\nu}\)). The method adaptively selects among analytic representations such as power series, large-\(|z|\) asymptotics, uniform asymptotics for large \(|\nu|\), and numerically stable forward recurrence with region boundaries tuned for accuracy and efficiency. A robust Fortran implementation supports double precision and quadruple precision. The use of quadruple precision extends the reliable computational domain and improves stability in challenging regimes. Accuracy is validated against high-precision Maple results, and benchmarks show runtimes significantly superior to those of established methods, in the literature, while avoiding their numerical failure modes across several decades of the parameter domain. Together with Part~I, this work provides a comprehensive, multiple-precision toolkit for \(\{I_{\nu},K_{\nu}\}\) across wide parameter ranges.

This paper develops a climate-aware pricing framework for excess-of-loss (XL) reinsurance contracts and catastrophe (CAT) bonds under non-stationary catastrophe risk. Catastrophe arrivals are modeled as a Cox process whose stochastic intensity depends exponentially on a temperature-related climate index. To represent climate dynamics, the index is modeled as a mean-reverting Ornstein–Uhlenbeck process around a time-dependent warming trend. Within this setting, aggregate losses follow a compound Cox structure with lognormal severities. Pricing is performed under a reduced-form risk-adjusted measure, which provides a tractable valuation approach for XL reinsurance layers and binary zero-coupon CAT bond payoffs in an incomplete market setting. Because catastrophe losses are not dynamically replicable, the framework emphasizes scenario-based valuation rather than model-independent no-arbitrage bounds. A Monte Carlo valuation scheme is implemented to quantify the economic implications of climate-dependent catastrophe intensity. The numerical results show that climate dependence materially changes the loss-generation mechanism and affects the valuation of catastrophe-linked contracts. In the baseline calibration, the climate-aware model increases the excess-of-loss reinsurance premium and lowers the CAT bond price relative to the stationary benchmark. Furthermore, our analysis of the 99.5% Tail Value-at-Risk (TVaR) indicates that stationary benchmarks may underestimate economic capital requirements by approximately 13.7% compared to the climate-aware framework, highlighting the potential regulatory relevance of the proposed model. This finding highlights that benchmark design is critical for interpreting climate-pricing effects.

We study the joint convergence in distribution of a sequence $X_N = I_p(f_N)$ of multiple Wiener–Itô integrals of order $p\geq 2$ that converges to a Gaussian limit $Z\sim N(0,\sigma^2)$, together with another sequence $Y_N = I_q(g_N)$ converging in law. The central finding is that the joint convergence of $(X_N, Y_N)$ is completely governed by the asymptotic behavior of the iterated Malliavin covariances $Y_{r+1,N} = \langle DX_N, DY_{r,N}\rangle_H$, $r\geq 0$: joint convergence holds as soon as these covariances converge jointly with $Y_N$, and the structure of the limiting distribution is then explicitly determined by their limits. Moreover, the convergence of the Malliavin covariances is necessary for joint convergence, as shown by a counterexample. When $q<p$, the sequence $X_N$ is asymptotically independent of any $Y\in L^2(\Omega)$, a result which strengthens the stable convergence results in [12] and extends the multidimensional Fourth Moment Theorem [9]. When $q \geq p$, genuine asymptotic dependence appears and its structure depends critically on the ratio $q/p$. Writing $q = ap + r'$ with $0\leq r' < p$, the iterated Malliavin covariances form a transport hierarchy of depth $a$ that terminates in both the non-critical regime $ap < q < (a+1)p$ and the critical regime $q = ap$, but with different structures: the hierarchy is nilpotent in the non-critical case and recurrent in the critical one, due to the non-vanishing limit $\rho_a = \lim_N \mathbf{E}[Y_{a,N}]$. In both cases, the limiting characteristic function admits an explicit series representation whose coefficients are determined by a simple recursion. Under exponential moment assumptions, the series closes in closed form, and the two regimes differ by exactly one additional factor that appears only in the critical case.

We study smooth functions on manifolds whose sets of critical points is a disjoint union of submanifolds each diffeomorphic to the same manifold P. These submanifolds are not necessarily non-degenerate. We obtain sufficent conditions for existence of functions with minimal number of such submanifolds. This a translation from Ukrainian of PhD defended in 1993 at the Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine. Scientific advisor: Volodymyr Sharko.

The integration of Foundation Models (FMs) and wireless communications is driving the evolution of image communication from bit-accurate transmission toward task-oriented transmission. However, existing task-oriented image communication methods still face three major challenges: insufficient task-oriented Token representation, inadequate collaboration between Visual Tokens and Task Tokens, and limited interpretability of task decisions. To address these challenges, we propose an Explainable Task-Oriented Token Communication (ET-TokenCom) framework. By treating Tokens as unified units for information representation and transmission, the proposed framework constructs an end-to-end communication link that spans visual perception, wireless transmission, and task reasoning. At the transmitter, the ET-TokenCom framework extracts Visual Tokens from images to preserve low-level visual information. Meanwhile, Task Tokens generated by the FM are introduced to represent the target information and decision intent required by the current task. A Cross-Modal Attention (CMA) fusion mechanism is further designed, enabling Task Tokens to explicitly guide the selection, weighting, and transmission of Visual Tokens. At the receiver, the framework integrates Token decoding with an explainable output mechanism, where attention heatmaps are generated to highlight critical perceptual regions under different task objectives and reveal the influence of Task Tokens on the outputs. Finally, simulation results validate the effectiveness and robustness of the proposed ET-TokenCom framework.

Standing on the shoulders of giants, we resolve the quantum pyramids conjecture, confirming the globally information-optimal measurement for an ensemble of equiangular equiprobable pure states, as conjectured by Englert and Řeháček (arXiv:0905.0510). We do so by proving the remaining entropy inequalities of Holevo and Utkin (arXiv:2506.06700), which certify optimality for obtuse and flat pyramids. For obtuse pyramids, our key contribution is a rigorous proof that local minimizers of the corresponding entropy inequality cannot have three distinct coordinate values. We show that eliminating this family can be reduced to a neat algebraic reciprocal inequality relating branches of the Lambert $W$ function, which may be of independent interest. For flat pyramids, we prove a tight $\ell^p$ inequality for zero-sum vectors that was recently conjectured, proved analytically in dimension $d=3$, and computationally verified for $d\leq 200$ by Holevo and Utkin (arXiv:2603.24017). We prove this bound for all $d\geq 2$ via a technique in symmetric inequalities known as the equal variables method.

We study a max-risk objective for active learning in a multi-group mean estimation $d$-armed bandits: a learner adaptively allocates a budget of $T$ samples across $d$ groups to minimize the worst-case uncertainty index $\max_{k\in[d]}\sigma_k^2/n_k$, where $\sigma_k$ is the standard deviation of the distribution of arm $d$, and $n_k$ is the number of times arm $d$ is sampled. We develop a local minimax framework and prove the first general lower bound for this objective, valid for any finite-variance hypothesis class. The bound separates difficulty into three orthogonal factors: a budget term, a heteroscedasticity index measuring how unevenly the uncertainty is spread across arms, and a model-dependent complexity measure, the Variance Local Curvature ($\mathrm{VLC}$), which captures how much information a local change of variance creates inside the hypothesis class. For smooth classes, the $\mathrm{VLC}$ is a reparametrization of a variance–Fisher information, with closed-form values for common families. Benchmarking against the strongest available upper bound shows near-optimality up to logarithmic factors in broad regimes, and pinpoints a systematic gap in highly heterogeneous instances. Our proof introduces two key ingredients: a loss-induced $\ell_1$ geometry on the decision space, and a representation-based instance generator that reduces hard-instance construction to an explicit random matrix calculation.

Lehner (1999) derived elegant formulas for the operator norm $\|\mathfrak{X}\|$ of operators of the form $\mathfrak{X} = \mathbf{A}_0 \otimes \mathfrak{1} + \sum_{i = 1}^n \mathbf{A}_i \otimes \mathfrak{m}_i$, also easily generalized to the spectral edge $\lambda_{\max}(\mathfrak{X})$, in terms of nonlinear optimization problems over positive definite matrices. Here the $\mathbf{A}_i$ are finite-dimensional Hermitian matrices, the $\mathfrak{m}_i$ are either free semicircular or free Rademacher families of operators, and $\mathfrak{1}$ is the identity operator. We first show that both of Lehner's nonlinear optimizations can be rewritten as linear semidefinite programs (SDPs), even in the Rademacher case where Lehner's optimization is not itself convex. We give the primal and dual forms of these SDPs, derive the complementary slackness relations and consequences thereof, and propose that the SDPs are more stable and accurate than the iterative numerical scheme proposed in Lehner's original work. We then apply the SDPs from the semicircular case to spiked matrix models, studied recently via Lehner's formula by Bandeira, Cipolloni, Schröder, and van Handel (2024). We give a new proof of the Baik–Ben Arous–Péché (BBP) transition they establish in models with isotropic (but possibly correlated) Gaussian noise by constructing feasible variables for the associated primal and dual SDPs. Combining our construction with a sensitivity interpretation of optimal dual variables, we study the fluctuations of leading eigenvectors of such models. We conjecture and give numerical evidence that these fluctuations are Gaussian but anisotropic and non-universal, and that their covariance may be computed in terms of the optimizer of the dual of Lehner's formula, which in turn is approximately the leading eigenmatrix of a completely positive operator associated to the covariance of the noise model.

In this paper, we obtain several Ramanujan-type congruences modulo 5 and 7 for sum of certain non-overlined parts in overpartitions classified by parity and sum of certain parts in partitions without repeated odd parts classified by parity. Our proofs for the congruences are elementary, depending only on classical theta function identities.

Online inventory optimization (OIO) is online convex optimization with physical memory: inventory carryover makes the feasible action set depend on the past. A natural principle, used in stochastic inventory learning and recently in OIO under a single linear capacity constraint, is to maintain a hidden target chosen by an online learner and implement its projection onto the currently feasible order-up-to set. We prove that this simple principle is optimal for OIO on arbitrary bounded convex capacity sets. With online gradient descent as the base learner, the method improves the best known regret guarantee for OIO on general convex sets from inverse to inverse-square-root dependence on the common-demand probability, and we prove a matching lower bound. The same principle gives the first polylogarithmic regret guarantee for strongly convex losses and the first dynamic regret guarantee adapting to Euclidean path variation on general convex capacity sets. The analysis introduces a norm alignment principle: the right state variable is the distance from the hidden target to the feasible set, measured in the same norm as the projection. Under norm alignment, this distance evolves pathwise as a scalar queue, with target movement as arrival and common demand as service. This reduction to one-dimensional queue control resolves the state dependence and extends the guarantees to general convex capacity sets, beyond the reach of prior productwise approaches. Experiments on synthetic and real-world inventory data corroborate the theory.

A Detector Error Model (DEM) is a structured representation of error mechanisms in quantum circuits, which has gained popularity in quantum compilation pipelines for its ability to capture fault-tolerance at a circuit level. It lists error mechanisms as instructions targeting detectors and observables, specifying for each physical fault channel the probability that the fault fires, the detectors it triggers, and the observables it flips. In this paper, we develop an equational theory for DEMs, with its associated categorical semantics. We present a sound, terminating, confluent rewriting system for DEM terms, formulating it as a symmetric monoidal theory (a PROP) over the Giry monad. We prove that every DEM term has a unique normal form, which can be computed efficiently in quasilinear time $O(k|E|\log|E|)$, where $|E|$ is the number of instructions and $k$ bounds the size of a target set. This provides a complete set of invariants (via Tanner graphs) for structural DEM equivalence. We provide the first static decision procedure for DEM equivalence, with rigorous correctness guarantees. It is complete (decides full decoder-equivalence exactly) for non-adaptive quantum error correction (QEC) pipelines, and scales to a sound and applicable decision procedure for partially-adaptive circuits (lattice surgery, distributed QEC, ...) without suffering exponential overhead. We discuss its application to the verification and optimisation of quantum compilers.

This paper introduces a model-theoretic generalization of the notion of forcing with random reals, in which forcing gives rise to random generic structures. Specifically, we consider forcing with $\kappa$-Borel probability measures on the space of $L$-structures with a (possibly uncountable) infinite set $X$, focusing on those that are invariant under the action of the symmetric group $Sym(X)$. We demonstrate how any $Sym(X)$-invariant measure where $X$ is countable can be uniquely extended to a $Sym(Y)$-invariant measure where $Y$ is uncountable, and prove that forcing with such measures satisfies the countable chain condition. We also show that we can uniformly distinguish between these random generic structures and the Cohen generic structures that arise from forcing with a strong Fraïssé class: There is a $\kappa$-Borel set of low complexity that contains every Cohen generic structure that is not highly homogeneous but contains no random generic structure, implying that a structure that is not highly homogeneous cannot be both Cohen generic and random generic. Finally, we answer an open question of Kostana in the case of $\omega_1$, by establishing a connection between forcing with a strong Fraïssé class and Cohen forcing.

We give a brief exposition of a result of Song, Nguyen, and Lin (2026) that every election (with ranked preferences) has a Condorcet winning set of at most five candidates.

Jackson and Tanigawa conjectured that the rigidity matroid $\mathcal{R}^d_n$ and the hyperconnectivity matroid $\mathcal{H}^d_n$ are the unique maximal matroids in the posets of $\{K_{d+2}, K_{d+2,d+2}\}$-matroids and $\{K_{d+2},K_{d+1,d+1}\}$-matroids, respectively. We disprove these conjectures by showing the existence of maximal matroids that are distinct from the proposed candidates.

We prove an analogue of Hilbert's basis theorem for Poisson Ore extensions and Poisson Laurent Ore extensions. We also obtain corresponding results for iterated Poisson Ore extensions and iterated Poisson Laurent Ore extensions associated to commuting Poisson-pairs. Finally, we give examples of Poisson Ore extensions that are Poisson-Noetherian without being Noetherian as ordinary algebras.

Pairwise compatibility graphs (PCGs) represent graph adjacency by an interval of leaf-to-leaf distances in a weighted tree. We study grid graphs under the PCG model and two natural extensions: multi-interval PCGs and OR-PCGs. First, we prove that every $d$-dimensional grid graph is a $(d-1)$-interval-PCG. The construction decomposes the grid into hyperplanes of constant coordinate sum and uses a large-base encoding so that distances between consecutive hyperplanes identify the coordinate direction of an edge. A pair of nearby code values is then merged into one interval, reducing the number of intervals from $d$ to $d-1$. Second, we prove that every $d$-dimensional grid is a $\lceil d/2\rceil$-OR-PCG by grouping coordinate directions into pairs; each paired-direction graph is a disjoint union of two-dimensional grid graphs and is therefore a PCG. Finally, an exact tree-metric satisfiability computation shows that $P_3\square P_3\square P_3$ is not a PCG. Consequently, the minimum number of intervals sufficient for all three-dimensional grid graphs is exactly two, resolving a previously posed open problem. The same obstruction shows that the OR-PCG bound is tight in dimension three and implies that every grid with at least three factors of order at least three is not a PCG.

The Jordan-Moore-Gibson-Thompson (JMGT) equation models nonlinear acoustic wave propagation in thermally relaxing media and in the vanishing relaxation limit approaches the damped Westervelt equation. We investigate discontinuous Galerkin spatial discretizations of the JMGT equation on simplicial meshes and analyze their behavior uniformly with respect to the relaxation parameter. Under practically relevant mixed Neumann and absorbing boundary conditions, we derive a priori error estimates independent of the relaxation parameter. These estimates enable a rigorous singular limit analysis, yielding convergence of the semi-discrete JMGT approximations to the corresponding Westervelt pressure profile at a linear rate. This also sheds light on the expected behavior of exact solutions in the vanishing relaxation limit. For the fully discrete problem, we propose a Newmark-type method based on a reformulation as a coupled second-/first-order system. Numerical experiments support the theoretical findings and demonstrate the robustness of the approach in the small-parameter regime.

We discuss three families of twisted graded Calabi–Yau algebras of dimension three that arise through the process of normal extensions. In addition to the twisted Calabi–Yau condition, we prove that these algebras are noetherian piecewise domains under a non-degeneracy condition. We show that these algebras can also be constructed as certain iterated skew polynomial rings. Finally, we discuss how these algebras shed light on possible types of twisted graded Calabi–Yau algebras of dimension four.

Robust machine learning and optimization rely on the uncertainty model choice. We investigate which uncertainty directions a model must cover when defined by a finite dictionary and a budget constraint. Selecting a subset forms an atomic uncertainty set with a closed form support function, yielding tractable robust programs for affine objectives. We propose a data driven selection rule based on a coverage objective over evaluation directions, including gradients, adversarial perturbations, or shifts observed on held out data. We prove this objective is monotone and submodular, supporting a greedy method with a $(1-1/e)$ approximation guarantee and a matching hardness barrier. We also provide a certificate bounding the loss from the selected subset and a radius calibration rule with out of sample control.

We study the gauge-invariant dynamically conserved charges, and their corresponding densities, for instanton solutions of Yang-Mills theories in four dimensional Euclidean space, for the gauge group $SU(2)$. Those charges were constructed in [1,2] through the integral equations of Yang-Mills theory, using techniques on generalized loop spaces. We use the integral non-Abelian Gauss law to evaluate the gauge-invariant flux of the magnetic and electric non-Abelian fields through spherical surfaces centered at the origin of the instanton solution. From such a flux, we define gauge-invariant charge densities by considering the charge within an infinitesimal spherical shell of radius $r\equiv\sqrt{x_i \, x^i}/\lambda$, with $\lambda$ being the parameter of the instanton solution, defining its size, and $x_i \, x^i = (x^1)^2 + (x^{2})^2 + (x^{3})^2$. We discuss the issue of the reparameterization invariance of the charges and densities, and show that the magnetic and electric fluxes for the instanton and anti-instanton, at $r=1$ and $x^4 = 0$, $x^4$ being the Euclidean time, are non-zero and observable. Our results give an interesting picture of the internal structure of the instanton, and may be important for the properties of the Yang-Mills $\theta$-vacuum.

We show that Braverman–Kazhdan normalized intertwiners for $SL_n(\mathbf{C})$ have a quasi-classical incarnation governed by type $A$ quiver varieties. More precisely, for standard parabolic subgroups $P$ and $P'$ with conjugate Levi subgroups, we construct $SL_n\times L^{\mathrm{ab}}$-equivariant isomorphisms $\Phi(P,P'):\overline{T^*(SL_n/[P,P])}^{\mathrm{aff}}\rightarrow\overline{T^*(SL_n/[P',P'])}^{\mathrm{aff}}$ between the affinizations of the cotangent bundles of the corresponding Braverman–Kazhdan spaces, and we prove that these isomorphisms satisfy Coxeter relations. The construction uses $SL$-gauge analogues of Lusztig–Maffei–Nakajima reflection functors, thereby extending Wang's quiver-variety realization of the quasi-classical Gelfand–Graev action from the Borel case to arbitrary parabolic subgroups. In this way, we complete the quasi-classical Braverman–Kazhdan intertwiner story for $SL_n(\mathbf{C})$ and obtain a systematic source of non-isomorphic varieties whose affinized cotangent bundles are isomorphic.

We adapt the recent theory for unique solvability of perturbed saddle-point problems in Banach spaces to the case of parameter-independent stability bounds. This also constitutes an extension to the Brezzi–Braess theory for parameter-robust stability of perturbed saddle-point problems from the Hilbert to the Banach setting. We apply the new abstract result to the mixed formulation of the advection-diffusion-reaction equation and tackle also its coupling with the Darcy–Forchheimer equations. The complete system is shown to be stable irrespective of the model parameters of permeability, Forchheimer coefficient, and reaction modulation. We discretise the problem with mixed finite element methods and show convergence in appropriately parameter-weighted norms. We also design operator-based preconditioners for the full system, utilising weighted norms that provide robustness with respect to some model parameters. We also present a few numerical examples that show the convergence and robustness of the proposed method in some regimes of practical interest.

John E. Wetzel conjectured that the 30-60-90 triangle T obtained by placing a square of side 1/3 on the hypotenuse covers every unit arc in the plane. We give a computer-assisted proof of this conjecture with independently checkable interval certificates. The proof reduces a hypothetical noncovered arc to a finite family of 599 closed second-order cone models, covering all representative and raw tail-order branches, and certifies a polygonal-chain lower bound greater than one in every model by interval validation of stored dual certificates. Since every certified lower endpoint exceeds 1.0048, the homothetic copy T/1.0048 still covers every unit arc. Its area is 0.260956..., below the area pi/12 approx. 0.261799 of the 30-degree unit sector, a certified area improvement over the sector cover within this convex Wetzel-cover setting.

Tensor networks provide a powerful framework for efficiently representing high-dimensional data and many-body quantum states. Endowing tensor networks with a Riemannian manifold structure provides a natural setting for numerical optimization and analysis. A central feature of tensor networks is their gauge freedom, whose characterisation (captured by so-called fundamental theorems) underlies both their intrinsic structure and the design of numerical algorithms. In this work, we study the interaction between the Riemannian manifold structure and the gauge freedom for several families of tensor networks. Using group actions and Riemannian submersions, we establish a Riemannian fundamental theorem for the tensor network families studied.

We study the data-to-solution value map for quasistatic nonlinear elasticity in the linearized-kinematics regime, allowing both generalized loads and variable Dirichlet data. Under standard direct-method hypotheses, the negative minimum potential energy is finite and locally Lipschitz, convex in the load variable, and concave in the Dirichlet datum. Its supporting slopes, and its first variations at differentiability points, are the equilibrium displacement and the Dirichlet reaction. This convex–concave structure leads to a mechanics-preserving saddle minmax architecture in which displacement atoms generate load slopes, reaction atoms generate Dirichlet slopes, and the coupling coefficients are the corresponding trace-reaction pairings. Manufactured samples are produced by prescribing displacement–reaction pairs and computing the associated ambient data and exact value labels. The resulting architecture directly approximates the negative minimum-potential-energy value map and provides mechanical subgradient readouts. Immersed representations and cell-center quadrature make the construction implementable on background grids and geometry-rich domains. We prove uniform convergence on compact data sets with respect to atom enrichment and quadrature refinement, and illustrate the method on elementary examples.

The paper is dedicated to calculate the number of semi-fast convergent representations of a set of subsums. The formula for specific irreductible sequences is obtained. We study the Central Cantor sets and consider complete sequences of natural numbers.