We show that the small-data scattering map uniquely determines the nonlinearity for a class of nonlinear Schrödinger equations with radial, Hartree-type nonlinearities. Our assumptions on the convolution kernel require only a mild decay condition at infinity and permit a locally integrable singularity at the origin.

We develop a data-driven framework for estimating optimal linear response of nonlinear dynamical systems. Our approach is based on kernel-smoothed approximations of the transfer/Koopman operators of the system, built from possibly high-dimensional observations along trajectories. Combining these operator approximations with the theory developed in [Antown et al. (2018), J. Stat. Phys., 170(6), 1051-1087], we formulate a computationally tractable optimization problem for the optimal infinitesimal perturbation realising any desired manipulation of the spectrum. We also introduce a notion of optimal-response vector fields for visualising, and physically interpreting, the system's response to the optimal perturbation under arbitrary observations. Our focus is on finding perturbations that optimally increase the frequency or optimally suppress the decay of correlations of almost-cycles or almost-invariant sets associated with the eigenvalues of the kernel-smoothed transfer operator. We illustrate our approach with applications to low-dimensional periodic and chaotic systems, as well as a high-dimensional example involving the El Nino Southern Oscillation in a comprehensive Earth system model. In these examples our approach discovers nontrivial optimal perturbations of the system, which are post hoc natural and consistent with the desired dynamical objectives.

We propose and study a simple and scalable Fourier-based spectral method for a continuum model of network formation under periodic boundary conditions. The model provides the evolution of the pressure $p$ and the conductivity $m$ over time. The evolution of $p$ is given by an anisotropic Poisson equation, while the equation for $m$ contains three terms corresponding to a diffusion and an activation term of the network -- that depends on the gradient of the pressure -- as well as a relaxation term that acts as a decaying term. This system arises as a formal $L^2$-gradient flow of a non-convex energy functional. Our algorithm combines two ingredients: (i) a splitting method for the equation for $m$, where the activation and relaxation parts are solved analytically, and the diffusion part is solved via Fast Fourier Transform (FFT), and (ii) an FFT combined with the Conjugate Gradient (CG) method applied to the equation for the pressure. This makes the scheme easy to implement compared to implicit schemes and naturally extensible to three dimensions on uniform periodic grids. To showcase the method, we recover the previously documented influence of the activation strength $c$, the diffusion coefficient $D$, and the metabolic exponent $γ$ on the morphology of emergent networks, and report grid convergence results.

We study a semidiscrete single-field Galerkin approximation of the Cahn-Hilliard equation using high-order conforming finite element spaces. More specifically, globally $C^1$ finite elements with $H^2$-conforming trial spaces, including Argyris, Bell, and Bogner-Fox-Schmit elements, allow a direct discretization of the fourth-order formulation and preserve mass exactly. The main structural result is an exact energy balance for the physical Cahn-Hilliard energy. Besides the expected discrete dissipation, the balance contains an explicitly computable energy defect. This defect vanishes for Laplacian-invariant periodic spaces, such as Fourier spaces, but is generally nonzero for classical $C^1$ finite elements. It therefore quantifies the precise loss of a discrete gradient-flow structure. We prove semidiscrete a priori error estimates by a relative-energy argument. The estimate is closed using an augmented relative energy and a discrete elliptic reconstruction bound for the inverse discrete Laplacian. The resulting convergence rates match the expected approximation orders. Numerical experiments with Bell and Argyris elements confirm the rates and demonstrate the defect mechanism by comparison with a Fourier reference discretization.

We give a product formula for the Tamagawa numbers of Jacobians over a discrete valuation field with perfect residue field k. It comes as a product of four terms - unipotent, toric, arithmetic and (somewhat intricate) cohomological. It is proved by (1) extending the classical flow-cut construction from semistable to arbitrary curves, which, by Raynaud's results, gives the formula when k is algebraically closed; (2) observing that this formula respects the natural metric on the edges of the dual graph, which allows to quotient out the Galois action; (3) extending Bosch-Liu's description of the cohomological term when k is finite. In particular, this answers a question of Bosch-Liu, and gives an alternative description of Poonen-Stoll's cohomological obstruction in terms of characters.

Gao and Wang proved a strong clopen marker theorem for finite generating sets of $\mathbb Z^n$ under the assumption that each generator has support of size either $1$ or $n$. We show that this support assumption can be removed. The proof is a short conjugacy argument: after a unimodular change of coordinates, any finite set of nonzero lattice vectors can be put in full-support position, allowing one to apply the theorem of Gao and Wang and conjugate the resulting marker set back.

Explicit arithmetic progressions modulo primes $p \equiv 1 \pmod{4}$ are derived in which the coefficients in the expansions of products of quintuple products vanish. In particular, if $p = m^{2} + n^{2}$, and $b$ is a positive integer, and $$\sum_{n=0}^{\infty} a_{n}q^{n} = \frac{(q^{2bm},q^{p-2bm};q^{2bn},q^{p-2bn};q^p)_{\infty}}{(q^p,-q^{b m},-q^{p-bm},-q^{bn},-q^{p-bn};q^p)_{\infty}^2},$$ we determine $α= α(m,n,p)$ such that $a_{pt+ α}=0$. Our results are proven using involutive transformations on integer lattices.

We generalize the Ahlswede--Daykin inequality (1978) to a Schur positive \emph{ADS inequality}, which also contains the Lam--Postnikov--Pylyavskyy inequality (2007) as a special case. We then present a number of further generalizations and applications. Notably, we resolve Mihalcea's conjecture on log-supermodularity of stable Grothendieck polynomials.

Let $D$ be a quaternion division algebra over a non-Archimedean local field $F$ of characteristic zero, and let $G_n=GL_n(D)$. Let $H_{1,n-1}$ denote the subgroup of $G_n$ consisting of block-diagonal matrices of the form $diag(g_1,g_2)$, where $g_1\in G_1$ and $g_2\in G_{n-1}$. In this article, we formulate a conjectural classification of irreducible smooth $H_{1,n-1}$-distinguished representations of $G_n$ for $n>2$. We prove this conjecture in the cases $n=3$ and $n=4$. When $n=2$, the results are well known due to the contributions by various authors.

We discuss self improving properties of some local and nonlocal, elliptic and parabolic, equations on bounded domains. We employ a functional analytic approach wherein the solution space sits in a suitable interpolation scale. Utilizing a classical analytic perturbation result, we extrapolate the invertibility of the main operator from the base space to nearby spaces within the interpolation family.

For a locally compact group G we introduce and study the reduced Beurling-Fourier-Stieltjes algebra, a weighted analogue of the reduced Fourier-Stieltjes algebra, together with the algebra of completely bounded multipliers of the associated weighted Fourier algebra. We show, in particular, that these two algebras coincide when G is amenable. For a general locally compact group G, we identify them as subspaces of the reduced Fourier-Stieltjes algebra and of the space of functions that locally belong to the Fourier algebra, respectively. Furthermore, we establish sufficient conditions on the group and the weight under which the algebra of completely bounded multipliers of the weighted Fourier algebra embeds into its unweighted counterpart.

Let $G$ be a compact abelian group whose dual group $Γ=\widehat{G}$ has bounded torsion. In 1967, Malliavin-Brameret and Malliavin proved that every Sidon set in $Γ$ is a finite union of quasi-independent sets when $Γ$ has prime exponent. This was later extended to squarefree exponents in work of Varopoulos and Bourgain. We prove the remaining bounded-torsion case. Consequently, if $\widehat{G}$ has bounded torsion, then a subset $Λ\subset \widehat{G}\setminus{0}$ is Sidon if and only if it is a finite union of quasi-independent sets.

This work considers combinatorial and statistical aspects of {\em{shifts of finite type}}, which are families of words over a finite alphabet which avoid a fixed class of {\emph{forbidden}} sub-words. The overarching question we are interested in is: how do local statistics of a uniformly random element of the shift space depend on combinatorial features of the forbidden set? We focus on the binary alphabet $\{0,1\}$, the class of shift spaces where a single pattern is forbidden, and the average frequency of $1$s (equivalently, the probability of observing $1$ at a given position). In this case, the relevant combinatorial information is encoded by a two-variable auto-correlation polynomial associated to the forbidden word, which we call the {\em{border polynomial}}. We present several results and examples characterizing the ordering of all words by their letter frequencies: for example, we describe the set of patterns which, when forbidden, cause the frequency of $1$s to increase, decrease, or stay exactly $1/2$. We conjecture that, among forbidden patterns of the same length (except for four exceptional words), the letter frequency is monotone with respect to the number of $1$s in the forbidden word. Our methodologies include novel explicit local injections and bijections, generating function analysis, and a connection with a probabilistic notion of letter frequency.

We address the Euler immersion problem, a fluid-structure interaction problem in which an elastic body is immersed in an incompressible inviscid fluid governed by the Euler equations. We show that the system exhibits a loss of one derivative and formulate the problem in analytic function spaces. We then prove local well-posedness in analytic spaces under velocity-matching boundary conditions. Finally, by means of an example, we show that existence fails in analytic spaces when both velocity and stress-matching boundary conditions are prescribed.

Let $\mathbf{k}$ be a field and let $Λ$ and $Γ$ finite dimensional $\mathbf{k}$-algebras. Assume that ${_Γ}X_Λ$ and ${_Λ}Y_Γ$ are bimodules that define a singular equivalence of Morita type with level (in the sense of Z. Wang) between $Λ$ and $Γ$ and which also induce an equivalence between the stable categories of finitely generated Gorenstein-projective modules $Λ$-$\underline{\text{Gproj}}$ and $Γ$-$\underline{\text{Gproj}}$. We prove that if $V$ is an indecomposable object in $Λ$-$\underline{\text{Gproj}}$ with $\underline{\mathrm{End}}_Λ(V)\cong \mathbf{k}$, then $X\otimes_ΛV$ is an object in $Γ$-$\underline{\text{Gproj}}$ such that $\underline{\mathrm{End}}_Γ(X\otimes_ΛV)\cong \mathbf{k}$ and the universal deformation rings (in the sense of F.M. Bleher and the second author) $R(Λ,V)$ and $R(Γ, X\otimes_ΛV)$ are isomorphic. This result generalizes the one obtained by the second author assuming that $Λ$ and $Γ$ are Gorenstein $\mathbf{k}$-algebras.

We introduce two new relations involving the pentagon and the quadrilateral for the evaluation of the Penrose polynomial at $n=4$ that is proven using a new type of ribbon graph polynomial. Additionally, we extend several relations for the evaluation of the Penrose polynomial at $n=3$ to all $n$.

Motivated by recent developments in Exceptional Orthogonal Polynomials (XOPs), which feature sequences of orthogonal polynomials missing finitely many degrees, we develop a construction of monic orthogonal polynomial sequences that omit a single degree using linear combinations of classical families. We then relate these polynomial families to quasi-orthogonal polynomials of order 2.

In this paper, we derive a new two-dimensional rod-Kadomtsev-Petviashvili (rod-KP) equation from the incompressible and irrotational three-dimensional Euler equation under the shallow water scaling. We establish the local well-posedness of the Cauchy problem in a suitable Sobolev space and derive the blow-up criterion for strong solutions via the energy method. Then, by exploring two appropriate conservation laws of the rod-KP equation, we are able to construct its global strong solution when the physical dimensionless parameter $σ=0$, and on the other hand produce the finite-time blow up solutions under some certain conditions when $σ\neq 0$. Furthermore, we present a uniqueness continuation property for the solutions. Finally, we investigate the existence of traveling-wave solutions in order to highlight the influence of weak transverse effects on wave stability, and we also exhibit the symmetry of solitary waves in the propagation direction.

In the round $N$-sphere, we calculate the Morse index and nullity of all immersed minimal 2-spheres having constant Gauss curvature. We also obtain bounds on the stability index of the associative cone in $R^7$ whose link is the Boruvka sphere in $S^6$.

Low-rank matrix denoising is a central primitive in patch-based image restoration and many other inverse problems. Classical SVD-based image denoising methods often choose a truncation rank by matching residual singular-value energy with an estimated noise energy, but this rule is not a finite-sample risk principle because a fitted low-rank approximation inevitably absorbs part of the noise. This paper develops a mathematically rigorous alternative based on Stein's unbiased risk estimate (SURE). Since singular value hard thresholding is discontinuous and does not satisfy the hypotheses of Stein's lemma, we introduce a logistic smooth hard-threshold spectral estimator. We prove that the smooth shrinker satisfies the regularity conditions required by a spectral-estimator version of Stein's lemma, and therefore admits an exactly unbiased fixed-threshold risk estimate under Gaussian noise. For a fixed observed matrix and a finite set of candidate thresholds separated from the observed singular values, the ordering of the fixed-threshold smooth SURE objective eventually agrees with a simple limiting score. The limiting score has the same algebraic form as the biased hard-threshold SURE formula, but here it is used only as a computational device for ranking finite candidates. Selecting the minimizing threshold is a data-adaptive tuning step; the selected SURE value should not be interpreted as an unbiased risk estimate of the finally selected estimator.

In this work, we establish compactness theorems for complete gradient $G_2$-solitons under the assumptions of a lower bound on the scalar curvature and a broad growth condition on the potential function associated with the gradient vector field. After first proving Gromov-Hausdorff convergence for such sequences, we sharpen this result by deriving epsilon-regularity estimates. As a consequence, we obtain smooth convergence provided there is a uniform energy bound at half the dimension.

Consider the $(2+1)$D Discrete Gaussian model (ZGFF) on an $L\times L$ box with a hard floor at height zero and zero boundary conditions, at low temperature. The second author, Martinelli and Sly (2016) showed that the surface has a plateau, filling nearly the full square, at height either $H$ or $H+1$ for an explicit function $H(L)$. In a companion paper, we studied the local laws of the top level lines near the four sides of the box, and showed that after rescaling each by $(L^{2/3-o(1)},L^{1/3-o(1)})$, they converge to a product of Ferrari--Spohn diffusions. Two key features of the top level lines remained unaddressed: their global limit shape, and the critical window marking the transition from a top plateau at height $H$ to one at height $H+1$. These features are intrinsically linked: deriving the global limit of the top level line is needed for determining whether it is preferable to be at height $H$ or $H+1$ near criticality. This work completes this picture as follows. First, we obtain the global limit of the top level lines: for every fixed $n$, the $n$-th from-the-top level line converges in Hausdorff distance to a deterministic shape $\mathscr{L}_n$ that features the Wulff shape at scale $N_n=L^{1-o(1)}$ near the four corners of the box. Second, we identify, for every $h$, the point of emergence of a macroscopic $h$ level line: the probability of this event is monotone increasing in $L$ (up to a $o(1)$ error), and undergoes a sharp transition from near $0$ to near $1$ in a critical window of width $\leq L^{1/2+o(1)}$ around a side length $L=L_c^{(h)}$. This transition is discontinuous in that, once a macroscopic level $h$ emerges, it immediately occupies nearly all the box, and the above global and local scaling limits (Wulff, Ferrari--Spohn) hold for it. The new results extend to the $(2+1)$D $|\nablaϕ|^p$-models (ZGFF is the case $p=2$) for every fixed $p> 1$.

A guiding principle in P. Scholze's $p$-adic geometry asserts that the points of $\operatorname{Spec}\mathbf{Z}$ over an algebraically closed perfectoid field of characteristic $p$ are classified, up to equivalence, by its untilts. In this paper, we give a concrete geometric realization and a generalization of this paradigm. We construct the absolute $\mathbf{F}_1$-arithmetic curve $(\operatorname{Spec}\mathbf{Z})_{\mathbf{F}_1}$ by pulling back the $\mathbf{F}_1$-structure sheaf of the arithmetic site to $\operatorname{Spec}\mathbf{Z}$. We demonstrate that $(\operatorname{Spec}\mathbf{Z})_{\mathbf{F}_1}$ provides a common geometric origin for fundamental structures in $p$-adic Hodge theory, complex analytic geometry, and the adelic scaling site. The moduli space of points of $(\operatorname{Spec}\mathbf{Z})_{\mathbf{F}_1}$ over an arbitrary perfectoid field, modulo intrinsic symmetries, canonically parameterizes the space of all perfectoid fields with the same tilt, providing a universal, characteristic-independent geometric realization of Scholze's heuristic. Evaluating the points of $(\operatorname{Spec}\mathbf{Z})_{\mathbf{F}_1}$ over the field $\mathbf{C}$ of complex numbers reveals, at each prime $p$, that the non-trivial points canonically form two principal homogeneous spaces (torsors) over the Weil groups $W_p=\mathbf{Q}_p^\times$ and $W_\infty=\mathbf{C}^\times$. Quotienting the archimedean orbit by the discrete Frobenius symmetries yields the complex Tate curve with modulus $q=p^{-1}$. We show that this elliptic curve canonically decomposes as the product of its real locus, which exactly recovers the adelic periodic orbit $C_p=\mathbf{R}_+^\times/p^\mathbf{Z}$, and a $p$-independent phase space that emerges naturally as a real analogue of the Fargues--Fontaine curve.

In this paper, we investigate uncertainty principles for random signals associated with the linear canonical transform (LCT). First, the LCT of random signals is formulated on the probability space. Based on this representation, the Heisenberg uncertainty principle is established to characterize the relationship between the expectations in the time and frequency domains. Furthermore, the Donoho-Stark uncertainty principle, developed from a measure theoretic perspective, reveals that a random signal cannot be simultaneously concentrated on arbitrarily small sets in both the time and frequency domains. The bounds obtained in these two uncertainty principles explicitly depend on the LCT parameters, indicating that the LCT offers greater flexibility than the Fourier transform (FT). The corresponding results in the fractional Fourier transform and FT domains are also given as special cases.

We prove that if $j>0$ is not an integer, then there is a choice of signs $\varepsilon_k\in\{\pm1\}$ such that the partial sums $ \sum_{k=1}^{N}\varepsilon_k k^j $ are dense in $\mathbb R$. The proof groups consecutive terms into Thue--Morse blocks, whose Prouhet--Tarry--Escott cancellation produces nonzero block sums tending to zero but with divergent total variation. A standard steering argument then chooses block signs so that the resulting partial sums visit arbitrarily small neighbourhoods of every real number.

For the better part of a century, isolated singularities were treated as pathological obstructions requiring elaborate avoidance strategies. Pierre Alphonse Laurent (1813--1854), a French military engineer at Le Havre, ended this avoidance in 1843 by extending Cauchy's Taylor-type theorem to doubly connected (annular) domains, producing the doubly infinite power series that now bears his name. Negative-power terms in the expansion encode precise geometric information about the singularity rather than signaling a breakdown of the formalism. Laurent's contribution arrived through an unhappy institutional trajectory -- submitted after a prize deadline, subjected to a priority claim by Cauchy, and issued in full only posthumously in 1863 -- yet it became indispensable to every branch of mathematics and mathematical physics that touches on complex function theory. We reconstruct the mathematical problem Laurent solved, place it within Cauchy's analytic program of the 1830s--1840s, examine the institutional failure that prevented publication, document the independent parallel proof by Weierstrass (1841, published 1894), and trace the series' absorption into the standard toolkit via Briot and Bouquet and the residue calculus. Drawing on Laurent's 1843 Comptes rendus notice, Cauchy's Academy report, Bertrand's memorial notice (1890), and the secondary literature (Neuenschwander 1978, 1981; Manning 1975; Bottazzini 1986; Gray 2015), we analyze the philosophical significance of the series, which we term ``exile mathematics'', and survey its reach into perturbation theory, number theory, probability, and quantum field theory. Readers familiar with the theorem but not its institutional history will find here a documented account of why a foundational result was withheld for two decades and how it nevertheless achieved canonical status.

Let $G$ be an $n$-vertex graph with $e(G)\ge n^2/k$. We prove a self-contained internal short-cycle core theorem at the threshold $k\le n^{1/3}$: the graph $G$ contains a subgraph $H_6$ with $Ω(n^2/k^3)$ edges in which every two distinct edges lie together on a cycle of length at most $6$ contained in $H_6$, and a subgraph $H_8$ with $Ω(n^2/k^2)$ edges in which every two distinct edges lie together on a cycle of length at most $8$ contained in $H_8$. In density notation $ρ=e(G)/n^2$, this gives internal cores of sizes $Ω(ρ^3n^2)$ and $Ω(ρ^2n^2)$ throughout the range $ρ\ge n^{-1/3}$. The $C_{\le6}$ conclusion above is an edge-connected statement and does not impose the adjacent-edge $C_4$ condition appearing in the strongest Duke--Erdős--Rödl formulation. We also include two complementary results clarifying this distinction. First, under the ambient-witness convention, every graph with at least $n^2/k$ edges and $k=o(n^{1/2})$ contains $Ω(n^2/k^3)$ selected edges whose pairs are witnessed by ambient cycles of length at most $6$, with adjacent pairs witnessed by ambient $C_4$'s. Second, under the standard internal strong $C_6$ convention, for every fixed $β\in[1/3,1/2)$ there is an infinite sequence of bipartite graphs $G$ with $n\to\infty$ and $e(G)=Θ_β(n^{2-β})$ such that every internally strongly $C_6$-connected subgraph has only $O_β(ρ(G)^3n^2/(\log n)^2)$ edges. The obstruction is a random cyclic shift-lift of $K_{q,q}$, together with an occupancy estimate excluding large aligned two-covers.

Majorization is a fundamental tool for comparing vectors, with connections to convexity, doubly stochastic matrices, eigenvalues, singular values, and zeros of polynomials. In matrix analysis, it plays a central role in the study of eigenvalue inequalities, particularly those arising from classical determinantal inequalities such as those attributed to Hadamard and Fischer in the context of positive semidefinite matrices. A result of Fischer and Holbrook shows that equality in the Hardy--Littlewood--Pólya theorem for non-affine convex functions is closely linked to block structure in the associated doubly stochastic transformations. Motivated by this, we introduce $*$-majorization, a structured extension of majorization that respects prescribed block decompositions of vectors. This framework naturally corresponds to block diagonal doubly stochastic matrices and provides a refinement of the classical Hardy--Littlewood--Pólya and Rado theorem. We show that such transformations are precisely the linear operators that preserve $*$-majorization, and we extend fundamental constructions such as $T$-transforms and convex combinations to this setting. In an application, we study the eigenvalue relations associated with the principal submatrices of positive definite matrices. Classical majorization does not, in general, capture determinantal inequalities such as those of Koteljanskii, whereas $*$-majorization provides a natural framework for structured comparisons of eigenvalue vectors. This leads to new insights into the interplay between majorization theory, determinantal inequalities, and spectral properties of matrices.

In this paper, we systematically develop the theory of mock-pre-Lie bialgebras from multiple perspectives. We introduce the notion of a phase space of a mock-Lie algebra, and show that a mock-Lie algebra admits a phase space if and only if it is sub-adjacent to a mock-pre-Lie algebra. We introduce the notions of Manin triples of mock-pre-Lie algebras and mock-pre-Lie bialgebras, and prove the equivalences between mock-pre-Lie bialgebras, Manin triples of mock-pre-Lie algebras, certain matched pairs of mock-pre-Lie algebras, certain matched pairs of mock-Lie algebras and phase spaces of a mock-Lie algebra, which lays a theoretical foundation for subsequent research. Next, we investigate coboundary mock-pre-Lie bialgebras, and derive an analogue of the classical Yang-Baxter equation. In addition, we introduce two important special classes of mock-pre-Lie bialgebras: quasi-triangular mock-pre-Lie bialgebras and factorizable mock-pre-Lie bialgebras. We show that quasi-triangular mock-pre-Lie bialgebras naturally induce relative Rota-Baxter operators of weight -1. Finally, we provide a new perspective for the study of triangular and factorizable mock-pre-Lie bialgebras by introducing the concept of quadratic Rota-Baxter mock-pre-Lie algebras of arbitrary weight.

QBism recasts quantum theory as a normative framework for an agent's probability assignments, with the Born rule taking the form of a consistency condition known as the Urgleichung. Motivated by this perspective, qplex theories provide a broader class of probabilistic models in which the sets of valid states and measurements are constrained by QBist-inspired geometric conditions. While qplexes have been extensively studied for single systems, their implications for bipartite correlations remain largely unexplored. In this work, we investigate bipartite correlations in qplex theories by expressing joint expectation values as inner products between suitably defined $C$-vectors. This geometric formulation allows Bell-type inequalities to be studied as optimization problems over qplex-compatible probability assignments. We first analyze the CHSH scenario and show that the shared inner-product structure of the $C$-vectors restricts the maximal value to the Tsirelson bound $2\sqrt{2}$. We then turn to the three-outcome CGLMP inequality $I_{2233}$ and find that the same qplex-derived norm and inner-product constraints allow the algebraic maximum of 4, thereby exhibiting superquantum correlations. These results show that qplex geometry captures enough structure to reproduce an important quantum bound in the two-outcome case, but not enough to recover the full set of quantum correlation constraints. The analysis therefore suggests that additional principles are needed to complete the QBist reconstruction of quantum theory.