Papers

We show that some classical identities valid for ordinary partitions have precise analogues for $n$-color partitions, that is partitions in which a part of size $n\geq 1$ can occur in colors $1, 2, \ldots, n$. For $r \ge 2$ and $j \ge 0$, we write $\mathcal{O}_{j,r}(m)$ and $\mathcal{D}_{j,r}(m)$ for the sets of $n$-color partitions of $m$ with, respectively, exactly $j$ different parts whose size and color are divisible by $r$, and exactly $j$ different parts occurring at least $r$ times. We prove an $n$-color version of Franklin's theorem, $|\mathcal{O}_{j,r}(m)| = |\mathcal{D}_{j,r}(m)|$, along with two Beck-type identities. We give both analytic and combinatorial proofs for all theorems.

We study exotic knottings of surfaces in 4-manifolds through their ambient symmetries. We first give a general recipe for producing projectively rigid surfaces, for which every smoothly extendable self-diffeomorphism acts on first homology by plus or minus the identity. For every integer g >0, a refinement of this construction yields a finite sequence of genus-g surfaces F_0, ..., F_2g contained in a 4-manifold X_g. These surfaces are topologically isotopic and topologically flexible: every orientation-preserving self-diffeomorphism of F_i can be realized by a self-homeomorphism of X_g preserving F_i. Successive knotting, however, rules out increasingly many projective homological symmetries, revealing a finer knottedness phenomenon. The first two constructions combine iterated rim surgery with the convex geometry of Newton polytopes of relative Seiberg-Witten invariants. We also use hyperbolic geometry to construct a totally geodesic surface of positive genus whose smooth and topological extendable mapping class groups are both trivial.

We show that for a hyperbolic curve X over an algebraically closed field k of characteristic zero and a finite set S of k-rational points of X, after replacing X by a finite étale cover, there exists a family of curves over X whose fibres over points in S have pairwise nonisogenous Jacobians. We also give an application of our results to a problem concerning Galois sections.

When dealing with nonlinear systems, classical covariance steering typically propagates uncertainty via first-order linearizations, discarding higher-order Taylor remainders. This truncation causes computed statistical moments to diverge from the true physical state distribution, often leading to chance constraint violations. This paper introduces a discrete-time Sequential Convex Programming (SCP) framework that casts the deterministic one-step nonlinear numerical map as a Linear Stochastic Inclusion. The The Taylor remainder is modeled as a state-independent unstructured uncertainty block, bounded over a uniform envelope. The second-moment tubes are propagated via what we refer to as a robust Stochastic Linear Matrix Inequality (S-LMI) derived from the Petersen's lemma, providing an upper bound on the expected uncentered second moment. Domain-exit risk is bounded analytically via a Markov trace inequality, and spatial chance constraints are enforced via Gauss unimodal second-moment bounds within a Difference-of-Convex program. Simulations on a state-dependent nonlinear dynamic system demonstrate constraint satisfaction.

Let $X$ be a smooth projective variety, $f:X\to X$ an int-amplified endomorphism, and $L$ any ample line bundle on $X$. We prove that, in every codimension, the total degree of prime cycles that become totally invariant under an iterate of $f$ satisfies an explicit Hilbert-function bound on their total $L$-degree. In particular, when $X=\mathbf{P}^n$, the number of totally invariant prime $(n-r)$-cycle is bounded by $\binom{n+1}{r}$, and this bound is optimal.

In this paper, we develop an asymptotic-preserving (AP) dynamical low-rank semi-Lagrangian method for multiscale linear kinetic transport equations. The method combines the large-time-step capability of semi-Lagrangian discretizations with the storage and cost reduction provided by low-rank representations. The proposed scheme couples an approximate macroscopic density update with the basis update Galerkin integrator for the kinetic distribution. To retain the reduced complexity in the semi-Lagrangian flux evaluation, the flux derivative is computed through a sampled angular quadrature strategy. We establish an unconditional stability analysis of the full-quadrature low-rank scheme in the constant-coefficient case. The error induced by angular sampling in the flux derivative is quantified. The resulting scheme is shown to be AP in the diffusive limit. Numerical experiments, including high-dimensional test cases, demonstrate that the proposed method is AP, stable under large time steps, and computationally efficient across kinetic and diffusive regimes.

Modern deep learning typically keeps the batch size static throughout training, thus overlooking the joint effect of learning rate and batch size on the training dynamics. In this paper, we study the deep learning dynamics through the lens of convex optimization and derive a joint characterization of loss in terms of both schedules, applicable to general optimizers and model architectures. This characterization yields a closed-form optimal batch size schedule for any prescribed learning rate schedule, and further leads to joint scaling laws that consistently outperform static batch size baselines, highlighting the significance of dynamic batch size schedule in large language model training.

A numerical framework is proposed for identifying governing partial differential equations (PDEs) from observational data by establishing a link between observation-driven and equation-driven Koopman operators in a common Chebyshev spectral domain. In contrast to data-driven approaches such as dynamic mode decomposition (DMD), which approximate Koopman operators without explicitly relating them to differential operators, the proposed framework constructs finite-dimensional Koopman operators using Chebyshev spectral representations, thereby enabling direct comparison between data-derived dynamics and candidate governing PDEs. A unified observation model together with a least-squares coefficient recovery formulation is introduced to recover Chebyshev spectral coefficients from observations obtained on arbitrary sampling grids. This provides a numerically consistent interface between practical observations and Chebyshev-based Koopman analysis. Numerical experiments under direct Chebyshev, uniform, and irregular sampling configurations demonstrate that the proposed framework accurately identifies the governing PDE from observations. An observation-density study shows that reliable PDE identification is consistently achieved once sufficient independent observations are available for stable coefficient recovery, providing a practical guideline.

Let $\Gamma_n(q)$ be the graph whose vertices are the invertible matrices in $\Mat_n(\F_q)$, with two distinct matrices adjacent whenever their sum is singular. A dissociation set is a vertex set inducing a graph of maximum degree at most one. We study the dissociation number of $\Gamma_n(q)$ by embedding it as an induced subgraph of the total graph $T_n(q)$ on all of $\Mat_n(\F_q)$. A general Laplacian inequality for $k$-independent sets, together with an explicit character computation for the additive group of the matrix ring, gives parity-sensitive upper bounds. For fixed $n$, the resulting bound is of order at most $q^{n^2-n+1}$ for odd $q$ and at most $q^{n^2-2n+2}$ for even $q$. In particular, \[ \diss(\Gamma_n(q))\le q^{n^2-n+1}-1. \] In the other direction, the regular representation of the extension field $\F_{q^n}$ gives $\diss(\Gamma_n(q))\ge q^n-1$. We give complete proofs, including a self-contained derivation of the required matrix character sum, and determine the smallest case: $\diss(\Gamma_2(2))=3$.

In this paper, we study the primitive equations in a collar neighborhood of a smooth closed hypersurface in ${\mathbb R}^3$. Using the framework of maximal $L_q$-regularity and the hydrostatic Helmholtz projection, we establish existence, uniqueness, and regularity results for strong solutions. Building on these local well-posedness results, we derive suitable a priori estimates and prove the global existence of strong solutions without imposing any smallness assumptions for the initial data. Moreover, we show that the solutions are $C^\infty$ jointly in time and space.

Estimating the Shannon entropy of discrete distributions with countably infinite support is a challenging problem. In this paper, we investigate the bias of the plug-in estimator $\hat{H}_n$ for the Shannon entropy $H(\boldsymbol{p})$ under an occupancy model whose frequency sequence exhibits regular variation with tail index $\alpha\in(0,1)$. Using Poissonization and the theory of regular variation, we establish the asymptotic relation $|\mathsf{E}[\hat H_n] - H(\boldsymbol{p})| \sim n^{\alpha-1}L(n)C_\alpha$, where $L$ is a slowly varying function and $C_\alpha$ is an explicit constant depending only on $\alpha$ that admits an integral representation. Our result shows that the asymptotic behavior of the bias of the plug-in estimator under power-law frequency distributions is determined by the tail behavior of the underlying distribution.

Determining the minimal number of intensity measurements required for phase retrieval in $\mathbb{C}^4$ has been a long-standing open problem. Prior to this work, the best-known results implied that this minimum was either $10$ or $11$. In this paper, we leverage characteristic classes and cohomology groups from differential topology to prove that no family of $10$ vectors in $\mathbb{C}^4$ possesses the phase retrieval property. Combining our lower bound with Vinzant's explicit eleven-vector construction establishes that the exact minimum is $11$. Our result yields a significant consequence for pure state quantum tomography, namely, a rank-one POVM on $\mathbb{C}^4$ requires exactly $11$ elements to be informationally complete for pure states. This further implies that three orthonormal bases are insufficient to uniquely distinguish all pure states in $\mathbb{C}^4$. Because four orthonormal bases are already known to be sufficient, we conclude that exactly four bases are required, thereby completely resolving the problem left in [C. Carmeli, T. Heinosaari, J. Schultz, A. Toigo, Eur. Phys. J. D].

Motivated by Mostow's decomposition theorem for positive definite matrices and by subsequent matrix factorizations involving geometric means, we establish a Mostow-type decomposition for arbitrary symmetric cones. As a consequence, we derive an analogous decomposition for the automorphism group of the cone. We also introduce a natural Hadamard metric on the symmetric cone by pulling back an $\ell^2$-product metric, together with its midpoint operation and associated Karcher mean.

We use inverse mean curvature flow to construct bi-Lipschitz maps that preserve normalized volume and decrease distances. These maps send a round sphere onto any smooth closed strictly convex hypersurface in a sphere and a flat disk onto any smooth strictly convex free-boundary disk in a Euclidean ball. In dimension two, this proves a conjecture of E. Milman for every smooth two-sphere with Gaussian curvature at least one and gives an analogous result for nonnegatively curved disks whose boundary has geodesic curvature one. The spherical result proves the two-dimensional case of the spectral comparison conjectured by Colding and Minicozzi. Counterexamples in dimensions $n\geq3$ show that the restriction to dimension two is sharp. Furthermore, an equivariant extension of this construction yields, for every $n\geq2$, a contracting transport map from the uniform probability measure on a round hemisphere to the uniform probability measure on any closed geodesically convex subset of positive volume. This settles the remaining uniform-target case of a question raised by Beck and Jerison. In dimension two, we also find geometric conditions under which the uniform measure on the hemisphere can be transported by a contracting map to a broad class of nonuniform probability measures supported on domains in a hemisphere.

We study structural averaged controllability for multi-input linear ensemble systems. In this problem, one asks whether a sparsity pattern admits a linear ensemble system that is averaged controllable. The single-input case has been characterized completely, while the general multi-input case has remained open. In this work, we give a complete characterization for the general multi-input case. In particular, we prove that in addition to accessibility, a necessary and sufficient condition for structural averaged controllability of ensemble systems is existence of a row-saturating matching for its core, an acyclic subgraph associated with the system which plays a central role in the result for the single-input case.

We introduce and study two partition-indexed families of quasimodular forms obtained from Schur functions: Schur Eisenstein series and Schur MacMahon series. An explicit transition between them can be interpreted as a convolution in a Faà di Bruno Hopf algebra of symmetric functions. We discuss the classical sl2-action and prove that Schur Eisenstein series for partitions with parts of size at most 3 give a basis for quasimodular forms. Further, we conjecture that the Schur MacMahon series span all quasimodular forms with integral coefficients.

We study the structure and regularity of higher rank graph $C^*$-algebras, with particular emphasis on their nuclear dimension. For a row-finite, locally convex $k$-graph $\Lambda$ with no sources, we characterise pure infiniteness of $C^*(\Lambda)$ in terms of generalised cycles, maximal tails, and strong aperiodicity, and we relate these conditions to topological dimension zero of the primitive ideal space and to the structure of gauge-invariant ideals. Our main application is that whenever $C^*(\Lambda)$ is purely infinite of topological dimension zero—in particular whenever its ideal lattice is finite—it is strongly purely infinite, $\mathcal O_\infty$-stable, and of nuclear dimension one, \emph{even when $C^*(\Lambda)$ is not simple}. This extends to the non-simple, higher-rank setting the nuclear-dimension-one computation known for simple UCT-Kirchberg $2$-graph algebras. Along the way we refine and correct several results in the existing graph $C^*$-algebra literature.

We perform a thorough analysis of the eigenvalues of tetradiagonal Toeplitz matrices of large order $n$ generated by the Laurent polynomial $a(t)=t^2+t^{-1}$. The spectra of these matrices are invariant under $2\pi/3$-rotation. They are contained in three segments of the complex plane and asymptotically fill these segments as $n$ tends to infinity. We apply Widom's formula for the determinants and transform the characteristic equation into a convenient form that can be solved by the fixed point iteration method. After that, we compute the asymptotic distribution of the eigenvalues. The main results are asymptotic formulas for the eigenvalues, both close to the origin and far from the origin. The main results are verified by numerical tests for moderate values of $n$.

We prove an improved restricted isometry bound for Gaussian partial circulant matrices with arbitrary prescribed sampling sets. There is a universal constant $C>0$ such that the following holds. Let $1\leq K\leq m\leq N$ be positive integers, let $\Omega\subset\mathbb Z_N$ be any fixed set with $|\Omega|=m$, and let $g\sim\mathcal N(0,I_N)$. For every $\delta,\eta\in(0,1)$, the normalized partial circulant matrix generated by $g$ has the RIP of order $K$ with constant at most $\delta$, with probability at least $1-\eta$ over the draw of $g$, provided \[ m\geq C\delta^{-2}K \max\{\log^2(eK)\log(2N)\log(em),\log(2/\eta)\}. \] The proof refines the Maurey entropy step in the chaos-process argument by combining a noncommutative Khintchine inequality with a Schatten moment estimate controlled by $m$, replacing one factor $\log(2N)$ in the Krahmer–Mendelson–Rauhut bound by $\log(em)$.

We establish a quantitative Landis estimate for the one-dimensional fractional Schrödinger equation $(-\Delta)^{1/4}u+V(x)u=0$ in $\mathbb R$ with a real-valued bounded potential. If $\|V\|_{L^\infty}\le 1$, $\|u\|_{L^\infty}\le C_0$, and $\|u\|_{L^2(-1,1)}\ge 1$, then \[ \inf_{|x_0|=R}\|u\|_{L^\infty(x_0-1,x_0+1)} \ge \exp(-CR\log R) \] for all sufficiently large $R$. After the Caffarelli–Silvestre extension and the substitution $y=z^2/2$, the equation becomes a Grushin equation with a weak Robin condition on the degeneracy line. The corresponding angular operator has the arithmetic spectrum $\kappa_n=2n+\tfrac12$ after half-density conjugation. The central spectral estimate is \[ \sup_{\xi\in\mathbb R} \bigl\|C\bigl((\tau+i\xi)^2-L_0\bigr)^{-1}C^*\bigr\| \le C\tau^{-1/2} \] for parameters separated from the angular lattice. It yields a linear-weight Carleman estimate for measurable Robin feedback with absorption threshold $\tau\ge C(1+\|V\|_\infty^2)$. Quantitative inward propagation, fixed-scale Grushin-ball propagation, and an interior-cylinder interpolation estimate then transfer bulk non-vanishing to the boundary. The Landis rescaling converts the local potential dependence $C\|q\|_\infty^2$ into the global rate $CR\log R$.

Let $G=\operatorname{SmallGroup}(128,859)$ and $k=\overline{k}$. The cohomology ring $H^*(G;k)$ has depth two, and we prove that the minimum quotient dimension of an associated prime is exactly three. Okuyama's theorem shows that an integer $r$ occurs as such a dimension exactly when there is an elementary abelian subgroup $E\leq G$ of rank $r$ with $\operatorname{depth} H^*(C_G(E);k)=r$. We use this equivalence to define the centralizer excess. If $d=\operatorname{depth} H^*(K;k)$, Carlson's equality holds precisely when some rank-$d$ subgroup has zero excess. For $G$, all rank-two centralizers have positive excess. A complete enumeration of the thirty-one actual rank-three elementary abelian subgroups finds six zero-excess witnesses. Hence $\omega_a\bigl(H^*(G;k)\bigr)=3$. Since $H^*(G\times(C_2)^n;k)\cong H^*(G;k)[u_1,\dots,u_n]$, the standard behavior of associated primes under polynomial extension gives $\omega_a\bigl(H^*(G\times(C_2)^n;k)\bigr)=n+3$ for $n\geq0$. We also study the class $\alpha_0=g+fc\in H^3(G;\mathbb F_2)$. It is killed by two degree-one classes but restricts nontrivially to a rank-four elementary abelian subgroup. It follows that $\dim H^*(G;\mathbb F_2)/\operatorname{ann}(\alpha_0)=4$. Thus two explicit linear annihilators do not force a two-dimensional cyclic support. The assertion is about Krull dimension; it does not say that the support is the whole spectrum.

We prove a sharp volume-stability theorem for closed hyperbolic three-manifolds. Let \((M,h)\) be closed hyperbolic with \(\operatorname{Ric}_h=-2h\), and let \(g_i\) be smooth metrics on \(M\) satisfying $R(g_i)\geq -6, \operatorname{Vol}_{g_i}(M)\longrightarrow \operatorname{Vol}_h(M)$. After passing to a subsequence, there exist \(Z_i\subset M\), smooth domains \(K_i\subset M\), and diffeomorphisms $\psi_i:K_i\longrightarrow M\setminus Z_i $ such that $\operatorname{Vol}_{g_i}(Z_i)\longrightarrow0, \operatorname{Vol}_h(M\setminus K_i)\longrightarrow0, $ and $ \|\psi_i^*g_i-h\|_{C^0(K_i,h)}\longrightarrow0. $ Thus near-equality in the sharp hyperbolic volume bound forces tensorial \(C^0\)-convergence to the hyperbolic metric outside regions of vanishing volume. As an application, we establish stability of the Fischer–Moncrief reduced Hamiltonian at the Lorentz-cone ground state: after CMC normalization, near-minimizing compact vacuum data in the hyperbolic topological class converge, modulo sets of vanishing volume, to the hyperbolic Lorentz-cone geometry in tensorial \(C^0\). This provides a rigorous volume-dominance formulation of the Fischer–Moncrief asymptotic picture.

This paper gives an algebraic description of rational higher order Whitehead products in mapping spaces. Given a Sullivan representative of the base map, we introduce defining systems in the corresponding derivation complex. The main result shows that, under the standard identification between rational homotopy groups of mapping spaces and the cohomology of the derivation complex, rational higher order Whitehead products are given exactly by the cohomology classes represented by the associated derivation brackets. The examples show how choices of defining systems produce non-trivial elements and indeterminacy.

This paper computes Poisson space noise functionals, $P^{\prime}(u)$, realised in a Gel'fand triple built from a Lévy measure $\lambda_{\beta}(u)du$. We isolate three discretisation parameters: a small-amplitude cut-off, a Donsker delta truncation M, and a chaos order N. For a stable-type intensity $\lambda_{\beta}(u)=cu^{-1-\alpha}$ ($0<\alpha<2$), replacing discarded small amplitudes with matched Gaussian space noise improves the Wasserstein-1 error from $O(\epsilon^{1-\alpha/2})$ to $O(\epsilon)$. The residual is asymptotically normal at $O(\epsilon^{\alpha/2})$. This compensation reduces computational complexity from $O(\tau^{-2\alpha/(2-\alpha)})$ to $O(\tau^{-\alpha})$. We also evaluate the Gamma-type boundary ($\alpha=0$) and exponential tempering. Truncations converge algebraically (M) and super-geometrically (N). All predicted rates are tightly confirmed by deterministic numerical experiments via Gil-Pelaez inversion, eliminating Monte Carlo noise.

For $1\le p\le\infty$ and $i=1,2$, let $W^{k_i,p}(\Omega_i)$ be the Sobolev space on a bounded open interval $\Omega_i$ with differentiability order $k_i$. We equip $W^{k_i,p}(\Omega_i)$ with an anchored Sobolev norm and the order $\ge_{k_i,p}$ defined by $f^{(j)}(x_i)\ge 0$ for each $j=0,\ldots,k_i-1$ and $f^{(k_i)}\ge 0$ a.e. We show that the positive unit spheres of $W^{k_1,p}(\Omega_1)$ and $W^{k_2,p}(\Omega_2)$ are surjectively isometric if and only if $k_1=k_2$. Every such isometry extends uniquely to a complex-linear isometric order isomorphism, for which we obtain a coordinate representation. The same conclusions hold for surjective phase-isometries. For $1<p<\infty$, they also hold for surjective norm-additive maps.

We introduce a general framework for curved Kakeya problems in $\mathbb{R}^n$, encompassing those arising from Hörmander-type oscillatory integrals. Every family of curves determines a spray geometry, which allows us to use the projective geometry of paths in the study of curved Kakeya problems. We focus on the two extremes of the "best" and "worst" possible behaviors of curved Kakeya sets. We characterize when the incidence structure underlying Wolff's hairbrush argument persists. In particular, we prove that the existence of many totally geodesic surfaces, as required by Wolff's hairbrush argument, is equivalent to projective flatness of the associated spray. Within this projectively flat class, Bourgain's condition provides a clean dichotomy: when it holds, the family is direction-equivalent to a Bochner–Riesz type family of lines and satisfies the Katz–Wolff condition, and thus the Wang–Zahl result is applicable; when it fails, every totally geodesic surface supports a two-dimensional Kakeya set. We also show that under an extra semi-algebraic assumption, a family of curves in $\mathbb{R}^3$ admits a curved Kakeya set of Hausdorff dimension $2$ if and only if it admits a curved Kakeya set contained in a surface. Equivalently, if this compression is absent, every associated curved Kakeya set has dimension strictly greater than $2$.

In this paper, we introduce a novel sampling method to recovering the trajectory of a moving point source in R^3, where both the spatial location and emission moment of the moving point source are unknown. Combining algebraic theory with geometric knowledge, we prove the uniqueness of the source location by using measured data from five observation points. Our sampling method constructs an indicator function based on the property that the residual of the time difference of arrival constraint formula vanishes at the true source location. It achieves the reconstruction of the spatial positions and emission moments of a moving point source only using data from five observation points and their corresponding arrival times. The algorithm not only reduces the required number of observation points, but also improves computational efficiency, stability, and noise resistance. Numerical experiments verify the effectiveness of the method.

In this note, we give a positive answer to K3 problem 4.28b, giving the first examples of homology 3-spheres that embed smoothly in $S^2 \times S^2$ but not $S^4$.

In this paper, we pose a new class of Minkowski problems corresponding to the mixed Euclidean-Gaussian volume. Using the flow method, we prove the existence of smooth normalized solutions for the corresponding Monge-Ampère-type equation in the symmetric case. Then, by approximation, we obtain the existence of origin-symmetric solutions to the mixed Euclidean-Gaussian Minkowski problem.

Fix $p\ge 3$. We establish a separation between regularized bulk universality and unrestricted annealed complexity for the pure spherical $p$-spin Hamiltonian with independent non-Gaussian tensor coordinates. There is a symmetric, compactly supported disorder law with a smooth density, matching the first $2p$ Gaussian moments, for which the expectation of a positive, frame-averaged regularization of the one-point Kac-Rice functional is asymptotic to its Gaussian counterpart, while the unrestricted critical-point count in a compact energy window has a lower exponential rate strictly above the Gaussian limit. The obstruction is a coherent block involving on the order of $N^{1/p}$ coordinates, so neither finite moment matching nor a uniform entry bound restores unregularized annealed universality. For uniformly subexponential disorder matching the first $m$ Gaussian moments, the regularized functional on incoherent frames satisfies $\left|\log\left(\mathbb{E}\mathcal{Z}_{N,L}^J/\mathbb{E}\mathcal{Z}_{N,L}^G\right)\right|\le C N q_N^{m-1}$, where $q_N=L^{p-1}N^{-(p-2)/2}(\log N)^{(p-1)/2}$. Thus mean and variance matching imply regularized pressure universality for every $p\ge 3$, and the expectation ratio tends to one when $(m-1)(p-2)>2$. We identify the Gaussian variational limit and prove the Gaussian quadratic energy-excursion upper bound uniformly over profiles asymptotically supported on $o(N)$ coordinates under a Gaussian-rate profile-tail condition. Finally, we give a conditional reduction to exact universality: once two one-sided de-regularization defects vanish, an exact max formula makes control of the localized complement necessary and sufficient. The defects vanish in the Gaussian model.