Papers

Let $R_2 = \mathbb{F}_{p^m} + u\mathbb{F}_{p^m}$ ($u^2 = 0$), where $p$ is an odd prime and $m, s \in \mathbb{N}$. For $p \equiv 3, 5 \pmod 7$ with $\gcd(m, 6) = 1$, the cyclotomic polynomial $\Phi_7(x)$ is irreducible over $\mathbb{F}_{p^m}$. This yields a direct sum decomposition $C = C_1 \oplus C_2$ for any cyclic code $C$ of length $7p^s$ over $R_2$, where $C_1$ has length $p^s$ and $C_2$ is a $7$-cyclotomic code of length $6p^s$. We classify $7$-cyclotomic codes into four disjoint generator-based types and calculate exact cardinalities using residue and torsion subcodes. Furthermore, explicit generators for the Euclidean dual codes $C^\perp$ are determined. As operational applications of these classified codes, we construct new families of quantum stabilizer codes via the CSS framework and establish parameter criteria for linear codes with complementary duals (LCD codes).

We establish a dual theory of Zhu's $A(\mathcal {V})$-theory, i.e., for a graded vertex operator coalgebra $V$, the coassociative coalgebra $C(V)$ are investigated, any admissible $V$-comodule gives a $C(V)$-comodule, and vice versa. We also prove that $V$ is corational, which means every admissible $V$-comodule is completely reducible, if and only if its dual vertex operator algebra $V'$ is rational.

Let $(M,J)$ be a compact almost hermitian $4$-manifold with a smooth embedded $J$-holomorphic curve $C$ representing the canonical class. Motivated by the symplectic Bogomolov–Miyaoka–Yau conjecture, we choose a twisted spin$^{\text{c}}$ Dirac operator ${\mathcal D}$ on $M$ satisfying $$ \operatorname{ind}{\mathcal D}=3c_2(M)-c_1^2(M). $$ We then use the curve $C$ to construct a complex-linear perturbation ${\mathcal A}$ of ${\mathcal D}$ whose singular set is $Z_{\alpha}\sqcup C$, where $Z_{\alpha}=\alpha^{-1}(0)$ for a transverse section $\alpha$ of $\Lambda^{1,0}M$ nonvanishing on $C$. Applying Maridakis' index localization theorem, we express $\operatorname{ind}{\mathcal D}$ as the sum of localized contributions: $3c_2(M)$ from $Z_{\alpha}$ and $-c_1^2(M)$ from $C$.

We develop semi-closed pricing formulas and lifted-model hedging methods for discretely monitored geometric and arithmetic Asian options in the Volterra-Heston stochastic volatility model. Exploiting the affine Volterra structure, we derive a tractable transform for the joint law of the terminal log-price and the discretely monitored geometric average. This transform yields semi-closed pricing formulas for geometric Asian options, which in turn provide effective control variates for Monte Carlo valuation of arithmetic Asian options. Under the stated real-moment and affine-transform hypotheses, we also derive the Galtchouk-Kunita-Watanabe decomposition for Fourier-representable payoffs and obtain a variance-optimal hedge in terms of the Riccati-Volterra equation and the forward-variance curve. Using N-factor Markovian approximations, we obtain a finite-dimensional numerical implementation for hedging Asian options. Our numerical experiments document factor convergence for a regular non-Markovian kernel and the effect of rebalancing frequency on hedging error. In the Heston benchmark, geometric Asian controls substantially reduce the variance of arithmetic-Asian price estimates and improve the finite-sample stability of regression-based hedging relative to direct regression.

Aggregation of estimation procedures has found important applications in econometrics, statistics, and machine learning. Classical statistical aggregation theory has mainly focused on a pure aggregation setting, where the candidate estimators are either deterministic or constructed using a held-out sample independent of that used for aggregation. When the candidate estimators and the aggregation weights are estimated from the same dataset without sample splitting, Bellec [Ann. Statist. 46 (2018), 30–59] established the optimal rate for model-selection aggregation of affine estimators. Beyond this regime, however, fundamental questions regarding the optimal aggregation rates and the construction of aggregation rules attaining these rates remain largely unresolved. In this paper, we consider the problem of aggregating a finite collection of affine estimators to learn an optimal convex combination of them. This framework encompasses a rich class of estimators widely used in statistics and machine learning, including least squares estimators, kernel ridge estimators, random feature regression estimators, and among many others. We establish the minimax rate for convex aggregation of affine estimators. In particular, we show that estimating the weights by minimizing a Mallows' $C_p$ criterion attains the optimal rate. We further study the linear aggregation regime with unrestricted weights and establish matching minimax lower and upper bounds over suitable classes of affine estimators.

A corollary of Hall's marriage theorem is that a sufficient condition for a list $(V_1, \ldots ,V_m)$ of sets to have a system of distinct representatives is that $|V_i|\ge deg_{\{V_1, \ldots ,V_m\}}(v)$ for every $i\in [m]$ and $v \in \bigcup_{i\in [m]}V_i$. This we dub a global condition. A folklore result is that a local condition - that the inequality holds for pairs $i,v$ for which $v \in V_i$ - suffices. These are special cases of a general type of results - large sets, whose elements are sparse in some sense, have a system of representatives that is independent in a related graph. We study two such scenarios, in both of which each $V_i$ is replaced by a $k$-uniform hypergraph $H_i$, the representatives are hyperedges, and distinctness is replaced by disjointness. In one setting the sparsity is measured by the degrees of vertices in the hypergraphs, in the other by the degrees of vertices in the line graph. The proofs use the topological version of Hall's theorem. In particular, we shall use a lower bound on the topological connectivity of the independence complex of a graph, defined by vector representations. We also provide short proofs of the local infinite version, known as the “Milner-Shelah theorem”.

We prove that the capillary $L_p$ dual Minkowski problem in $\mathbb{R}^{n+1}$, $n\geq3$, admits a unique positive smooth strictly convex solution for any prescribed positive smooth function $f$ when $p>q$ and the contact angle satisfies $\theta\in(0,\pi/2)$. Earlier results for $p>q$ either required the additional condition $q\leq1$ or were restricted to $n=2$. We reduce the problem to a Monge–Ampère type equation with a Robin boundary condition on a spherical cap. By constructing a new auxiliary function for the global $C^2$ estimate, we overcome these limitations and completely resolve the case of $p>q$ in higher dimensions, i.e., prove existence and uniqueness throughout the case of $p>q$, without any further restriction on $p$ or $q$.

We establish conditional central limit theorems in Fisher information under log-concavity in every fixed dimension. For conditionally centered normalized sums, after whitening by the averaged conditional covariance, the averaged conditional Fisher information converges to the dimension if and only if it is finite at one convolution level. The scalar criterion follows as the one-dimensional case; we also provide an independent scalar proof based on a second-order continuity theorem for Fisher production on Gaussian-smoothed, tail-controlled classes. For the original sums, the averaged Fisher information matrix converges in operator norm to the inverse averaged conditional covariance. Consequently, the conditional relative Fisher information with respect to the limiting Gaussian law vanishes, and the Gaussian logarithmic Sobolev inequality yields convergence in conditional relative entropy and conditional entropy. We give two operational consequences. For any fixed finite-constellation low-power input, the first-order conditional mutual-information slope converges to the Gaussian-noise benchmark. For Gaussian signaling at any fixed signal covariance, the mutual-information gap from that benchmark is bounded by the conditional relative Fisher deficit and hence vanishes asymptotically.

This paper studies the nondominated set for topology optimization of compliance and volume. In the multiobjective topology optimization literature, scalarization is a popular method to approximate this set, which is better known as the Pareto frontier. However, previous work indicates that the pointwise approximation obtained by scalarization obfuscates its underlying mathematical structure: under an imposed length scale, it consists of locally nondominated curve segments. To investigate this structure, the paper uses a simple methodology to (i) generate local optima with a tightly controlled topology and (ii) use continuation to extend these points to locally nondominated curves. The paper then performs two rounds of numerical experiments. The first round applies the methodology to several cantilever examples. This leads to novel insights regarding topological complexity and symmetry and confirms prior conclusions regarding continuity, smoothness and convexity. The second round investigates two numerical instabilities: (i) a lack of uniqueness and (ii) discontinuities of the locally nondominated curves. The former is linked to an observed flatness of the optimization landscape, whereas the latter is attributed to branch splitting and linked to bifurcation theory. Namely, we classify these discontinuities as higher-dimensional analogues of the subcritical, symmetry-breaking pitchfork bifurcation. The paper concludes with suggestions for future research to address these features of the multiobjective optimization landscape.

We answer a question of C. S. Herz about the number of integral $k\times m$ matrices $T$ such that $T^tT\leq R$ for a fixed positive definite $m\times m$ matrix $R$. We give an asymptotic formula for the count. This can be seen as a generalization of the Gauss circle problem, as counting representation of quadratic forms by the identity $I_k$, and as counting integral points bounded by the Stiefel manifold $T^tT=R$. The main tool is a bound for Bessel functions of matrix argument that was proved over Jordan algebras.

In 1940, Levin\footnotemark[1] and Milman proved that a closed linear subspace of $C[0,1]$ whose elements all have bounded Jordan variation must be finite-dimensional. We prove its analogue for variation in the sense of Young and, more generally, for every finite-valued nondecreasing gauge $\varphi:[0,\infty)\to[0,\infty)$ with $\varphi(0)=0<\varphi(t)$ for $t>0$. If $E$ is a closed linear subspace of $C[0,1]$ and every $f\in E$ satisfies $\Var_\varphi(\lambda_f f)<\infty$ at some scale $\lambda_f>0$, then $E$ is finite-dimensional. No continuity, convexity, or doubling condition is needed. The proof combines a lower-semicontinuous regularization that preserves the scaled class, Baire uniformization, Helly selection, and a quantitative nested-peaks construction for nonhomogeneous gauges. Consequently, for every infinite-dimensional closed subspace $F\subset C[0,1]$, the set $F\cap\cV_\varphi([0,1])$ is a meagre $F_\sigma$ subset of $F$. For every Young function, both the scaled and raw finite-variation families are maximal dense-lineable but not spaceable. The scaled family is itself a dense vector subspace of Hamel dimension $\mathfrak c$, whereas without local doubling the raw family need not itself be linear.

In this paper, we study the Kähler–Ricci flow on $\mathbb{C}P^m$-bundles over a product of Kähler–Einstein manifolds, starting from an initial metric with Calabi symmetry. We prove that every finite-time singularity arising along the flow must be of Type~I.

We prove a Legendrian Reidemeister theorem for Legendrian knots in thickened convex surfaces.

The Bingham distribution is widely used to model directional data with antipodal symmetry, and its normalizing constant $Z$ and moments play a central role in statistical inference and closure models. Their behavior becomes singular when one or more eigenvalue gaps of the parameter matrix become unbounded. In this work, we develop a uniform asymptotic framework for the Bingham distribution in arbitrary dimensions. Starting from an inverse-Laplace integral representation, we derive asymptotic expansions with explicit remainder estimates that are uniform with respect to arbitrary relative scales among multiple diverging eigenvalues. In a particular regime, the asymptotic series becomes absolutely convergent with an exponentially small remainder. These results yield precise asymptotic profiles of the Bingham moments and characterize the degeneration of higher-dimensional distributions to lower-dimensional counterparts. As an application to the Bingham closure, we prove that the entropy admits a decomposition into an explicit logarithmic leading term and a residual that is uniformly Lipschitz continuous on the moment simplex. Moreover, on each boundary face, the residual agrees with its lower-dimensional version up to an explicit additive constant. The results provide a unified description of the singular structure of the Bingham distribution and have implications for numerical closure and $Q$-tensor models of nematic liquid crystals.

We study near alternating sign matrices with prescribed row and column weights. After recalling the basic definitions and the necessary conditions coming from the Gale–Ryser theorem, we give a graph-theoretic characterization of the existence problem. More precisely, we show that a near alternating sign matrix with prescribed row and column weight sequences exists if and only if there exists a binary array with the same weights whose associated adjacency graph is bipartite. This reformulation shows that the Gale–Ryser conditions are not sufficient in general. We then formulate a natural conjecture up to permutations of the row and column weight sequences, and present some positive results, based on convex binary arrays and composition constructions.

We give a \(3\times 3\) counterexample, valid over every field, to a rank-minimization conjecture of Lin and Wimmer (Bull. Aust. Math. Soc., 84 (3) (2011), 441–443) related to Roth's similarity theorem for the Sylvester matrix equation. We also prove that, over the complex field, no counterexample can occur when one of the two matrix sizes is less than \(3\). Hence the example is dimensionally minimal over the complex field.

For every $n\geq1$, we classify monotone rotation-invariant real-valued valuations on closed spherical convex sets, without assuming continuity or measurability. On proper sets, namely those contained in an open hemisphere, these are precisely the nonnegative linear combinations of the normalized spherical quermassintegrals. On all closed spherical convex sets, they are precisely the linear combinations of the spherical intrinsic volumes with nonnegative, nondecreasing coefficients. The representations are unique, and all such valuations are continuous and invariant under the full orthogonal group. In hyperbolic space, an isometry-invariant real-valued valuation on compact convex sets is continuous if and only if it is a linear combination of the Euler characteristic and the hyperbolic quermassintegrals. This representation is unique. Monotonicity is equivalent to nonnegative coefficients and implies continuity. If monotonicity is required only between nonempty sets, the Euler coefficient is unrestricted in the proper spherical and hyperbolic cases, whereas the classification on all closed spherical convex sets is unchanged. We also obtain corresponding classifications for valuations on closed convex cones that vanish at the zero cone and monotone classifications on compact projectively convex sets contained in an affine chart of real elliptic space.

Every quadratic Hessian slice of a Lorentzian polynomial yields bounded monomial ratios among the normalized coefficients of the polynomial. We determine exactly for which pairs $(n,d)$ these quadratic-slice ratios generate the full bounded-ratio cone for every $M$-convex support $S\subseteq\Delta_n^d$. For $d\geq 2$, this quadratic local-to-global principle holds universally if and only if \[ n\leq 3,\qquad d=2,\qquad\text{or}\qquad (n,d)=(4,3). \] In every remaining case, the principle fails already for Lorentzian polynomials with full support: for cubics in $n\geq 5$ variables and for polynomials of degree $d\geq 4$ in $n\geq 4$ variables. We identify the two minimal obstructions, at $(n,d)=(4,4)$ and $(n,d)=(5,3)$, and propagate them throughout the failure region by degree and variable aggregation.

We give an approach to discover continued fractions for series of the form $$\sum_{k=0}^\infty \frac{\epsilon^k}{(x+k)^s},$$ where $\epsilon = \pm 1$. We find a continued fraction of the form \begin{equation*} \frac{a_1}{b_1(x)} \fplus \frac{a_2}{b_2(x)} \fplus \fdots \end{equation*} where $a_k$ are constants and $b_k(x)$ are polynomials. Our technique involves telescoping continued fractions. This provides a discovery approach to continued fractions given by Ramanujan for $$2\sum_{k=1}^\infty \frac{(-1)^{k+1}}{x+2k-1}, 2\sum_{k=0}^\infty \frac{1}{(x+2k+1)^2}, 2\sum_{k=0}^\infty \frac{(-1)^k}{(x+2k+1)^2}, \sum_{k=1}^\infty \frac{1}{(x+k)^3}, $$ and the like. We display the first few terms of several continued fractions obtained in this manner for larger values of $s$, including $s=5, 7, 9, 11$. They do not follow as simple a pattern as Ramanujan's continued fractions.

A Hadamard difference set (HDS) $D$ of order $u^2$ in an abelian group $G$ satisfies $|\chi(D)|=u$ for every nontrivial character $\chi$ of $G$. We call such a character value naive if it is divisible by $u$, i.e., if it is equal to $u$ times a root of unity. All previously known abelian HDSs only have naive character values. We show that for $d\ge 1$ and $u=3d$, a group $Z_3^2\times H$, with $H$ an abelian group of order $2^{2d+2}$, contains a HDS of order $u^2$ with non-naive character values if and only if $8\le\exp H\le 2^{d+2}$. All difference sets obtained are new. The proof rests on a specific HDS in $Z_3^2\times Z_8\times Z_2$, a covering extended building set on $Z_3^2\times Z_8\times Z_4$, and a variation of the Davis-Jedwab recursive construction.

Fluid antenna systems (FASs) provide reconfigurable spatial resources within compact apertures. In this paper, we introduce code-domain grouped index modulation (CGIM) and its spatial-code-domain extension, termed SCGIM, for FA-assisted transceivers. CGIM partitions the available orthogonal spreading codes into multiple subsets and jointly maps information onto their in-phase and quadrature indices and constellation symbols. In an Rx-FAS-assisted single-input multiple-output (SIMO) link, group-wise despreading separates the orthogonal code groups for parallel detection, while receive-port selection provides spatial diversity. SCGIM further associates interleaved Tx-FA port subsets with the code subsets, with the Tx-FAS conveying spatial-index information and the Rx-FAS providing selection diversity in a multiple-input multiple-output (MIMO) link. For SCGIM, we develop maximum-likelihood (ML), staged greedy (GD), and cross-domain index message-passing (CD-IMPD) detectors. CD-IMPD exchanges soft information over a cycle-free factor graph to account for the coupling between the spatial and code indices, requiring only one inward and one outward message pass. For CGIM, the BER is derived from the joint decision regions of the despread-domain observations and averaged over the Rx-FAS selected-gain distribution under Rayleigh, Nakagami-m, and additive white Gaussian noise channels. For SCGIM, an average-BER approximation is derived from a full-pair union bound using the selected-gain density ratio and exponentially tilted quadratic-form Laplace transforms. Simulation results validate the BER analysis and show that the proposed schemes achieve lower BER and higher throughput than the considered IM schemes, while CD-IMPD achieves near-ML BER performance with lower detection complexity.

This paper presents a physics-aware reduced-order method (ROM) for the efficient computation of wave-scattering problems. Standard model order reduction techniques, typically treating scattering as generic parameterized systems, frequently overlook the underlying physical structure, limiting their effectiveness in practice. To address this limitation, we propose an algorithmic framework that utilizes the intrinsic low-rank structure of the induced contrast source density. By modeling the incident wave as a random variable governed by a specified prior probability measure, we formulate the contrast source as a spatial random field whose covariance function captures essential spatial correlation and physical interactions. The reduced-order basis is then constructed via the Karhunen-Loève (KL) expansion, effectively extracting the dominant features from the scattering process to resolve multiple scattering scenarios. A central algorithmic contribution is the efficient reconstruction of the covariance matrix for arbitrary scatterer geometries and specified incident wave priors. To circumvent the prohibitive computational cost of assembling high-fidelity covariance matrices, we introduce a non-intrusive, physics-informed graph neural network (GNN) surrogate to learn the complex mapping from scatterer geometry to the source correlation kernel, enabling a highly efficient offline-online computational paradigm suitable for large-scale scattering configurations. Extensive numerical experiments demonstrate that the proposed framework achieves robust computational acceleration over full-order models without sacrificing accuracy.

We establish logarithmic upper bounds for nonnegative multiplicative functions evaluated at values of multivariable polynomials. The polynomial contribution is encoded by the permutation character on the geometric irreducible components of the corresponding hypersurface, while the arithmetic contribution is described by a class function on a joint Chebotarev–Sato–Tate group. This yields a unified framework for symmetric-power coefficients of CM and non-CM modular forms, Dedekind zeta-function coefficients, and mixed automorphic–Galois weights. As a further application, we obtain quantitative divisibility results for Fourier coefficients along polynomial values, including explicit formulas in the full residual-image and Eisenstein congruence cases.

We prove an impossibility theorem for uniformly consistent recovery of labeled edge probabilities from complete aggregated relational data, even when the population count law identifies every probability. For any known partition into two equal trait groups, we consider an independent-edge logistic network with a balanced rank-one signal and unknown activity effects with bounded total dyadic energy. The signal amplitude is known and fixed at a small positive value. Every node reports its counts to both groups. The minimax mean squared error for the probability matrix remains bounded away from zero as the network grows, whereas it tends to zero under full adjacency on the same parameter class. The lower bound accounts for the dependence across the entire count array through a two-node oracle from which all observed counts can be reconstructed. Conditional binomial smoothing bounds the information about a local sign orientation, and an anchored many-bit construction converts these ambiguities into a nonvanishing normalized matrix loss. A mixed-cumulant identity recovers every edge probability from the population degree law, locating the obstruction in estimation from a single aggregated network rather than in population identification.

The Bott-Samelson varieties provide natural desingularizations of Schubert varieties, and their generalizations, called generalized Bott-Samelson varieties, were constructed by Nicolas Perrin as towers of locally trivial fibrations with fibers isomorphic to Schubert varieties. In this article, we give a complete characterization of Fano and weak Fano generalized Bott-Samelson varieties. As a consequence, we recover the corresponding results for ($G$-)Bott-Samelson varieties and minuscule generalized Bott-Samelson varieties.

We establish the asymptotic stability of planar viscous shock waves for the outflow problem of the three-dimensional barotropic compressible Navier–Stokes equations in a half-space, with periodic boundary conditions imposed in the transverse directions. For a weak shock located sufficiently far from the boundary, we prove that, under small perturbations in $H^{2}$, the outflow problem admits a unique global-in-time solution and that the solution converges uniformly to the viscous shock, up to a dynamical shift, whose velocity time-asymptotically decays. This provides the first shock-stability result for the multidimensional outflow problem. Our proof combines the $a$-contraction method and higher-order energy estimates adapted to the half-space boundary. The boundary trace remaining in the zeroth-order estimate is controlled by differentiating the perturbation system in the tangential directions and exploiting the favorable sign of the outflow flux. The highest-order normal derivatives are then recovered from the time and tangential derivatives through the momentum equation, while the boundary terms generated by the dynamical shift are controlled using the exponential decay of the viscous shock tail.

In this paper, we discuss recent developments and open problems concerning the geometry of compact Kähler varieties whose tangent sheaves are pseudo-effective in a strong sense. As our main result, we prove that, after passing to a finite quasi-étale cover, any compact Kähler variety with quotient singularities and pseudo-effective tangent sheaf admits a flat fibration onto a complex torus, with rationally connected fibers. We also obtain an analogous structure theorem for klt compact Kähler varieties, assuming the existence of minimal models in numerical dimension zero; consequently, the result holds unconditionally in the projective setting and for low-dimensional compact Kähler varieties.

Let $W=n^{-1/2}\sum_{i=1}^n X_i$, where the $X_i$ are independent centered random vectors in ${\mathbb R}^p$ with $|X_{ij}|\le B$ almost surely. Suppose that $\text{Cov}(W)$ has unit diagonal and smallest eigenvalue at least $b^2>0$. We prove that the distance between $W$ and a Gaussian vector with the same covariance, uniformly over axis-aligned rectangles, is at most $C\min\{1,b^{-2}Bn^{-1/2}\log^{3/2}(ep)\}$. For fixed $b$, the dependence on summand size and dimension matches known lower bounds in growing-dimensional regimes. The proof combines a concentration estimate near rectangle boundaries with a carefully chosen Gaussian comparison.

Let $(F_n)$ be a sequence of random vectors with identity covariance matrix whose components belong to the same fixed Wiener chaos, and assume that $F_n$ converges in law to a standard Gaussian vector. We prove that, for every integer $m\geq0$ and every $p\in[1,\infty]$, the optimal rate of convergence of the densities in the Sobolev space $W^{m,p}$ is given by the maximum of the absolute third-order cumulants and the diagonal fourth-order cumulants. The same quantity also gives the optimal rates in total variation, Kolmogorov and $1$-Wasserstein distances. Our proof first derives, by Gaussian interpolation and Gaussian convolution, a cumulant expansion in the space of tempered distributions without imposing Malliavin nondegeneracy at the endpoint. Finite-order Malliavin density estimates then upgrade this identity to Sobolev spaces. Matching lower bounds follow from a finite-dimensional argument in which parity separates the third-order and fourth-order Gaussian corrections, while norm equivalence rules out cancellations among mixed cumulants. A superconvergence theorem supplies the required finite negative moments of the Malliavin determinant along a sufficiently far tail of the approximating sequence, so that no Malliavin nondegeneracy assumption is required.

We show that the \(q\)-Gaussian von Neumann algebras associated with an infinite-dimensional real Hilbert space are pairwise nonisomorphic as \(q\) ranges over \((-1,1)\).