The excess growth rate, defined as the gap in Jensen's inequality for the logarithm, is a fundamental functional in portfolio theory. In this paper, we present a mathematical study motivated by information theory. We begin by establishing its properties and showing that it has rich connections with information theoretic concepts such as the Helmholtz free energy, L. Campbell's measure of average code length and large deviations. Our main results consist of three axiomatic characterization theorems of the excess growth rate, in terms of (i) the relative entropy, (ii) the gap in Jensen's inequality, and (iii) the logarithmic divergence that generalizes the Bregman divergence. Furthermore, we study maximization of the excess growth rate and compare it with the growth optimal portfolio. Our results not only provide theoretical justifications of the significance of the excess growth rate, but also establish new connections between information theory and quantitative finance.

About ten years ago, Katz, Klemm and Huang conjectured that topological string amplitudes on compact, elliptically fibered Calabi-Yau threefolds at fixed base degree could be expressed in terms of meromorphic Jacobi forms for $SL(2,\mathbb{Z})$, giving access to Gromov-Witten invariants at arbitrary genus. This was later generalized to torus-fibered CY threefolds with $N$-sections, where topological string amplitudes are conjecturally governed by meromorphic Jacobi forms under the congruence subgroup $\Gamma_1(N)$. In this work, we show that these modularity properties follow from (and are equivalent to) the wave-function property of the topological string partition function $Z_{\rm top}$ under a relative conifold monodromy, implementing a particular Fourier-Mukai transformation on the derived category of coherent sheaves. In particular, we introduce a variant of $Z_{\rm top}$ which is both holomorphic and modular covariant. Under the same relative conifold monodromy, the generating series of genus 0 Gopakumar-Vafa invariants at fixed base degree is mapped to the generating series of rank 0 Donaldson-Thomas indices counting D4-D2-D0-brane bound states wrapped on the torus fiber. We show that the quasimodularity of the generating series of GV invariants matches the expected mock-modular behavior of the generating series of D4-D2-D0 indices, despite having different multi-cover contributions. We analyze and tabulate a large number of CY threefolds fibered over del Pezzo surfaces, with an $N$-section for $N\leq 5$, including several new examples beyond the realm of toric geometry.

Model Predictive Control (MPC) is well understood in the deterministic setting, yet rigorous stability and performance guarantees for stochastic MPC remain limited to the consideration of terminal constraints and penalties. In contrast, this work analyzes stochastic economic MPC with an expected cost criterion and establishes closed-loop guarantees without terminal conditions. Relying on stochastic dissipativity and turnpike properties, we construct closed-loop Lyapunov functions that ensure $P$-practical asymptotic stability of a particular optimal stationary process under different notions of stochastic convergence, such as in distribution or in the $p$-th mean. In addition, we derive tight near-optimal bounds for both averaged and non-averaged performance, thereby extending classical deterministic results to the stochastic domain. Finally, we show that the abstract stochastic MPC scheme requiring distributional knowledge shares the same closed-loop properties as a practically implementable algorithm based only on sampled state information, ensuring applicability of our findings. Our findings are illustrated by a numerical example.

A system $\boldsymbol\lambda_{\theta}$ is developed that combines modal logic and simply-typed lambda calculus, and that generalizes the system studied by Montague and Gallin. Whereas Montague and Gallin worked with Church's simple theory of types, the system $\boldsymbol\lambda_{\theta}$ is developed in the typed base theory most commonly used today, namely the simply-typed lambda calculus. Further, the system $\boldsymbol\lambda_{\theta}$ is controlled by a parameter $\theta$ which allows more options for state types and state variables than is present in Montague and Gallin. A main goal of the paper is to establish some basic metatheory of $\boldsymbol\lambda_{\theta}$: (i) an Andrews-like characterization of its models in terms of combinatory logic is given, and this combinatory logic involves a $\mathsf{BCKW}$-like basis rather than an $\mathsf{SKI}$-like basis and (ii) semantic conservation and expressibility results relating $\boldsymbol\lambda_{\theta}$ to the maximal system $\boldsymbol\lambda_{\omega}$ are proven. Similar results are proven for the relation between $\boldsymbol\lambda_{\omega}$ and$\boldsymbol\lambda$, the corresponding ordinary simply-typed lambda calculus. This answers a question of Zimmermann in the semantics of the simply typed setting. In a companion paper this is extended to Church's simple theory of types. We further develop a partial correspondence between a pure combinatory logic centered on the $\mathsf{BCKW}$-like basis and the weak deductive system for $\boldsymbol\lambda_{\omega}$ wherein $\beta$-reduction is not allowed under a lambda abstract, and we use this to show partial deductive conservation between the maximal system $\boldsymbol\lambda_{\omega}$ and the intermediary systems $\boldsymbol\lambda_{\theta}$.

We study the asymptotic behavior of random dimer coverings of the one-periodic Aztec diamond in random environment. We investigate quenched limit theorems for the height function and we extend annealed limit theorems that were recently studied in [ arXiv:2507.08560 ]. We consider more general choices of random edge weights (independence is not assumed) and we distinguish two cases where the random edge weights satisfy the Central Limit Theorem (CLT) under different scalings. For both cases, we prove convergence to the Gaussian Free Field for the quenched fluctuations. For the annealed version, it had been shown in [ arXiv:2507.08560 ], that Gaussian Free Field fluctuations can be dominated by the much larger fluctuations of the random environment. To access quenched fluctuations we analyze the Schur process with random parameters in a way that allows to prove the annealed CLT for the height function for non i.i.d. weights. We consider specific examples where we determine the asymptotic fluctuations.

A second-order block coordinate descent method is proposed for the unconstrained minimization of an objective function with a Lipschitz continuous Hessian. At each iteration, a block of variables is selected by means of a greedy (Gauss-Southwell) rule which considers the amount of first-order stationarity violation, then an approximate minimizer of a cubic model is computed for the block update. In the proposed scheme, blocks are not required to have a predetermined structure and their size may change during the iterations. For non-convex objective functions, global convergence to stationary points is proved and a worst-case iteration complexity analysis is provided. In particular, given a tolerance $\epsilon$, we show that at most ${\cal O(\epsilon^{-3/2})}$ iterations are needed to drive the stationarity violation with respect to a selected block of variables below $\epsilon$, while at most ${\cal O(\epsilon^{-2})}$ iterations are needed to drive the stationarity violation with respect to all variables below $\epsilon$. Numerical results are finally given, comparing the proposed approach with other second-order methods and block selection rules.

We introduce coarse scrambling, a novel randomization for digital sequences that permutes blocks of digits in a mixed-radix representation. This construction is designed to preserve the powerful $(0,\mathbb{e},d)$-sequence property of the underlying points. For sufficiently smooth integrands, we prove that this method achieves the canonical $O(n^{-3+\epsilon})$ variance decay rate, matching that of standard Owen's scrambling. Crucially, we show that its maximal gain coefficient grows only logarithmically with dimension, $O(\log d)$, thus providing theoretical robustness against the curse of dimensionality affecting scrambled Sobol' sequences. Numerical experiments validate these findings and illustrate a practical trade-off: while Owen's scrambling is superior for integrands sensitive to low-dimensional projections, coarse scrambling is competitive for functions with low effective truncation dimension.

Since the 1970s, it has been known that any open connected manifold of dimension 2, 4 or 6 admits a complex analytic structure whenever its tangent bundle admits a complex linear structure. For half a century, this has been conjectured to hold true for manifolds of any dimension. In this paper, we extend the result to manifolds of dimension 8. To prove the result new $\Gamma_n^\mathbb{C}$-structures on $\mathbb{CP}^n$ are constructed. As a consequence we derive a theorem concerning the homology of Haefligers classifying space, $B\Gamma_n^\mathbb{C}$. The result then follows from obstruction theory.

A real number is called left-computable if there exists a computable increasing sequence of rational numbers converging to it. In this article we investigate the Kolmogorov complexity and the binary expansions of a very specific subset of the left-computable numbers. We show in our main result that a real number is reordered computable if, and only if, it is left-computable and not strongly Kurtz random. In preparation of this, we characterize strong Kurtz randomness by a suitable notion of randomness tests. We also look at the binary expansions of reordered computable numbers and clarify whether they can be immune, hyperimmune, hyperhyperimmune, strongly hyperhyperimmune, or cohesive. Then, we investigate the effective Hausdorff and packing dimensions of reordered computable numbers. Finally, we have a short look at regular reals in the context of immunity properties, Kolmogorov complexity and (strong) Kurtz randomness.

We prove optimal non-commutative analogues of the classical Law of Iterated Logarithm (LIL) for both martingales and sequences of independent (non-commutative) random variables. The classical martingale version was established by Stout [Sto70b] and the independent case by Hartman-Wintner [HW41]. Our approach relies on a key exponential inequality essentially due to Randrianantoanina [Ran24] that improves that from Junge and Zeng [JZ15]. It allows to derive an optimal non-commutative Stout-type LIL just as in [Zen15], from that martingale result we then deduce a non-commutative Hartman-Wintner type LIL for independent sequences of random variables.

Fighting fish is a combinatorial configuration introduced by Duchi, Guerrini, Rinaldi and Schaeffer as a new model of branching surfaces that generalizes directed convex polyominoes. We come up with an alternative construction of fighting fish, using a tree structure built on the so-called stem cells of fighting fish. From this perspective, we establish a bijection between ternary trees and fighting fish with a marked strip of cells, which specializes to a direct bijection between left ternary trees and fighting fish. Using these results, we obtain a combinatorial enumeration of the fighting fish of size $n$ by establishing an $(n+1)$-to-2 bijection with the ternary trees with $n$ nodes. We present some additional enumerative results including that fighting fish with a marked tail and horizontally symmetric fighting fish are equinumerous with ordered pairs of ternary trees having a total of a given number of nodes.

We show that the statistics of chaotic systems can be predicted by constructing an associated sequence of periodic differential operators and computing their densities of states. For such operators, the density of states is well understood and can be computed straightforwardly, often yielding explicit formulas. As a case study, we investigate a nonlinear recursion relation that maps naturally onto a family of periodic operators generated by a Fibonacci tiling rule. This correspondence enables us to derive an explicit formula for the limiting statistics of the chaotic system and to demonstrate that the clustering near to critical values is equivalent to the van Hove singularities in the operators' densities of states.

We study the dynamics of multidimensional contracted rotations and address a problem posed by Y. Bugeaud and J-P. Conze in \textit{Acta Arithmetica} in 1999. More precisely, we show that if $A$ is an invertible linear contraction of $\mathbb{R}^d$, then the map $f: [0,1)^d\to [0,1)^d$ defined by $f(x) = Ax +b\,\,(\textrm{mod}\,\mathbb{Z}^d)$ is asymptotically periodic for Lebesgue almost all $b\in\mathbb{R}^d$. We also include an example of a family of multidimensional contracted rotations $(d>1)$ not conjugate to the product of one-dimensional contracted rotations $(d=1)$, showing that our result cannot be reduced to or derived from the one-dimensional result of Bugeaud and Conze.

We investigate the mean value of the first moment of primitive quartic $L$-functions over $\mathbb{F}_q(T)$ in the non-Kummer setting. Specifically, we study the sum \begin{equation*} \sum_{\substack{\chi\ primitive\ quartic\\ \chi^2 primitive\\ genus(\chi)=g}}L_q(\frac{1}{2}, \chi), \end{equation*} where $L_q(s,\chi)$ denotes the $L$-function associated with primitive quartic character $\chi$. Using double Dirichlet series, we derive an error term of size $q^{(\frac{3}{5}+\varepsilon)g}$.

In this article, we study a degenerate version of Ramsey's theorem for pairs and two colors ($\mathsf{RT}^2_2$), in which the homogeneous sets for color 1 are of bounded size. By $\mathsf{RT}^2_2$, it follows that every such coloring admits an infinite homogeneous set for color 0. This statement, called $\mathsf{BRT}^2_2$, is known to be computably true, that is, every computable instance admits a computable solution, but the known proofs use $\Sigma^0_2$-induction ($\mathsf{I}\Sigma_2^0$). We prove that $\mathsf{BRT}^2_2$ follows from the Erdős-Moser theorem but not from the Ascending Descending sequence principle, and that its computably true version is equivalent to $\mathsf{I}\Sigma_2^0$ over $\mathsf{RCA}_0$.

We study symplectic groups and indefinite orthogonal groups over involutive, possibly noncommutative, algebras $(A, \sigma)$. In the case when the algebra $(A, \sigma)$ is Hermitian, or the complexification $(A_{\mathbb C}, \sigma_{\mathbb C})$ of a Hermitian involutive algebra, one can identify maximal compact subgroups of such groups, and consider their associated Riemannian symmetric spaces. This new perspective allows for the realization of various geometric models for the symmetric space. We describe explicitly the complexified tangent space for each of the models, as well as the diffeomorphisms between them and their differentials. In the second part of the article, we give a number of applications of this theory. The geometric realizations of the Riemannian symmetric spaces described in the first part provide new geometric interpretations of Higgs bundle data that can be used for the study of fundamental group representations into symplectic or into indefinite orthogonal groups over Hermitian involutive algebras. We give an exact component count for the moduli spaces of $\rm{Sp}_2(A_{\mathbb C}, \sigma_{\mathbb C})$-Higgs bundles and of $\rm O(A_{\mathbb C}, \sigma_{\mathbb C})$-Higgs bundles, using the topology of the corresponding maximal compact subgroups rather than Morse-Bott theory techniques. Furthermore, we use the noncommutative symmetric-space models to construct a factorization of the Hitchin morphism for $\rm{Sp}_2(A_{\mathbb C},\sigma_{\mathbb C})$-Higgs bundles, together with analogous factorizations for the real groups $\rm{Sp}_2(A,\sigma)$ and $\rm O_{(1,1)}(A,\sigma)$. These factorizations are induced by quadratic norm maps from the corresponding tangent models to Jordan-algebraic targets and pass through intermediate affine GIT quotients. As a consequence, they reduce the algebraic complexity required in order to characterize the Hitchin base explicitly.

We introduce triple quadratic residue symbols $[\mathfrak{p}_1, \mathfrak{p}_2, \mathfrak{p}_3]$ for certain finite primes $\mathfrak{p}_i$'s of a real quadratic field $k$ with trivial narrow class group. For this, we determine a presentation of the Galois group of the maximal pro-2 Galois extension over $k$ unramified outside $\mathfrak{p}_1, \mathfrak{p}_2, \mathfrak{p}_3$ and infinite primes, from which we derive mod 2 arithmetic triple Milnor invariants $\mu_2(123)$ yielding the triple symbol $[\mathfrak{p}_1, \mathfrak{p}_2, \mathfrak{p}_3] = (-1)^{\mu_2(123)}$. Our symbols $[\mathfrak{p}_1, \mathfrak{p}_2, \mathfrak{p}_3]$ describes the decomposition law of $\mathfrak{p}_3$ in a certain dihedral extension $K$ over $k$ of degree 8, determined by $\mathfrak{p}_1, \mathfrak{p}_2$. The field $K$ and our symbols $[\mathfrak{p}_1, \mathfrak{p}_2, \mathfrak{p}_3]$ are generalizations over real quadratic fields of Rédei's dihedral extension of $\mathbb{Q}$ and Rédei's triple symbol of rational primes. We give examples of Rédei type extensions $K$ over real quadratic fields. We also give a cohomological interpretation of our symbols in terms of Massey products.

We prove a differential transcendence result of type "Ax-Lindemann-Weierstrass" for Euler's gamma function. Given meromorphic functions $\zeta_1,\dots,\zeta_n$ of a complex variable $\nu$ that are pairwise distinct modulo $\mathbb Z$ and algebraic over the field $k$ of meromorphic $1$-periodic functions, the functions $ \Gamma(\nu-\zeta_1(\nu)),\dots,\Gamma(\nu-\zeta_n(\nu))$ are differentially independent over the field $k(\nu)$. We determine the structure of certain difference field extensions related to the torsion of an avatar of the Carlitz module over meromorphic functions. These extensions are abelian and purely transcendental, the latter property being crucial in our main result, and obtained applying a criterion of differential algebraicity of Hardouin and Singer.

Quantum entanglement lies at the heart of quantum information science, yet its reliable detection in high-dimensional or noisy systems remains a fundamental computational challenge. Semidefinite programming (SDP) hierarchies, such as the Doherty-Parrilo-Spedalieri (DPS) and Extension (EXT) hierarchies, offer complete methods for entanglement detection, but it is well known that their practical use is limited by exponential growth in problem size if implemented naively. We make three contributions. First, we introduce a new SDP hierarchy, PST, that is sandwiched between EXT and DP -- offering a tighter approximation to the set of separable states than EXT, while incurring significantly lower computational overhead than DPS. Second, we explicitly construct compact, polynomially-scalable descriptions of EXT and PST using partition mappings and operators. These descriptions in turn yield formulations that satisfy desirable properties such as the Slater condition and are well-suited to both first-order methods (FOMs) and interior-point methods (IPMs). Third, we design a suite of entanglement detection algorithms: three FOMs (Frank-Wolfe, projected gradient, and fast projected gradient) based on a least-squares formulation, and a custom primal-dual IPM based on a conic programming formulation. These methods are numerically stable and capable of producing entanglement witnesses or proximity measures, even in cases where states lie near the boundary of separability. Numerical experiments on benchmark quantum states demonstrate that our algorithms improve the ability to solve deeper levels of the SDP hierarchy.

Complete intersections may be unexpectedly simple over fields of positive characteristic: for instance, they may be unirational despite being of general type. One explanation is given by profiles, structure that tracks the special shape of polynomials, refining the degree. The aim of this work is to show that complete intersections with small profile should be considered simple by generalizing two classical results on low degree complete intersections: First, the basic geometry of Fano schemes associated with complete intersections depends only on the profile, so that complete intersections with small profile contain many linear spaces. Second, a general complete intersection is unirational once its dimension is sufficiently large compared to its profile.

The determinant of a tournament $T$, denoted by $\det(T)$, is defined as the determinant of the skew-adjacency matrix of $T$. It is well-known that $\det(T)$ is equal to $0$ if $n$ is odd, and $\det(T)$ is the square of an odd integer if $n$ is even. For a positive odd integer $k$, let $\mathcal{D}_k$ be the set of tournaments whose all subtournaments have determinant at most $k^2$. Former studies showed that for $k \in \{1,3,5\}$, a tournament $T \in \mathcal{D}_k \backslash \mathcal{D}_{k-2}$ ($T \in \mathcal{D}_1$ when $k=1$) if and only if $T$ is switching equivalent to a transitive blowup of $L_{k+1}$, where $L_{k+1}$ is a tournament of order $k+1$ with a specific structure. For $k \geq 7$, no characterization results are known. A natural problem is to characterize tournaments in $\mathcal{D}_{k} \backslash \mathcal{D}_{k-2}$ that can be switching equivalent to a transitive blowup of $L_{k+1}$ for $k \geq 7$. To address this problem and to further explore the structural properties of tournaments in $\mathcal{D}_{k}$, we introduce CR tournaments, strong CR tournaments, basic tournaments and $Z$-matrices, and investigate their properties. We use these properties to characterize those tournaments $T \in \mathcal{D}_{k} \backslash \mathcal{D}_{k-2}$ where $T$ contains a subtournament switching isomorphic to a basic strong CR tournament in $\mathcal{D}_{k} \backslash \mathcal{D}_{k-2}$. This result implies former characterizations of $\mathcal{D}_3\backslash \mathcal{D}_1$ and $\mathcal{D}_5 \backslash \mathcal{D}_3$. Using $Z$-matrices, we also show that for even $n$, $L_{n}$ is a basic strong CR tournament, and thus solve the open problem posed in [Discrete Math. 349 (2) (2026) 114766].

We first obtain a simpler proof of the main results in [IEOT, {\bf 93}(2021), 17], which characterized the bounded and compact differences $C_{u,\varphi}-C_{v,\psi}$ of two weighted composition operators acting between different Bergman spaces. More importantly, we get some characterizations for the difference of two weighted composition operators belonging to Schatten class. Futhermore, the compact difference of two weighted composition operators acting on Hardy-Hilbert spaces is also studied.

A novel, stochastic approach to amputation, the process of introducing missing values to a complete dataset, is presented. It allows one to construct a wide variety of missingness patterns by only having to specify distributions of missingness indicators as opposed to specifying each missingness pattern manually. Missingness indicators are modeled in a principled way via copulas and Bernoulli margins, thus allowing one to incorporate dependence in missingness patterns. Besides more classical missingness mechanisms such as missing completely at random, missing at random, and missing not at random, the approach is able to model structured missingness such as block missingness and, via mixtures, monotone missingness, which are patterns of missing data frequently found in real-life datasets. Properties such as joint missingness probabilities or missingness correlation are derived mathematically. The flexibility of the approach in capturing different missingness patterns while only requiring to specify distributional assumptions on missingness indicators is demonstrated with mathematical examples and empirical illustrations in terms of a well-known example dataset of sufficiently small sample size that allows to identify each missing data point visually. Finally, an example application to multivariate financial time series is provided.

The hexablock is a domain arising from a special case of the $\mu$-synthesis problem. We study the commuting operator tuples having the hexablock as a spectral set. Such a tuple is called a hexablock-contraction or simply $\mathbb H$-contraction. We characterize the unitaries and isometries associated with $\mathbb H$-contractions. Two different types of dilation results for $\mathbb H$-contractions are obtained. We find connection of this theory with the operators associated with the symmetrized bidisc and tetrablock, two other domains related to the $\mu$-synthesis problem.

We establish a connection between barcode entropy and metric entropy. Namely, we show that the barcode entropy bounds the metric entropy from below for a measure from a specific class of invariant measures associated with a pair of Lagrangian or Legendrian submanifolds, which we call pseudo-chord measures. This inequality refines, via the variational principle, the previously known upper bound of barcode entropy by topological entropy.

We ask whether a stationary lattice in dimension $d$ whose points are shifted by identically distributed but possibly dependent perturbations remains hyperuniform. When $d = 1$ or $2$, we show that it is the case when the perturbations have a finite $d$-moment, and that this condition is sharp. When $d \geq 3$, we construct arbitrarily small perturbations such that the resulting point process is not hyperuniform. As a side remark of independent interest, we exhibit hyperuniform processes with arbitrarily slow decay of their number variance.

Let $C$ be a genus one nodal curve over a local artinian base and let $E$ be a proper subcurve of genus one. We define residues for curves over local artinian rings, then define generalized residues with values in line bundles over the local artinian ring that arise from tropical data on the curve. We then use these residues to construct a contraction of $C$ that collapses $E$ to a Gorenstein genus one singularity.

In this paper, we study the almost everywhere convergence of sequences of two-parameter ergodic averages over rectangles in the plane. On the one hand, we show that if the rectangles we consider have their sides with slopes in a finitely lacunary set, then the averages converge almost everywhere in all $L^p$ spaces, $1 < p < \infty$. On the other hand, given some non-lacunary sets of directions, we construct sequences of rectangles oriented along these directions for which the associated ergodic averages fail to converge almost everywhere in any $L^p$ space, $1 < p < \infty$.

The Fantappiè and Laplace transforms realize isomorphisms between analytic functionals supported on a convex compact set $K\subset{\mathbb C}^n$ and certain spaces of holomorphic functions associated with $K$. Viewing the Bergman space of a bounded domain in ${\mathbb C}^n$ as a subspace of the space of analytic functionals supported on its closure, the images of the restrictions of these transforms have been studied in the planar setting. For the Fantappiè transform, this was done for simply connected domains (Napalkov Jr--Yulumukhamtov, 1995), and for the Laplace transform, this was done for convex domains (Napalkov Jr--Yulumukhamtov, 2004). In this paper, we study this problem in higher dimensions for strongly convex domains, and establish duality results analogous to the planar case. We also produce examples to show that the planar results cannot be generalized to all convex domains in higher dimensions.

The finite element solution of Helmholtz problems near resonant or quasi-resonant frequencies poses significant challenges, as iterative solvers typically suffer from severely degraded convergence. We analyze the convergence behavior of GMRES applied to linear systems arising from such configurations. Theoretical convergence estimates are derived based on harmonic Ritz values, highlighting their proximity to small eigenvalues as a key determining factor. We further examine deflation strategies and their interplay with preconditioning techniques, using the Complex Shifted Laplacian preconditioner as a case study. Numerical experiments on resonant and quasi-resonant test cases validate the theoretical framework and demonstrate the effectiveness of deflation strategies. This study provides new insights and practical guidance for analyzing and improving iterative solvers for time-harmonic problems near resonances.