Papers

We calculate the R-matrix for the exceptional Lie superalgebra $\mathfrak{d}(2,1;\kappa)$ in the smallest representation of its quantum supergroup, using a method of Hasse diagrams.

In this paper, we study a system in which the Cahn–Hilliard system is imposed in the bulk domain and an Allen–Cahn type equation is prescribed on the boundary, connected through a third type transmission condition characterized by a permeability parameter. This setting is closely related to the theory of transmission problems, and the third type transmission condition can be viewed as a remnant of a thin-boundary description, with the permeability parameter controlling the degree of coupling between the bulk and the boundary. The main objective is to perform a rigorous asymptotic analysis with respect to the permeability parameter. We investigate two limits: the parameter tending to zero, corresponding to a completely impermeable boundary, and the parameter tending to infinity, corresponding to a perfectly permeable boundary where the bulk and boundary phases are fully coupled. For each limiting regime, we establish the convergence of solutions to the respective limit problems and characterize the resulting equations.

We prove that a possibly unbounded Reinhardt domain in $\mathbb{C}^n$ with smooth boundary and a complete Bergman-Einstein metric is biholomorphic to the complex unit ball. In particular, no unbounded smooth Reinhardt domain admits a complete Bergman-Einstein metric.

This paper proposes a framework for treating the inverse obstacle problem for nonlinear elliptic equations with nonlinear materials. The problem is challenging because nonlinear materials exhibit a rich diversity of scenarios to consider, since nonlinearity can take different forms. In this article, after categorizing the nonlinearities into a few fundamental classes, a dedicated imaging method is proposed for each class, derived by combining two powerful concepts: the Monotonicity Principle (MP) and the $p-$Laplace Signature (pLS). The Monotonicity Principle (MP), recently extended to nonlinear materials, provides a monotonic relationship between the material property and the measured quantity (the Average Dirichlet-to-Neumann map) that can be \lq\lq inverted\rq\rq \ to find the shape of anomalies. The $p-$Laplace Signature (pLS) allows for modelling the solution of an elliptic PDE with nonlinear materials, for large or small boundary data, in terms of a proper $p-$Laplace equation that captures the essence (the signature) of the problem. For example, pLS with $p=2$ allows the reduction of a nonlinear elliptic PDE to a linear one, providing a powerful bridge for applying imaging methods and algorithms developed for linear materials. In this contribution, the two pillars of MP and pLS are combined in new imaging methods to enlarge the class of nonlinearity that can be treated within the inverse obstacle problem. Moreover, the theoretical limits of the methods are provided in the ideal case of noise-free measurements: outer-support reconstruction when $p=2$, and convex-hull reconstruction when $p\neq2$.

Factorization for positive semidefinite matrix-valued polynomials over a nonsingular compact affine real surface is established. Corollaries include Fejér-Riesz factorization for bivariate matrix polynomials that take positive semidefinite values and resolutions to questions posed by Mehta-Slofstra-Zhao and Savchuk-Schmüdgen. An explicit example shows a conclusion of [Dri25, Theorem, p. 519] that arises organically from its proof need not hold. The difficulty is traced to [Dri25, Theorem 3.7].

We study a class of nonlinear elastic energies whose constitutive structure is naturally expressed in terms of a stretch variable but is not directly covered by the standard polyconvex theory formulated in the deformation gradient. The problem is lifted by introducing an independent microrotation $ \overline R \in {\rm SO}(3)$ and the relative stretch $ \overline U = \overline R ^T{\rm D}\varphi$. The curvature variable $ \overline R ^T\operatorname{Curl} \overline R $ controls the full first-order variation of the rotation field and yields strong compactness of minimizing sequences. The identities $\operatorname{Cof}( \overline R^T{\rm D}\varphi) = \overline R^T\operatorname{Cof}{\rm D}\varphi$ and $\det( \overline R^T{\rm D}\varphi)=\det{\rm D}\varphi$ then allow the minors of the lifted stretches to be identified through the weak continuity of the corresponding minors of the deformation gradients. We prove existence in two regimes. The first allows convex dependence on $( \overline U ,\operatorname{Cof} \overline U ,\det \overline U )$ and assumes separate cofactor coercivity. The second depends only on $( \overline U ,\det \overline U )$ and requires no independent cofactor bound. For the constrained model, the weakly closed condition $ \overline R ^T{\rm D}\varphi\in\operatorname{Sym}^{+}(3)$, together with $\det{\rm D}\varphi>0$, implies $ \overline R =R:=\operatorname{polar}({\rm D}\varphi)$ for every admissible pair. The lifted problem is therefore equivalent to a deformation problem containing the curvature of the polar factor. The resulting model is a selectively rotationally regularized nonlinear elastic model of couple-stress type, rather than a purely first-gradient Biot model.

In this paper, we study the renormalization of a three-dimensional quartic model on a compact connected Riemannian manifold with a boundary. We show that, in addition to the standard shift of the mass parameter, it is necessary to take into account boundary contributions involving singular coefficients. This indicates the need to extend the classical action within the framework of standard renormalization theory. The requirements of consistency regarding the gluing of manifolds lead to renormalization of an operator on the boundary.

We study a time-discrete, finite-particle Consensus-Based Optimization (CBO) algorithm in a separable Hilbert space. Our analysis provides convergence guarantees directly for the computable particle system, complementing recent continuous-time and mean-field results in infinite dimensions. Using a common-noise formulation with trace-class covariance, we first establish quantitative pairwise contraction, exponential decay of the expected swarm variance, and almost-sure convergence of all particles to a common consensus state. We then combine estimates on the exponentiated objective functional with a quantitative Laplace principle to show that, for sufficiently large inverse temperature and suitably prepared initial data, the energy of the limiting consensus state can be made arbitrarily close to the global minimum over the active subspace. A key feature of the Hilbert-space formulation is that the stochastic contribution to the convergence estimates is controlled by the trace of the covariance operator and is therefore uniform with respect to the Galerkin dimension. Numerical experiments on an elliptic energy minimization problem with mixed boundary conditions and a PDE-constrained inverse source problem validate the theoretical convergence results and demonstrate stable performance under increasing spatial resolution, in contrast with CBO based on isotropic finite-dimensional noise.

Escobar (J Funct Anal 165(1):101-116, 1999) conjectured that for every $n\ge 3$, an $n$-dimensional compact Riemannian manifold with nonnegative Ricci curvature and all boundary principal curvatures bounded below by $\kappa>0$ must satisfy $\sigma_1\geq \kappa$. We disprove this conjecture for every $n\geq 3$ by constructing conformal deformations of the Euclidean unit ball. We first establish a perturbative criterion, then construct explicit polynomial conformal factors satisfying this criterion. For every sufficiently small $t>0$, the resulting metrics $g_t=e^{2t\Phi}g_{\mathbb{R}^n}$ have positive Ricci curvature, every boundary principal curvature is strictly larger than $1$, and $\sigma_1(\mathbb{B}^n,g_t)<1$. The proof requires several computations, some of which were carried out in Mathematica. The Mathematica code is attached to this submission.

In this paper, we construct deterministic matrices from subspaces of orthogonal spaces over finite fields of odd characteristic and investigate their applicability to compressed sensing. The construction is based on incidence relations among three types of subspaces, yielding families of matrices with explicitly computable dimensions and coherence. Using coherence-based estimates, we establish sufficient conditions under which these matrices satisfy the Restricted Isometry Property for prescribed sparsity levels. We also provide numerical comparisons with DeVore's deterministic construction to illustrate the trade-off between the number of measurements, coherence, and sparse recovery guarantees.

We consider the two-dimensional dissipative generalized surface quasi-geostrophic equations given by \[\partial_t \theta+u\cdot \nabla \theta+\nu (-\Delta)^{\alpha}\theta=0,\quad u=\nabla^{\perp}\Lambda^{\beta-2}\theta\] and establish global existence of weak solutions with initial data in $\dot{H}^{\frac{\beta}{2}-1}(\mathbb{R}^2)$. Our result extends the framework of Marchand's solutions (Comm. Math. Phys. 277 (2008), no. 1, 45-67) to this larger family of active scalar equations.

We prove the existence of small-amplitude traveling shock profiles for the one-dimensional Boltzmann equation of a binary gas mixture with angular cutoff potentials in the full range $-3<\gamma\le 1$. The result extends the classical construction of Caflisch and Nicolaenko from hard potentials to the cutoff soft-potential regime. Indeed, the argument of proofs combines a Lyapunov–Schmidt reduction of the macroscopic component to a Burgers equation with an accelerated backward bi-characteristic method and a weighted $L^2$–$L^\infty$ iteration. Acceleration restores a uniformly positive collision frequency, compensating for the lack of a spectral gap for soft potentials, while the $L^2$–$L^\infty$ framework accommodates the absence of velocity smoothing induced by the cutoff, including a possible singularity along the grazing characteristic $v_1=s$. The shock profile tends to the Rankine–Hugoniot bi-Maxwellians at a mixed exponential rate as $|x|\to \infty$, with a sub-exponential remainder of order $|\varepsilon x|^{2/(3-\gamma)}$.

Let $C\subset X(1)\times X(1)$ be a smooth curve defined over $\mathbb{C}_p$ with irreducible reduction. We study the intersection loci of $C$ with the modular subvarieties $Y_0(n)$. If the curve $C$ avoids points where both coordinates have supersingular reduction, or if a sequence $Y_0(a_n)$ is taken where the numbers $a_n$ have increasing divisibility by $p$, then these loci equidistribute to the unique canonical point of the analytification, $C^\text{Berk}$. If neither condition is satisfied, we expect equidistribution to fail and give a family of examples whose behaviour is believed to be typical. We also study the accumulation points of the set $C\cap\bigcup_nY_0(n)$ and show that this set is always non-discrete.

Freshness is a key attribute of perishable goods, particularly in the food sector, and strongly influences consumer purchasing behavior. Retailers often prioritize high freshness to remain competitive, yet maintaining this standard under demand uncertainty increases the likelihood of unsold inventory and waste. Retailers use markdowns to mitigate this risk, but the timing and depth of discounts must be carefully coordinated to avoid lost revenue or excess waste. This paper studies how retailers can jointly optimize order and markdown pricing decisions to balance profitability and waste reduction for perishable products within an integrated decision-making framework. The joint ordering-markdown pricing problem is formulated as a stochastic dynamic programming model that accounts for the demand cannibalization between regular price and markdown sales. The conflicting objectives of profit maximization and waste reduction are captured through a weighted objective function that reflects the retailer's trade-off preferences. To manage the complexity of the model, we propose an efficient solution approach and conduct numerical experiments to assess the performance of alternative markdown policies. Our results show that implementing multiple early price reductions with successive price reductions can simultaneously increase revenue and reduce waste. Our findings generate several key managerial insights: (i) firms can tailor their markdown policies according to their strategic priorities of profit maximization, waste reduction, or a balanced combination of both, (ii) early and well-timed markdowns based on available inventory levels allow firms to simultaneously attract quality-seeking customers and price-sensitive segments, (iii) implementing a second or multiple markdowns can further reduce waste while also increasing revenue.

Texture is commonly described through statistical descriptors, filter-bank responses, or multiscale representations, yet its functional-analytic structure remains largely unexplored. In this work we investigate texture from the viewpoint of Gaussian scale-space evolution, interpreting it as the persistence of local structures across scales. This perspective naturally leads to a multiscale texture operator that quantifies the evolution of local image structures under Gaussian diffusion. We establish the fundamental analytical properties of this operator, proving its boundedness on $L^p(\mathbb{R}^n)$, its infinite-order smoothing effect in the Hilbert setting, and an explicit spectral characterization showing that it acts as a localized Gaussian band-pass filter. Motivated by the logarithmic organization of Gaussian scales, we introduce an $r$-adic Difference-of-Gaussians discretization and prove Littlewood–Paley-type stability estimates together with an explicit reconstruction formula. Building on this multiscale representation, we define multiscale texture norms that quantify the persistence of local structures across scales. These norms induce a natural family of Banach spaces, with a distinguished Hilbertian case, and we prove that they provide an equivalent characterization of the classical Besov and Sobolev scales. Consequently, classical functional regularity can be described intrinsically through the persistence of texture under Gaussian scale evolution. These results establish the first functional-analytic characterization of texture persistence under Gaussian scale evolution, revealing that classical Besov and Sobolev regularity admit an equivalent description in terms of multiscale texture persistence. This provides a rigorous bridge between Gaussian scale-space theory, harmonic analysis, and variational models based on texture.

We consider compact immersed surfaces of genus zero in a three-dimensional smooth metric measure space $(M^3,g,e^{-f}\mathrm{d}V_{g})$. We introduce a new weighted mean curvature and prove that any such surface whose weighted mean curvature is constant must be totally umbilical, provided that a condition relating the Ricci curvature of $M$ and the weight function $f$ is satisfied. We also give an application of this result to surfaces in $\mathbb{R}^3$ satisfying $|x|^{2} H + 2\langle x,\nu \rangle = c$.

The notation $\psi(x,t)$ is among the first pieces of quantum mechanics that students learn. It is also among the easiest to over-interpret. Because $x$ and $t$ occur as arguments of the same function, students may ask whether they have the same mathematical status. They may also ask whether $\psi(t)$ should require a generalized bra $\bra{t}$ in the same way that $\psi(x)=\braket{x}{\psi}$ is often written. A related question is whether the absence of a universal time operator follows simply from Pauli's argument. These questions mix several structures that are usually introduced in different parts of the curriculum. We present a unified pedagogical treatment built around two maps hidden in $\psi(x,t)$. Time evolution selects a state along a trajectory in Hilbert space. A spectral representation then maps that state to amplitudes labelled by outcomes of a chosen observable. We formulate the position representation without generalized eigenkets. We recover Dirac's $\ket{x}$ notation as a controlled continuum shorthand and use a finite-grid limit to show where delta normalization enters. We distinguish background coordinates, translation parameters, observables, spectral labels, and physical records. We also clarify the Stone-theorem analogy, compare prescribed-time position measurements with arrival-time measurements, state what the strong form of Pauli's argument excludes, and exhibit an exactly solvable boundary case in which a canonical self-adjoint time observable exists. Spin, circuit-QED, and optical-clock examples provide experimentally grounded checks. The aim is not a new interpretation of time. It is a reusable teaching framework for separating mathematical role from notation.

We prove that if a smooth affine complex surface has the same rational mixed Hodge structure as a smooth projective curve of positive genus, then it admits a smooth morphism to the curve with fibers isomorphic to affine line. In particular, every such surface is the total space of a torsor for a line bundle on the curve. The motivation comes from Jouanolou's trick, to which this result gives a kind of converse in dimension 2.

We prove that if $x\in (X,\Delta,\mathbb{T})$ is a log Fano cone singularity over an uncountable algebraically closed field, and $\nu_0$ is a $\mathbb{T}$-invariant valuation with center $x$ and $A_{X,\Delta}(\nu_0)<\infty$, then the value $$ \delta(X,\Delta;\nu_0):=\inf_{\nu\in \mathrm{Val}^{\mathbb{T},*}_{X,\ni x}}\frac{A_{X,\Delta}(\nu)}{S(\nu_0;\nu)}$$ admits a minimum. The proof uses the generic limit argument. Note that there is a counterexample if we discard the log Fano cone structure.

We establish that only finitely many powers of $3$ can have a simple binary representation.

In this paper, we provide a complete classification of real three-dimensional simple Bol algebras $(B, [.,.], \langle .,.,. \rangle )$. The main part of the classification concerns the case $[B,B]=B$. In addition, we identify two isomorphism classes with a nonzero binary product, both satisfying $\dim[B,B]=2$, as well as four simple classes of Lie triple systems in which the binary product vanishes.

We study decentralized stochastic optimization over a network of $N$ agents under compressed communication. We propose CED-EF, an exact diffusion-based method with error feedback that directly accommodates biased $\delta$-contractive compressors while communicating one compressed model-sized vector per node per iteration. For smooth nonconvex objectives with unbiased stochastic gradients whose variance is bounded by $\sigma^2$, where $\sigma\geq0$, we establish a convergence rate whose leading stochastic term is $\mathcal O(\sigma/\sqrt{NK})$. For $\sigma>0$, the dominant dependence of the corresponding transient time on the number of agents, compression level, and spectral gap $\Delta_\lambda$ is $\mathcal O(N^3/(\delta^4\Delta_\lambda^4))$, with fixed problem-dependent factors suppressed. Under the Polyak–Łojasiewicz condition, CED-EF attains a leading stochastic term $\widetilde{\mathcal O}(\sigma^2/(NK))$ with transient time on the order of $\widetilde{\mathcal O}(N/(\delta^2\Delta_\lambda^2))$. These dependencies improve the compression and/or network dependence of existing results. Numerical experiments on least-squares and logistic-regression problems illustrate the performance advantages of CED-EF.

Tuza's conjecture states that for any graph $G$, the minimum size of a triangle transversal $\tau(G)$ is at most twice the maximum size of a set of edge-disjoint triangles $\nu(G)$. In this note, we prove $\tau(G) \leq \frac{63}{22}\nu(G)$, improving the previous bound $\tau(G) \leq \frac{66}{23}\nu(G)$ established by Haxell in 1999. The key observation is that for a "2-colorable" family of triangles $\mathcal{F}$, where each triangle has two blue edges and one red edge, we can obtain $\tau(\mathcal{F})\leq (1+\sqrt{3})\nu(\mathcal{F})$.

We demonstrate a new lower bound on the sensitivity of $\phi$-OTDR by estimating phase using Kalman filtering. Numerical analysis shows a $\sim$35-dB reduction in the minimum detectable SNR, compared to when using conventional phase estimation.

A formal large-degree phase expansion associated with the zeros of the Jacobi polynomial $P_n^{(\alpha,\beta)}$ contains a sequence of constants $\kappa_1,\kappa_2,\dots$, each of which is a polynomial in $\alpha^2$ and $\beta^2$ with rational coefficients; for instance $\kappa_1=(\alpha^2-\beta^2)/4$. This paper asks a simple arithmetic question: which primes, and to which powers, occur in the denominators of these coefficients? We prove that the odd part of the common denominator of $\kappa_r$ divides $\operatorname{lcm}(1,3,\ldots,2r-1)$, and that the power of $2$ is at most $2^{E_r}$ with $E_r=3r-1+\nu_2((r-1)!)$, where $\nu_2$ is the $2$-adic valuation. The study of the leading coefficient leads to the formula \[ \nu_2\!\left(\sum_{j=0}^{m}\binom mj\frac1{2j+1}\right)=m+\nu_2(m+1)\qquad(m\ge0), \] which we prove by showing that the series $\sum_{k\ge0}k!/(2k+1)!!$, whose real sum is $\pi/2$, converges to $0$ in the field $\mathbb Q_2$ of $2$-adic numbers. Finally, we show that the statement “the exponent $E_r$ is attained for every $r$” is equivalent to an integrality property of the Taylor coefficients of a single explicit power series in one variable. That property is established in a separate paper, and is confirmed here independently, by exact computation, for $r\le48$.

The classical Hamburger moment problem can be viewed as an interpolation problem for Nevanlinna functions. A closely related problem is the Nevanlinna–Pick interpolation problem. Each of these problems is either determinate, with a unique solution, or indeterminate, with infinitely many solutions. Hamburger established a criterion for the indeterminacy of the moment problem in terms of the convergence of two series involving the classical polynomials of the first and second kind. In this paper, we establish an analogous criterion for the matrix Nevanlinna–Pick interpolation problem. The criterion is formulated in terms of the convergence of matrix series involving the rational functions of the first and second kind.

Two-stage sample robust optimization with linear decision rules is a standard data-driven approach to two-stage stochastic linear programs with unknown distributions. \cite{bertsimas2022two} established that this approach is asymptotically optimal under several conditions. Most conditions are mild, but one imposes a technical requirement on the recourse system. Whether this condition is implied by the others was left as an open question. We resolve this open question through a geometric feature of the support set: simpleness, the property that every vertex is incident to exactly as many edges as the dimension of the support. For simple support sets, the technical condition is automatic, and asymptotic optimality holds under the remaining standard assumptions. For every non-simple support, we construct a recourse system that is feasible everywhere in the support but violates the technical condition, and this construction resolves the open question. We further characterize the simpleness of one-sided and two-sided budgeted uncertainty sets and identify when budget perturbations can restore simpleness. We also develop a polynomial-time algorithm that certifies the technical condition at a given vertex using its incident edge directions. Numerical experiments on newsvendor and network lot-sizing problems illustrate the characterization and the effectiveness of the verification algorithm.

We prove a positive mass theorem for complete Riemannian manifolds with noncompact boundary, a distinguished asymptotically flat half-space end, and finitely many additional complete ends with no prescribed asymptotics. If $3\leq n\leq7$, $R_g\geq0$, and $H_{\partial M}\geq0$, then $ \mathfrak m(M,g,\mathcal E)\geq0$. Moreover, equality holds if and only if $(M,g)$ is isometric to the Euclidean half-space. The proof combines a density deformation near the distinguished end with doubling across the noncompact boundary, local smoothing, and a conformal correction. We also obtain the sharp Riemannian Penrose inequality when a compact outermost minimal hypersurface separates $\mathcal E$ from all the remaining ends; equality holds precisely when the exterior region is a Schwarzschild half-space exterior.

Kellogg's checkerboard interface problem is a classical benchmark for robust adaptive finite element methods. Its successful adaptive meshes are radial: they refine strongly toward the interface crossing but show no angular structure, despite the large contrast and the asymmetric solution. We explain this by proving that the singular solution $u(r,\theta)=r^\gamma\mu(\theta)$ satisfies the exact identities $\kappa|\nabla u|^2=\Lambda r^{2\gamma-2}$ and $\kappa|\nabla^2 u|_F^2=2(1-\gamma)^2\Lambda r^{2\gamma-4}$, with $\Lambda=\gamma^2\cos^2(\pi\gamma/4)$ and the Hessian taken separately in each quadrant. The point is what has disappeared: the right-hand sides depend on $r$ alone, although $\kappa$ and $u$ each depend on the angle as well. Combined with equal discretization-error distribution, this shows that the target element density is radial, so a correct mesh should display nothing but refinement toward the center, and a Kellogg mesh that is not radial is visible evidence that the computation is not following the coefficient-weighted local difficulty. The reading is specific to this benchmark: on a second interface problem the same estimator correctly produces a strongly material-biased mesh, with a computed element-count ratio of $3.934$ against the predicted $4$. A byproduct gives the benchmark constants in closed form, so the problem data can be generated from $\gamma$ alone at any precision.

Sparse bounds have proved to be a powerful tool which simplify the understanding of many important operators in harmonic analysis. The existence of a sparse bound for an operator implies the weighted boundedness of the operator for weights within the Muckenhoupt classes. As in the example of Calderón-Zygmund operators, this implication can prove weighted boundedness with the best possible dependency on the characteristic constant of the Muckenhoupt weight. In this paper, we prove a partial converse for sub-additive operators by showing that weighted boundedness, with an arbitrary a priori control of the operator norm in terms of the characteristic constants of the weight, implies the existence of sparse form bounds. This result implies that a near-optimal dependency of the operator norm on characteristic constants of the weights follows from the weighted boundedness itself, rather than relying on the particular structure of the sub-additive operator. Moreover, it allows Rubio de Francia's extrapolation theorem to be extended to the weak-type end-point for sub-linear operators and leads to an interpolation result for sparse form bounds for analytic families of operators. As an application, we obtain new sparse form bounds for rough singular integral operators.