Papers

Let $\mathcal{H}_d(M)$ denote the space of harmonic functions with polynomial growth of degree at most $d$ on a complete Riemannian manifold $(M,g)$. Yau raised two fundamental questions regarding $\mathcal{H}_d(M)$ on complete manifolds with nonnegative Ricci curvature. The first question is the finite dimensionality of $\mathcal{H}_d(M)$, which was confirmed by Colding and Minicozzi. The second question asks whether a sharp upper bound given by its Euclidean analog $\operatorname{dim}\mathcal{H}_{d}(\mathbb{R}^n)$ holds. We verify that the second question is true on locally conformally flat manifolds. Indeed, one can precisely determine the value of $\dim \mathcal{H}_d(M)$ case by case.

In this paper, we investigate the relationships between the $\beta$-$n$-sensitivity and other forms of sensitivity in discrete dynamical systems. And we present some different sufficient conditions (equivalent condition) to be (strongly) $\beta$-$n$-sensitive. These results improve and extend some existing ones. Moreover, we introduce and study the semi-strong $\beta$-$n$-sensitivity and (strong) multi-$\beta$-$n$-sensitivity with respect to $\alpha$. Besides, we proved that strongly multi-transitivity implies strongly multi-sensitivity, which improves some existing ones. Finally, we obtain that (semi-)strongly $\beta$-$n$-sensitivity is preserved under iterations but $\beta$-$n$-sensitivity fails.

In this paper, we first construct many simple restricted modules for the $N=2$ Neveu-Schwarz algebra $\mathcal{L}$. These modules contain the highest weight modules and Whittaker modules. Then we precisely characterize the simple restricted modules over $\mathcal{L}$ under certain conditions, and establish several equivalent conditions for a simple \(\mathcal{L}\)-module to be restricted. Finally, we prove that if one specific positive root element of $\mathcal{L}$ acts locally finitely on a simple $\mathcal{L}$-module $V$, then $V$ must be restricted.

We study the Gromov width of smooth generalized Bott-Samelson varieties, a class of projective varieties constructed by Perrin in \cite{Per07} as a generalization of classical Bott-Samelson resolutions of Schubert varieties. We show that the Gromov width of such a variety equipped with a rational Kähler form is given by the symplectic area of its minimal rational curves. As a consequence, we obtain upper bounds for the Seshadri constants of these varieties with respect to ample line bundles.

With regard to the Hadwiger-Nelson problem, several 5-chromatic unit distance graphs in the Euclidean plane have been discovered in recent years. While most constructions rely heavily on the Moser spindle, a few recent examples completely avoid it, the smallest one consisting of 1441 vertices. In this note, we introduce an original geometric approach to constructing such graphs by utilizing the arcs of a 7-fold symmetric unit distance graph on 21 vertices, and obtain a Moser-spindle-free 5-chromatic unit distance graph on 2131 vertices. While this is not a record small result, it arises from a straightforward, structured rule rather than a purely automated or brute-force search.

We prove an infinitesimal circular version of Morera's theorem. Let $D\subset\mathbb{C}$ be a domain and let $f\in C(D)$. If, at every $a\in D$, $\int_{\vert{}\zeta-a\vert{}=r}f(\zeta)\,d\zeta=o(r^2)$ as $r\to0^+$, then $f$ is holomorphic in $D$. In particular, exact vanishing of all sufficiently small centered circular integrals implies holomorphicity. The proof uses a local distributional $\partial$-primitive, a circular identity for weak $\partial$-derivatives, and a pointwise asymptotic mean-value criterion for harmonicity.

In this paper, we introduce the notion of four-angle Hopf modules over a BiHom-Hopf algebra $H$. We show that the category ${}_{H}^{H}\mathfrak{M}_{H}^{H}$ of BiHom-four-angle Hopf modules admits a strict monoidal category structure with respect to either the BiHom-tensor product $\otimes_{H}$ or the BiHom-cotensor product $\square_{H}$ as its monoidal product. We prove that the category $\mathcal{YD}_{H}^{H}(m,n,p,q)$ of BiHom-$(m,n,p,q)$-Yetter-Drinfel'd modules with parameters $m,n,p,q\in\mathbb{Z}$ forms a strict braided monoidal category equipped with a new monoidal product. Furthermore, we establish monoidal equivalences between the monoidal categories $\mathcal{YD}_{H}^{H}(m,n,p,q)$ and ${}_{H}^{H}\mathfrak{M}_{H}^{H}$, where ${}_{H}^{H}\mathfrak{M}_{H}^{H}$ carries either $\otimes_{H}$ or $\square_{H}$ as its monoidal product. Finally, we construct braiding structures for the monoidal categories $({}_{H}^{H}\mathfrak{M}_{H}^{H},\otimes_{H})$ and $({}_{H}^{H}\mathfrak{M}_{H}^{H},\square_{H})$.

We give an explicit coordinate description of the Hamiltonian isotopy classes of the regular Lagrangian torus fibers of the smoothing \(\widehat{F}_4(0)\) of the \(F_4(0)\) toric degeneration, expressed in the explicit Oakley–Usher coordinates on \(\CP^2(\sqrt2)\). For the wall fibers, they are not Hamiltonian isotopic to standard toric fibers, and no two distinct wall fibers are Hamiltonian isotopic. For the off-wall fibers, we find the standard toric fibers they are Hamiltonian isotopic to.

Functional flow matching learns to generate random functions, such as velocity fields or solutions of partial differential equations, by continuously transforming a simple probability distribution into a target distribution. In practice, however, training and sampling use only finitely many grid values or coefficients. Does refining these observations recover the intended function-space model, and how do observation, learning, and numerical errors affect the generated distribution? We address these questions by first proving that the velocity targets used for training converge in mean square when the finite reconstructions converge to the underlying functions and remain uniformly bounded. The observations need not be nested: each refinement may replace earlier sensor locations. We also give quantitative bounds for orthogonal projections and extend the result to point observations under suitable regularity and reconstruction assumptions. We then bound the discrepancy between the generated and target distributions in Wasserstein distance, separating observation, learning, numerical, and endpoint errors without requiring uniqueness of the population finite-dimensional dynamics. For a normalized quadrature neural operator representing the learned velocity field, we establish stability and learning bounds with constants independent of the number of sensors. Under fixed model dimension and boundedness assumptions, the statistical excess risk decreases at an inverse-sample-size rate, up to logarithmic factors. A Gaussian example combines these estimates into an explicit sampling-error rate.

We record a general condition guaranteeing that the animation of a small $1$-category remains a $1$-category. The proof is extremely elementary universal algebra. This recovers as a very special case the striking observation of Antieau, \cite{Ant}, that the animation of $\fnt^{\mathrm{op}}$ is a $1$-category. Our result is a fair amount more general, and for example applies to $\fnt^{\mathrm{op}}_{\mathcal{G}}$ with $\mathcal{G}$ a groupoid, as well as beyond this.

Category theory is a mathematical theory of composition, widely used in logic, computing, and physics. Here we apply it to give a theory of secure composition. In particular, we provide a categorical treatment of Canetti's Universal Composability (UC) framework for systems with a static number of parties and sessions, often termed UC for static systems, yielding four benefits. First, we present our results graphically yet retain rigor by applying a standard categorical technique known as string diagrams. In particular, our formulation of the composition theorem can be graphically verified with a short sequence of diagrams, while remaining translatable to equations and amenable to formal verification. Second, categories let us generalize so that our results extend beyond interactive Turing machines to other forms of computation, such as quantum computation or domain-specific languages. Third, categories help us drop some unnecessary restrictions of UC (e.g., our adversary can be a computational network rather than a single Turing machine); we prove equivalence between our variant and the usual UC, showing no expressiveness is lost. Finally, the categorical perspective leads us to identify and correct some minor technical oversights in the standard formulation of simple UC.

In this article, we investigate two notions of genericity for triples of antipodal ideal chambers in a locally finite affine building $X$: one defined at the ideal boundary $X^{\infty}$, which we call ideal-genericity, and the other defined from within the affine building \(X\), which we call affine-genericity. While ideal-genericity implies affine-genericity, the latter is the more suitable notion for constructing a barycenter map associated with affine-generic triples of antipodal ideal chambers. This perspective also allows us to establish that this barycenter map is locally constant, and hence continuous. Finally, we provide sufficient geometric conditions on the affine Weyl group associated with $X$ that guarantee both ideal- and affine-genericity for all triples of antipodal ideal chambers of $X^\infty$. These conditions yield an algorithmic method for constructing geometric configurations in $X \cup X^{\infty}$ (when they exist) that correspond to non-generic triples of antipodal ideal chambers of $X^{\infty}$. Furthermore, our computations for the irreducible finite Weyl groups of rank at least two show that automatic ideal-genericity holds for all triples of antipodal ideal chambers only in types $B_2 = C_2$, $G_2$ and $B_3$.

Given finitely many compact convex bodies in $\R^n$, one seeks translates maximizing the volume of their common intersection. A lazy solver merely translates each body so as to place its centroid at the origin. We prove that the lazy strategy always captures strictly more than $\left(\tfrac{2}{n+1}\right)^n$ of the optimal volume, and that this constant is sharp: families of cones over tangent disks, indexed by finite nets on the sphere, approach it. The infimum is not attained. For two convex bodies in the plane the resulting sharp constant $4/9$ closes a gap open since 1996, when de Berg, Cheong, Devillers, van Kreveld and Teillaud proved that centroid alignment of two convex polygons captures at least $9/25$ of the maximum overlap and exhibited examples capturing only $4/9$. The proof rests on the following identity: for a convex body $K$ with centroid at the origin, the intersection of all centroid-recentered compact convex supersets of $K$ equals $\tfrac{1}{n+1}(K-K)$. We close with a promise-problem variant in which the lazy strategy captures at least $\left(\tfrac{n}{n+1}\right)^n > \tfrac1e$ of the optimum, uniformly in the dimension. All results below have been checked in Lean~4; the one classical input quoted rather than proved is the equality case of the Brunn–Minkowski inequality, which enters only for $n\ge2$.

Classical definitions of envelopes are vague for singular curves in the Lorentz-Minkowski plane. To study envelopes of such curves, we introduce a family of smooth curves that may admit singular points. Notably, the curves considered here may contain both non-lightlike and lightlike points. We then propose a new definition of envelopes for such curves and investigate their properties using lightlike tangential data. Furthermore, we examine the contact between the curves and their envelopes at singular points.

We consider a quantum system on the real line obeying a deformed Heisenberg algebra originally proposed by Wigner in 1950. Its explicit coordinate representation was provided by Yang in 1951 and in essence is identical in form with Dunkl's difference-differential operator introduced in 1989 in connection with roots systems of refection groups. Under certain conditions such quantum systems exhibit a supersymmetric (SUSY) structure where the reflection operator acts as the grading operator. We present a generalisation of Yang's representation by first considering only of one the two equations of motion in phase space. The corresponding non-interacting system is found to represent Witten's model of SUSY quantum mechanics. Imposing also the second equation of motion the original result of Wigner and Yang is reconsidered by extending their discussion to general symmetric potentials on the real line. As explicit example we discuss the harmonic oscillator and an attractive Coulomb-like potential $V(x)=-\gamma/|x|$. We also establish a Hooke-Newton duality between this Coulomb-like system and the original Wigner-Yang harmonic oscillator system.

For a group $G$, let $S(G)$ be the set of elements $g \in G$ such that $\langle g,x\rangle$ is solvable for all $x \in G$. We study when $S(G)$ coincides with the locally solvable radical $R_{\mathrm{L}\mathfrak{S}}(G)$. Using Wilson's profinitely convergent word sequences, we show that, for every locally (solvable-by-finite) group $G$ and every Wilson sequence $\omega$, $$ R_{\mathrm{L}\mathfrak{S}}(G) = R_{\mathrm{L}\mathfrak{R}}(G) = S(G) = W_{\omega}(G).$$ We also obtain four-conjugate and seven-commutator descriptions of radical membership, together with a two-conjugate result for torsion elements of order coprime to $6$. The same radical identity holds for locally linear groups, and hence for subgroups of $\mathrm{GL}_{\infty}(D)$ when $D$ is a locally finite-dimensional division ring. Independently, we prove that groups with nearly modular subgroup lattice satisfy $$R_{\mathrm{L}\mathfrak{S}}(G) = R_{\mathrm{L}\mathfrak{R}}(G) = S(G).$$

We present three proofs of the generalized Griesmer bound together with the corresponding proofs of the generalized antiGriesmer bound. The first proof follows from inequalities relating consecutive minimum and maximum subcode support weights. We also write the projective construction of Tsfasman and Vlăduţ as a residual code argument and express the geometric proof of Kurz, Landjev, and Rousseva in terms of shortened subcodes. In addition, complements in repeated simplex codes show that the two bounds are equivalent. The residual and shortening arguments also determine the consequences of equality for the resulting residual codes and shortened subcodes. Finally, the complement relation transfers known divisibility results for Griesmer codes to antiGriesmer codes.

We develop a direct analytic theory of stuffle-regularized multitangent functions and prove their reduction to finite linear combinations of monotangent functions, without using mould calculus. We first establish an asymptotic comparison between one-sided truncated multiple Hurwitz zeta functions and their stuffle regularizations at the natural parameter \(T_N-H(s)\), where \(T_N\) is the harmonic truncation and \(H(s)\) is the harmonic-number function. Applying this comparison to symmetric multitangent truncations yields meromorphic, \(1\)-periodic regularized multitangent functions. An explicit partial-fraction decomposition of each summand, combined with asymptotic estimates for moving truncation ranges, gives formulas for the reduction coefficients in terms of stuffle-regularized multiple zeta values. The constant term and the coefficient of the monotangent \(\mathcal T(1;s)\) are determined from the limits as \(\operatorname{Im}s\to\pm\infty\): both vanish whenever the index contains an entry greater than \(1\), whereas the exceptional indices \(\{1\}^r\) are evaluated through a sine-quotient generating function. As a consequence, we obtain a family of relations among regularized multiple zeta values.

Let $R$ be a complete Cohen–Macaulay local ring and $\Lambda$ an $R$-order. We study the Krull–Gabriel dimension of the functor category $\mathrm{mod}\,\underline{\mathcal C}(\Lambda)$, where $\underline{\mathcal C}(\Lambda)$ is the stable category of maximal Cohen–Macaulay $\Lambda$-modules. This work is motivated by the non-existence theorem of Herzog and Krause for Krull–Gabriel dimension $1$ over artin algebras. We first prove that, if $\Lambda$ is Gorenstein and $\mathrm{KGdim}\,\mathrm{mod}\,\underline{\mathcal C}(\Lambda)\leq 1$, then $\Lambda$ is an isolated singularity. We then show that, if $\Lambda$ is an isolated singularity of uncountable Cohen–Macaulay representation type, then $\mathrm{KGdim}\,\mathrm{mod}\,\underline{\mathcal C}(\Lambda)\neq 1$.

Shimada recently showed that Frobenius–Witt cotangent complex, the animation of Saito's Frobenius–Witt differentials, vanishes on perfectoid rings, thus it serves as a candidate of “absolute” cotangent complex. In this article, we give a pullback description of their arithmetic extension, and as a consequence, we give a direct description of Frobenius–Witt cotangent complex, which leads to generalizations to derived rings and animated pre-log rings. This description allows us to compute Frobenius–Witt cotangent complex of derived $\delta$-rings as well. We also propose a version of Frobenius–Witt cotangent complex relative to derived $\delta$-rings, and establish a vanishing result for prisms, which generalizes vanishing of Frobenius–Witt cotangent complex of perfectoid rings. Finally, independently of previous considerations, we give a regularity criterion via Frobenius–Witt cotangent complex for $p$-local Noetherian (not necessarily local) rings without $F$-finiteness. We also record the flatness of Frobenius–Witt cotangent complex of valuation rings, with essential ideas due to ChatGPT-6 Astra.

We study edge-spectral supersaturation for two families of color-critical graphs with chromatic number three. For an integer $r\geq 1$, we define the spectral threshold \[ g_r(m):=\frac{r-1+\sqrt{4m-r^2+1}}{2}, \] which is the tight upper bound on the spectral radius of graphs avoiding $K_{s,t}^+$ (when $t+1\geq s\geq 3$) and $C_{2k+1}$ (when $r=k$), realized by split-graph constructions. First, let $t+1 \geq s\geq 3$ be fixed integers, and let $K_{s,t}^{+}$ be obtained by adding an edge to the part of size $s$ in $K_{s,t}$. We prove that every sufficiently large $m$-edge graph $G$ with $\rho(G)>g_{s-1}(m)$ contains $\Omega(m^{(s+t-1)/2})$ copies of $K_{s,t}^{+}$. Second, for any fixed $k\geq 2$, the condition $\rho(G)>g_k(m)$ forces $N(C_{2k+1},G)=\Omega(m^k).$ We also construct graphs showing that both lower bounds are tight up to constant factors. These results establish that exceeding the tight spectral Turán threshold $g_r(m)$ forces not just a single copy, but the optimal polynomial number of copies of these color-critical graphs. Thus, crossing the relevant split-graph spectral threshold forces the optimal polynomial order of copies, extending edge-spectral existence theorems to supersaturation results in the delicate three-chromatic regime.

In this paper, we study entire solutions and periodic correctors for the coupled-drift Monge-Ampère equation \[ \det D^2u = \exp\{-a\cdot Du+b\cdot x+V(x)-c_0\}, \quad D^2u>0. \] For $V\equiv0$, we obtain a sharp classification of the whole-space solvability regimes: all entire smooth strictly convex solutions are quadratic when $a=b=0$; no such solution exists when $a\neq0$ and $a\cdot b\le0$; and non-quadratic entire solutions exist when $a=0$ and $b\neq0$, or when $a\cdot b>0$. For the null case $a\neq0$, $a\cdot b=0$, we give a scalar maximum-principle argument in every dimension $n\ge2$. For periodic $V$, we prove existence and uniqueness of the normalized pair $(\psi_A,c_A)$ solving the drifted cell problem \[ \det(A+D^2\psi) = \exp\{-a\cdot D\psi+V-c_A\}, \quad A+D^2\psi>0 \quad\text{on }\mathbb T^n. \] We also prove that any asymptotically quadratic entire solution must satisfy $b=Aa$. If its remainder is bounded, then the solution is the corresponding quadratic-periodic corrector up to an additive constant.

Data-driven decision pipelines combining predictive machine learning models with downstream optimization software are increasingly used to make high-stakes operational decisions. Certifying the safety, fairness, and reliability of these decisions is essential, yet traditional scenario generation methods rely on repeated random testing, which becomes computationally prohibitive when failure events are rare and offers little insight into why failures occur. We present a framework for local violation certification designed specifically for linear decision pipelines under input uncertainty. We mathematically demonstrate that standard sampling methods fail efficiently for rare violations, motivating a direct structural approach. By analyzing the fixed decision boundary of a deployed pipeline, we show that the local risk of failure can be calculated directly in closed form using a single optimization solve. Furthermore, we introduce an exact sampling procedure and closed-form risk statistics that provide feature-level attributions (identifying which input characteristics contribute most to potential non-compliance) without requiring repetitive random trials or complex sampling algorithms. We demonstrate our approach on an economic power dispatch system subject to emissions regulations, delivering precise, auditable risk assessments at a fraction of the traditional computational cost.

Starting from an integer composition, form its consecutive absolute differences, provided that they are nonzero and pairwise distinct, and then permute these differences arbitrarily before repeating the operation. The depth of the composition is the maximum possible number of successive iterations. We determine the least positive integer admitting a composition of any prescribed depth. If a(n) is the least positive integer having a composition of depth n, then $a(n)=n+1+\lceil n(n+1)/4\rceil+\lfloor n/2\rfloor$. We also prove that every integer k at least a(n) has a composition of depth at least n. Consequently, if d(k) denotes the maximum depth of a composition of k, then $d(k)=\max\{n\geq 0:a(n)\leq k\}$. Finally, we classify and enumerate all compositions attaining the minimum a(n). Their number is given by four factorial formulas according to n modulo 4.

Blind deconvolution has been widely applied in system identification and signal processing. While joint sparsity commonly arises in practical scenarios, effectively exploiting this structure to enhance recovery performance remains a challenging and largely open problem. In this paper, we propose a joint-sparsity-promoting optimization problem and develop a Riemannian optimization algorithm for its accurate and efficient solution. We further establish theoretical guarantees that characterize the non-asymptotic relationship between the estimation error and the sample complexity, showing that exploiting joint sparsity can significantly reduce the sample complexity required for successful recovery. Numerical experiments are provided that validate the theoretical results and demonstrate the effectiveness of the proposed approach.

In this paper, we study modular fusion categories whose mapping class group representations are trivial on the Torelli groups, with particular emphasis on the congruence properties of the resulting representations of $\mathrm{Sp}(2g, \mathbb{Z})$. We prove that, for any $g \ge 3$, the Torelli group is contained in the kernel of the genus-$g$ mapping class group representation $\rho_g$ associated with a modular fusion category $\mathcal{C}$ if and only if $\mathcal{C}$ is pointed. This refines the corresponding result in \cite{MW25}. We also characterize the modular fusion categories for which the genus-$2$ Torelli group is contained in $\ker \rho_2$; in particular, categories with isotropic adjoint subcategories belong to this class. Finally, we give a complete classification of modular fusion categories with isotropic adjoint subcategories, and determine the groups $G$ satisfying $\mathcal{Z}(\mathrm{Vect}_G)$ is 2-Torelli trivial but $\mathcal{Z}(\mathrm{Vect}_G)_{\mathrm{ad}}$ is not isotropic are precisely 2-Engel groups of nilpotent class 3.

We study the simplicial boundary $\partial\mathcal{FS}$ of Culler-Vogtmann Outer space via thickenings defined by graph-theoretic connectivity. Let $C'$ be the subcomplex of the free splitting complex obtained from $\partial\mathcal{FS}$ by adding all stable graphs that are not $3$-edge connected, together with their faces. We prove that the inclusion $\partial\mathcal{FS}\hookrightarrow C'$ is $(2n-3)$-connected. The proof shows, more precisely, that adding graphs with cut vertices is a homotopy equivalence, while the only non-contractible fibres in the $2$-bond thickening occur over $\theta$-graphs. The result gives further evidence that $\partial\mathcal{FS}$ may be $(2n-3)$-spherical, an $\operatorname{Out}(F_n)$-analogue of Rognes's connectivity conjecture for the common basis complex. It also gives a topological, universal-cover perspective that unifies several existing results about the commutative graph complex.

We give a negative solution to MAIS-O60. We first construct an example in which an initially active ReLU neuron becomes completely inactive in finite time and thereafter remains frozen at a limit whose Fourier energy is equally distributed among all nonzero real frequency classes. The counterexample holds on an open set of initial conditions and therefore occurs with positive probability under Gaussian initialization. An appendix prepared by GPT-5.6 Sol strengthens the counterexample by showing that the same failure can occur for every Clarke trajectory from an open set of initial conditions, under the convention $\mathrm{ReLU}'(0)=0$, for smooth dead-zone approximations of ReLU, and for fixed-step full-batch gradient descent. Thus, single-frequency alignment is not a general consequence of training a single neuron on modular addition.

We examine the equation $y^{a}=\sum\limits_{i=1}^{k}F_{n_{i}}$ for positive integers $y,a\geq 2$ and $k\geq3$. This equation can be expressed as a problem in terms of the Zeckendorf representations of integers. Using bounds on linear forms in logarithms and Baker-Davenport reduction methods, we are able to completely solve the equation for $ y\leq 25000$ when $k=3$, for $y\leq 1000$ when $k= 4$, for $y\leq 40$ when $k= 5$, and for $y\leq 3$ when $k=6$.

We consider the semilinear heat equation in the unit ball with the exponential nonlinearity and an inhomogeneous term $f$. When $f=0$, it is known that the bifurcation structure of the stationary problem undergoes a qualitative change at the critical dimension $N=10$. This change affects the large-time behavior of solutions to the heat equation, and in particular, the grow-up phenomenon occurs for $N\ge 10$. In this paper, we show that once $f$ exceeds a threshold, the bifurcation structure changes to a type that does not appear in the case $f=0$. The change in the bifurcation structure leads to the disappearance of the grow-up phenomenon beyond the threshold. Moreover, we provide a quantitative characterization of this transition by determining the sharp grow-up rates for $N\ge 11$. In particular, we identify a new dimension-specific phenomenon in the threshold case: a log-log type correction term emerges in the grow-up rate only for $N=11$.