arXiv Math @arxiv_math@qoto.org

We consider a family of metric generalized connections on transitive Courant algebroids, which includes the canonical Levi-Civita connection, and study the flatness condition. We find that the building blocks for such flat transitive Courant algebroids are compact simple Lie groups. Further, we give a description of left-invariant flat Levi-Civita generalized connections on such Lie groups, which, in particular, shows the existence of non-flat ones.

We prove that the groups $\mathrm{Aut}(F_n)$ satisfy the Boone-Higman conjecture for all $n$, meaning each $\mathrm{Aut}(F_n)$ embeds in a finitely presented simple group. In fact, we prove that each $\mathrm{Aut}(F_n)$ satisfies the "permutational" Boone-Higman conjecture, which means the simple group in question can be taken to be a twisted Brin-Thompson group. A far-reaching consequence of our approach is that finitely presented twisted Brin-Thompson groups are universal among finitely presented simple groups that are highly transitive. This is evidence toward the Boone-Higman conjecture being equivalent to its permutational version. Proving the conjecture for $\mathrm{Aut}(F_n)$ also confirms the conjecture for all groups (virtually) embedding into some $\mathrm{Aut}(F_n)$, such as mapping class groups of non-closed surfaces, braid groups, loop braid groups, ribbon braid groups and certain Artin groups. This answers several questions of the first and fourth authors with Bleak and Matucci. Yet another consequence of our approach is that satisfying the permutational Boone-Higman conjecture is closed under free products.

We prove the slogan, promoted by Walker and Freed-Teleman twenty years ago, that "The Witten-Reshetikhin-Turaev 3-TQFT is a boundary condition for the Crane-Yetter 4-TQFT" and generalize it to the non-semisimple case following ideas of Jordan, Reutter and Walker. To achieve this, we prove that the Crane-Yetter 4-TQFT and its non-semisimple version arXiv:2306.03225 are once-extended TQFTs, using the main result of arXiv:2412.14649. We define a boundary condition, partially defined in the non-semisimple case, for this 4D theory. When the ribbon category used is modular, possibly non-semisimple, we check that the composition of this boundary condition with the values of the 4-TQFT on bounding manifolds reconstructs the Witten-Reshetikhin-Turaev 3-TQFTs and their non-semisimple versions arXiv:1912.02063, in a sense that we make precise.

This work investigates an efficient solution to two fundamental problems in topology optimization of frame structures. The first one involves minimizing structural compliance under linear-elastic equilibrium and weight constraint, while the second one minimizes the weight under compliance constraints. These problems are non-convex and generally challenging to solve globally, with the non-convexity concentrated in a polynomial matrix inequality. We solve these problems using the moment-sum-of-squares (mSOS) hierarchy and improve the scalability by enhancing (mSOS) with the Term Sparsity Pattern (TSP) technique. Additionally, we exploit the unique polynomial structure of our problems by adopting a reduced monomial basis containing only non-mixed terms. These modifications significantly enhance computational efficiency. Extensive numerical experiments demonstrate that our approach achieves global solutions for instances twice as large as those previously solved while substantially accelerating the solution process.

In this work, we present the first algorithm for identifying the minimum reboiler vapor duty requirement for a general multi-feed, multi-product (MFMP) distillation column separating ideal multicomponent mixtures. This algorithm incorporates our latest advancement in developing the first shortcut model for MFMP columns. We demonstrate the accuracy and efficiency of this algorithm through case studies. The results obtained from these case studies also provide valuable insights on optimal design of MFMP columns. Many of these insights are against the existing design guidelines and heuristics. For example, placing a colder saturated feed stream above a hotter saturated feed stream sometimes leads to higher energy requirement. Furthermore, decomposing a general MFMP column into individual simple columns may lead to incorrect estimation of the minimum reflux ratio for the MFMP column. Thus, the algorithm presented here offers a fast, accurate, and automated approach to synthesize new, energy-efficient, and cost-effective MFMP columns.

The non-Hermitian Bethe-Salpeter eigenvalue problem is a structured eigenproblem, with real eigenvalues coming in pairs $\{λ,-λ\}$ where the corresponding pair of eigenvectors are closely related, and furthermore the left eigenvectors can be trivially obtained from the right ones. We exploit these properties to devise three variants of structure-preserving Lanczos eigensolvers to compute a subset of eigenvalues (those of either smallest or largest magnitude) together with their corresponding right and left eigenvectors. For this to be effective in real applications, we need to incorporate a thick-restart technique in a way that the overall computation preserves the problem structure. The new methods are validated in an implementation within the SLEPc library using several test matrices, some of them coming from the Yambo materials science code.

We investigate the relationship between analysts' one-year consensus forecasts for the S&P 500 index and its current level. Contrary to the conventional view that sentiment drives price forecasts, our analysis predicts that the average consensus has no effect on forecasts, while the current S&P 500 index level alone is sufficient to anticipate analysts' price expectations. Employing a kinetic theory framework, we model the dynamics of analysts' opinions, by taking into account both the mutual influences shaping price consensus and the dynamics of the actual S&P 500 index level. Testing the model on real data on a long-time series of data shows that just three free parameters are enough to accurately describe the one-year average price forecasts of analysts.

In this article, we give a characterisation of crossed homomorphisms on Lie superalgebras as a Maurer-Cartan element of a graded Lie algebra. Using this characterisation we study cohomology of these crossed homomorphisms. As an application of this cohomology we study formal deformation of crossed homomorphisms. We show that linear deformations of these homomorphisms are characterised by one cocycles.

We present an extension of the Prouhet-Tarry-Escott problem by demonstrating that signed sums of noninteger powers of consecutive integers can be made arbitrarily close to zero.

Proof-theoretic semantics (PTS) is normally understood today as Base-Extension Semantics (B-eS), i.e., as a theory of proof-theoretic consequence over atomic proof systems. Intuitionistic logic (IL) has been proved to be incomplete over a number of variants of B-eS, including a monotonic one where introduction rules play a prior role (miB-eS). In its original formulation by Prawitz, however, PTS consequence is not a primitive, but a derived notion. The main concept is that of argument structure valid relative to atomic systems and assignments of reductions for eliminating generalised detours of inferences in non-introduction form. This is called Proof-Theoretic Validity (P-tV), and it can be given in a monotonic and introduction-based form too (miP-tV). It is unclear whether, and under what conditions, the incompleteness results proved for IL over miB-eS can be transferred to miP-tV. As has been remarked, the main problem seems to be that the notion of argumental validity underlying the miB-eS notion of consequence is one where reductions are either forced to be non-uniform, or non-constructive. Building on some Prawitz-fashion incompleteness proofs for IL based on the notion of (intuitionistic) construction, I provide in what follows a set of reductions which are surely uniform (however uniformity is defined) and constructive, and which make atomic Kreisel-Putnam rule logically valid over miP-tV, thus implying the incompleteness of IL over a Prawitzian (monotonic, introduction-based) framework strictly understood.

This paper introduces a novel class of initial data for which the three-dimensional incompressible Navier--Stokes equations yield unique global-in-time solutions. Building on a logarithmically improved regularity criterion, we impose a logarithmically subcritical condition on the initial data. Specifically, if \[ u_0 \in L^2(\mathbb{R}^3) \quad \text{and} \quad \|(-Δ)^{s/2}u_0\|_{L^q(\mathbb{R}^3)} \le \frac{C_0}{\Bigl(1+\log\bigl(e+\|u_0\|_{\dot{H}^s}\bigr)\Bigr)^δ}, \] for some $s \in (1/2,1)$ under appropriate scaling, then the corresponding solution exists globally and is unique. The proof employs refined commutator estimates for the fractional Laplacian together with new energy methods that exploit this logarithmic improvement to prevent singularity formation. Furthermore, we establish links between these improved criteria and turbulence theory. We derive precise relationships connecting the regularity conditions with turbulent intermittency, showing that the logarithmic enhancements correspond to anomalous scaling exponents in the turbulent energy spectrum. Additionally, we characterize the local structure of potential singularities and provide tight bounds on the energy flux in turbulent cascades. This approach bridges the gap between subcritical and critical regularity for the Navier--Stokes equations and offers a robust mathematical foundation for key phenomena observed in turbulence.

Let $H_g$ denote the 4-dimensional handlebody of genus $g$ and $U_g$ its boundary. We show that for all $g \ge 0$ the map from $B Homeo(H_g)$ to $B Homeo(U_g)$ induced by restriction to the boundary admits a section.

Contextual Stochastic Bilevel Optimization (CSBO) extends standard Stochastic Bilevel Optimization (SBO) by incorporating context-specific lower-level problems, which arise in applications such as meta-learning and hyperparameter optimization. This structure imposes an infinite number of constraints - one for each context realization - making CSBO significantly more challenging than SBO as the unclear relationship between minimizers across different contexts suggests computing numerous lower-level solutions for each upper-level iteration. Existing approaches to CSBO face two major limitations: substantial complexity gaps compared to SBO and reliance on impractical conditional sampling oracles. We propose a novel reduction framework that decouples the dependence of the lower-level solution on the upper-level decision and context through parametrization, thereby transforming CSBO into an equivalent SBO problem and eliminating the need for conditional sampling. Under reasonable assumptions on the context distribution and the regularity of the lower-level, we show that an $ε$-stationary solution to CSBO can be achieved with a near-optimal sampling complexity $\tilde{O}(ε^{-3})$. Our approach enhances the practicality of solving CSBO problems by improving both computational efficiency and theoretical guarantees.

This paper describes extension of the Master Stability Function for arrays of non-smooth oscillators. This extension is based on the previously introduced Jacobi matrix estimation method, which can be applied to networks of non-smooth coupled oscillators. Proposed method and its limitations were described. Then the new algorithm was applied to calculate MSF for an impact oscillator. The results were presented and validated using an alternative MSF estimation method.

We study the injectivity property of certain actions of higher Chow groups on refined unramified cohomology. As an application for every $p\geq1$ and for each $d\geq p+4$ and $n\geq2,$ we establish the first examples of smooth complex projective $d$-folds $X$ such that for all $p+3\leq c\leq d-1,$ the higher Chow group $\text{CH}^{c}(X,p)$ contains infinitely many torsion cycles of order $n$ that remain linearly independent modulo $n$. Our bounds for $c$ and $d$ are also optimal. A crucial tool for the proof is morphic cohomology.