arXiv Math @arxiv_math@qoto.org

We study non-perturbative superpotential generated by D(-1)-branes in type IIB compactifications on orientifolds of Calabi-Yau threefold hypersurfaces. To compute D-instanton superpotential, we study F-theory compactification on toric complete intersection elliptic Calabi-Yau fourfolds. We take the Sen-limit, but with finite $g_s,$ in F-theory compactification with a restriction that all D7-branes are carrying SO(8) gauge groups, which we call the global Sen-limit. In the global Sen-limit, the axio-dilaton is not varying in the compactification manifold. We compute the Picard-Fuchs equations of elliptic Calabi-Yau fourfolds in the global Sen-limit, and show that the Picard-Fuchs equations of the elliptic fourfolds split into that of the underlying Calabi-Yau threefolds and of the elliptic fiber. We then demonstrate that this splitting property of the Picard-Fuchs equation implies that the fourform period of the elliptic Calabi-Yau fourfolds in the global Sen-limit does not contain exponentially suppressed terms $\mathcal{O}(e^{-π/g_s})$. With this result, we finally show that in the global Sen-limit, superpotential of the underlying type IIB compactification does not receive D-instanton contributions.

By studying the properties of $q$-series $\widehat Z$-invariants, we develop a dictionary between 3-manifolds and vertex algebras. In particular, we generalize previously known entries in this dictionary to Lie groups of higher rank, to 3-manifolds with toral boundaries, and to BPS partition functions with line operators. This provides a new physical realization of logarithmic vertex algebras in the framework of the 3d-3d correspondence and opens new avenues for their future study. For example, we illustrate how invoking a knot-quiver correspondence for $\widehat{Z}$-invariants leads to many infinite families of new fermionic formulae for VOA characters.

We introduce an epistemic information measure between two data streams, that we term $influence$. Closely related to transfer entropy, the measure must be estimated by epistemic agents with finite memory resources via sampling accessible data streams. We show that even under ideal conditions, epistemic agents using slightly different sampling strategies might not achieve consensus in their conclusions about which data stream is influencing which. As an illustration, we examine a real world data stream where different sampling strategies result in contradictory conclusions, explaining why some politically charged topics might exist due to purely epistemic reasons irrespective of the actual ontology of the world.

Airfoil shape design is a classical problem in engineering and manufacturing. Our motivation is to combine principled physics-based considerations for the shape design problem with modern computational techniques informed by a data-driven approach. Traditional analyses of airfoil shapes emphasize a flow-based sensitivity to deformations which can be represented generally by affine transformations (rotation, scaling, shearing, translation). We present a novel representation of shapes which decouples affine-style deformations from a rich set of data-driven deformations over a submanifold of the Grassmannian. The Grassmannian representation, informed by a database of physically relevant airfoils, offers (i) a rich set of novel 2D airfoil deformations not previously captured in the data, (ii) improved low-dimensional parameter domain for inferential statistics informing design/manufacturing, and (iii) consistent 3D blade representation and perturbation over a sequence of nominal shapes.

In this paper, we prove a quantitative regularity theorem and a blow-up criterion of classical solutions for the three-dimensional Navier-Stokes equations. By adapting the strategy developed by Tao in [20], we obtain an explicit blow-up rate in the setting of critical Lorentz spaces $L^{3, q_{0}}(\mathbb R^3)$ with $3 \leq q_0 < \infty $. Our results improve the previous regularity in critical Lebesgue spaces $L^3(\mathbb R^3)$ in [20] and quantify the qualitative result by Phuc in [16].

In this article, we study a semi-linear heat equation with the nonlinearity which is the product of polynomial and logarithmic functions. Using the invariance of the potential well(s), we have established the global existence and exponential decay estimates of solutions in $L^2$ - norm without having any restriction on the exponent in the source term under suitable conditions on the initial data. Moreover, finite time blow up of solutions at subcritical, critical and supercritical initial energy levels is also discussed.

We prove an analyticity result for the Dirichlet-Neumann operator under space periodic boundary conditions in any dimension in an unbounded domain with infinite depth. We derive an analytic bifurcation result of analytic Stokes waves -- i.e. space periodic traveling solutions -- of the water waves equations in deep water.

We analyze the convergence and approximation error of the inverse Born series, obtaining results that hold under qualitatively weaker conditions than previously known. Our approach makes use of tools from geometric function theory in Banach spaces. An application to the inverse scattering problem with diffuse waves is described.

The role of consistent measures in the rigorous construction of nonabelian lattice theories is analized. General conditions that measures must fulfill to insure consistency, positivity and a mass gap are obtained. The impact of nongeneric strata on the nature of the Hamiltonian lattice potential is also discussed.

We consider the continuous Anderson operator $H=Δ+ξ$ on a two dimensional closed Riemannian manifold $\mathcal{S}$. We provide a short self-contained functional analysis construction of the operator as an unbounded operator on $L^2(\mathcal{S})$ and give almost sure spectral gap estimates under mild geometric assumptions on the Riemannian manifold. We prove a sharp Gaussian small time asymptotic for the heat kernel of $H$ that leads amongst others to strong norm estimates for quasimodes. We introduce a new random field, called Anderson Gaussian free field, and prove that the law of its random partition function characterizes the law of the spectrum of $H$. We also give a simple and short construction of the polymer measure on path space and relate the Wick square of the Anderson Gaussian free field to the occupation measure of a Poisson process of loops of polymer paths. We further prove large deviation results for the polymer measure and its bridges.

Planar holomorphic systems $\dot{x}=u(x,y)$, $\dot{y}=v(x,y)$ are those that $u=\operatorname{Re}(f)$ and $v=\operatorname{Im}(f)$ for some holomorphic function $f(z)$. They have important dynamical properties, highlighting, for example, the fact that they do not have limit cycles and that center-focus problem is trivial. In particular, the hypothesis that a polynomial system is holomorphic reduces the number of parameters of the system. Although a polynomial system of degree $n$ depends on $n^2 +3n+2$ parameters, a polynomial holomorphic depends only on $2n + 2$ parameters. In this work, in addition to making a general overview of the theory of holomorphic systems, we classify all the possible global phase portraits, on the Poincaré disk, of systems $\dot{z}=f(z)$ and $\dot{z}=1/f(z)$, where $f(z)$ is a polynomial of degree $2$, $3$ and $4$ in the variable $z\in \mathbb{C}$. We also classify all the possible global phase portraits of Moebius systems $\dot{z}=\frac{Az+B}{Cz+D}$, where $A,B,C,D\in\mathbb{C}, AD-BC\neq0$. Finally, we obtain explicit expressions of first integrals of holomorphic systems and of conjugated holomorphic systems, which have important applications in the study of fluid dynamics.

We solve the Dirichlet problem in the unit disc and derive the Poisson formula using very elementary methods and explore consequent simplifications in other foundational areas of complex analysis.

We define the s-order principal component of the jets of a graph and give a description of the primary decomposition of its edge ideal in terms of the minimal vertex covers of the base graph. As an application, we show the s-order principal component of the jets of a cochordal graph is cochordal, and connect this to Froberg's theorem on the linear resolution of edge ideals of cochordal graphs. An appendix is provided describing some computations of jets in the computer algebra system Macaulay2.

Model analogies and exchange of ideas between physics or chemistry with biology or epidemiology have often involved inter-sectoral mapping of techniques. Material mechanics has benefitted hugely from such interpolations from mathematical physics where dislocation patterning of platstically deformed metals [1,2,3] and mass transport in nanocomposite materials with high diffusivity paths such as dislocation and grain boundaries, have been traditionally analyzed using the paradigmatic Walgraef-Aifantis (W-A) double-diffusivity (D-D) model [4,5,6,7,8,9]. A long standing challenge in these studies has been the inherent nonlinear correlation between the diffusivity paths, making it extremely difficult to analyze their interdependence. Here, we present a novel method of approximating a closed form solution of the ensemble averaged density profiles and correlation statistics of coupled dynamical systems, drawing from a technique used in mathematical biology to calculate a quantity called the {\it basic reproduction number} $R_0$, which is the average number of secondary infections generated from every infected. We show that the $R_0$ formulation can be used to calculate the correlation between diffusivity paths, agreeing closely with the exact numerical solution of the D-D model. The method can be generically implemented to analyze other reaction-diffusion models.

We present an algorithm for computing the barcode of the image of a morphisms in persistent homology induced by an inclusion of filtered finite-dimensional chain complexes. These algorithms make use of the clearing optimization and can be applied to inclusion-induced maps in persistent absolute homology and persistent relative cohomology for filtrations of pairs of simplicial complexes. They form the basis for our implementation for Vietoris-Rips complexes in the framework of the software Ripser.

We show that the discrete approximate volume preserving mean curvature flow in the flat torus $\mathbb{T}^N$ starting near a strictly stable critical set $E$ of the perimeter converges in the long time to a translate of $E$ exponentially fast. As an intermediate result we establish a new quantitative estimate of Alexandrov type for periodic strictly stable constant mean curvature hypersurfaces. Finally, in the two dimensional case a complete characterization of the long time behaviour of the discrete flow with arbitrary initial sets of finite perimeter is provided.

For a function $f:X \rightarrow (-\infty,+\infty]$ on a normed space $(X,\Vert \cdot \Vert)$ and given parameters $p>1$ and $\varepsilon>0$, we investigate the properties of the generalized Moreau-Yosida regularization given by \begin{align*} f_\varepsilon(u)=\inf_{v\in X}\left\lbrace \frac{1}{p\varepsilon} \Vert u-v\Vert^p+f(v)\right\rbrace \quad ,u\in X. \end{align*} We show that the generalized Moreau-Yosida regularization satisfies the same properties as in the classical case for $p=2$ provided $X$ is not a Hilbert space.

Maintenance planning plays a key role in power system operations under uncertainty by helping system operators ensure a reliable and secure power grid. This paper studies a short-term condition-based integrated maintenance planning with operations scheduling problem while considering the unexpected failure possibilities of generators as well as transmission lines. We formulate this problem as a two-stage stochastic mixed-integer program with failure scenarios sampled from the sensor-driven remaining lifetime distributions of the individual system elements whereas a joint chance-constraint consisting of Poisson Binomial random variables is introduced to account for failure risks. Because of its intractability, we develop a cutting-plane method to obtain an exact reformulation of the joint chance-constraint by proposing a separation subroutine and deriving stronger cuts as part of this procedure. To solve large-scale instances, we derive a second-order cone programming based safe approximation of this constraint. Furthermore, we propose a decomposition-based algorithm implemented in parallel fashion for solving the resulting stochastic program, by exploiting the features of the integer L-shaped method and the special structure of the maintenance and operations scheduling problem to derive stronger optimality cuts. We further present preprocessing steps over transmission line flow constraints to identify redundancies. To illustrate the computational performance and efficiency of our algorithm compared to more conventional maintenance approaches, we design a computational study focusing on a weekly plan with daily maintenance and hourly operational decisions involving detailed unit commitment subproblems. Our computational results on various IEEE instances demonstrate the computational efficiency of the proposed approach with reliable and cost-effective maintenance and operational schedules.

Tether-net launched from a chaser spacecraft provides a promising method to capture and dispose of large space debris in orbit. This tether-net system is subject to several sources of uncertainty in sensing and actuation that affect the performance of its net launch and closing control. Earlier reliability-based optimization approaches to design control actions however remain challenging and computationally prohibitive to generalize over varying launch scenarios and target (debris) state relative to the chaser. To search for a general and reliable control policy, this paper presents a reinforcement learning framework that integrates a proximal policy optimization (PPO2) approach with net dynamics simulations. The latter allows evaluating the episodes of net-based target capture, and estimate the capture quality index that serves as the reward feedback to PPO2. Here, the learned policy is designed to model the timing of the net closing action based on the state of the moving net and the target, under any given launch scenario. A stochastic state transition model is considered in order to incorporate synthetic uncertainties in state estimation and launch actuation. Along with notable reward improvement during training, the trained policy demonstrates capture performance (over a wide range of launch/target scenarios) that is close to that obtained with reliability-based optimization run over an individual scenario.

The probability that a random permutation in $S_n$ is a derangement is well known to be $\displaystyle\sum\limits_{j=0}^n (-1)^j \frac{1}{j!}$. In this paper, we consider the conditional probability that the $(k+1)^{st}$ point is fixed, given there are no fixed points in the first $k$ points. We prove that when $n \neq 3$ and $k \neq 1$, this probability is a decreasing function of both $k$ and $n$. Furthermore, it is proved that this conditional probability is well approximated by $\frac{1}{n} - \frac{k}{n^2(n-1)}$. Similar results are also obtained about the more general conditional probability that the $(k+1)^{st}$ point is fixed, given that there are exactly $d$ fixed points in the first $k$ points.