arXiv Math @arxiv_math@qoto.org

Aircraft design is an iterative process that requires an estimation of aerodynamic characteristics including drag and lift coefficients, stall behavior, velocity, and pressure profiles repeatedly, especially in the conceptual design phase. PanAir is a high-order aerodynamic panel method-based algorithm developed as a part of the Public Domain Aeronautical Software program mostly with NASA sponsorship. It is based upon potential flow theory and is used to numerically compute lift, induced drag, and moment coefficients of the aircraft in both subsonic and supersonic flight regimes. Quoting from developers it is the most versatile and accurate of all the linear theory panel codes. PanAir is a classical software that requires geometric data as an input in the form of a PanAir-compatible format; however, commonly used Computer-Aided Design Software packages no longer conform to the PanAir input format. Likewise, PanAir produces its output in a PanAir-specific output file which is not compatible with commonly used visualization software. The input geometry required by PanAir and its output, therefore, involves significant manual preand post-processing using intermediary software. The work proposed here is an automated pre and post-processor to be used together with PanAir. With the environment proposed in this work, manipulation of input and output data using several intermediary software to and from PanAir is bypassed successfully. The proposed environment is validated over a Cessna 210 aircraft geometry with a modified NLF (1)-0414 airfoil. The aircraft is numerically analyzed using PanAir together

This article solves the Hume's problem of induction using a probabilistic approach. From the probabilistic perspective, the core task of induction is to estimate the probability of an event and judge the accuracy of the estimation. Following this principle, the article provides a method for calculating the confidence on a given confidence interval, and furthermore, degree of confirmation. The law of large numbers shows that as the number of experiments tends to infinity, for any small confidence interval, the confidence approaches 100\% in a probabilistic sense, thus the Hume's problem of induction is solved. The foundation of this method is the existence of probability, or in other words, the identity of physical laws. The article points out that it cannot be guaranteed that all things possess identity, but humans only concern themselves with things that possess identity, and identity is built on the foundation of pragmatism. After solving the Hum's problem, a novel demarcation of science are proposed, providing science with the legitimacy of being referred to as truth.

Flat origami refers to the folding of flat, zero-curvature paper such that the finished object lies in a plane. Mathematically, flat origami consists of a continuous, piecewise isometric map $f:P\subseteq\mathbb{R}^2\to\mathbb{R}^2$ along with a layer ordering $λ_f:P\times P\to \{-1,1\}$ that tracks which points of $P$ are above/below others when folded. The set of crease lines that a flat origami makes (i.e., the set on which the mapping $f$ is non-differentiable) is called its \textit{crease pattern}. Flat origami mappings and their layer orderings can possess surprisingly intricate structure. For instance, determining whether or not a given straight-line planar graph drawn on $P$ is the crease pattern for some flat origami has been shown to be an NP-complete problem, and this result from 1996 led to numerous explorations in computational aspects of flat origami. In this paper we prove that flat origami, when viewed as a computational device, is Turing complete. We do this by showing that flat origami crease patterns with \textit{optional creases} (creases that might be folded or remain unfolded depending on constraints imposed by other creases or inputs) can be constructed to simulate Rule 110, a one-dimensional cellular automaton that was proven to be Turing complete by Matthew Cook in 2004.

In this paper, we introduce a new heuristics for global optimization in scenarios where extensive evaluations of the cost function are expensive, inaccessible, or even prohibitive. The method, which we call Landscape-Sketch-and-Step (LSS), combines Machine Learning, Stochastic Optimization, and Reinforcement Learning techniques, relying on historical information from previously sampled points to make judicious choices of parameter values where the cost function should be evaluated at. Unlike optimization by Replica Exchange Monte Carlo methods, the number of evaluations of the cost function required in this approach is comparable to that used by Simulated Annealing, quality that is especially important in contexts like high-throughput computing or high-performance computing tasks, where evaluations are either computationally expensive or take a long time to be performed. The method also differs from standard Surrogate Optimization techniques, for it does not construct a surrogate model that aims at approximating or reconstructing the objective function. We illustrate our method by applying it to low dimensional optimization problems (dimensions 1, 2, 4, and 8) that mimick known difficulties of minimization on rugged energy landscapes often seen in Condensed Matter Physics, where cost functions are rugged and plagued with local minima. When compared to classical Simulated Annealing, the LSS shows an effective acceleration of the optimization process.

We extend previous results due to Ding and Zhuang in order to prove that a phase transition occurs for the long range Ising model in lower dimensions. By making use of a recent argument due to Affonso, Bissacot and Maia from 2022 which establishes that a phase transition occurs for the long range, random-field Ising model, from a suggestion of the authors we demonstrate that a phase transition also occurs for the long range Ising model, from a set of appropriately defined contours for the long range system, and a Peierls' argument.

We investigate eigenvalue attraction for open quantum systems, biophysical systems, and for Parity-Time symmetric materials. To determine whether an eigenvalue and its complex conjugate of a real matrix attract, we derive expressions for the second derivative of eigenvalues, which is dependent upon contributions from inertial forces, attraction between an eigenvalue and its complex conjugate, as well as the force of the remaining eigenvalues in the spectrum.

We present a physics-informed machine-learning (PIML) approach for the approximation of slow invariant manifolds (SIMs) of singularly perturbed systems, providing functionals in an explicit form that facilitate the construction and numerical integration of reduced order models (ROMs). The proposed scheme solves a partial differential equation corresponding to the invariance equation (IE) within the Geometric Singular Perturbation Theory (GSPT) framework. For the solution of the IE, we used two neural network structures, namely feedforward neural networks (FNNs), and random projection neural networks (RPNNs), with symbolic differentiation for the computation of the gradients required for the learning process. The efficiency of our PIML method is assessed via three benchmark problems, namely the Michaelis-Menten, the target mediated drug disposition reaction mechanism, and the 3D Sel'kov model. We show that the proposed PIML scheme provides approximations, of equivalent or even higher accuracy, than those provided by other traditional GSPT-based methods, and importantly, for any practical purposes, it is not affected by the magnitude of the perturbation parameter. This is of particular importance, as there are many systems for which the gap between the fast and slow timescales is not that big, but still ROMs can be constructed. A comparison of the computational costs between symbolic, automatic and numerical approximation of the required derivatives in the learning process is also provided.

A pure fermionic state with a fixed particle number is said to be correlated if it deviates from a Slater determinant. In the present work we show that this notion can be refined, classifying fermionic states relative to $k$-${\rm \textit{body}}$ correlations. We capture such correlations by a family of measures $ω_k$, which we call twisted purities. Twisted purity is an explicit function of the $k$-fermion reduced density matrix, insensitive to global single-particle transformations. Vanishing of $ω_k$ for a given $k$ generalizes so-called Plücker relations on the state amplitudes and puts the state in a class ${\cal G}_k$. Sets ${\cal G}_k$ are nested in $k$, ranging from Slater determinants for $k = 1$ up to the full $n$-fermion Hilbert space for $k = n + 1$. We find various physically relevant states inside and close to ${\cal G}_{k=O(1)}$, including truncated configuration-interaction states, perturbation series around Slater determinants, and some nonperturbative eigenstates of the 1D Hubbard model. For each $k = O(1)$, we give an explicit ansatz with a polynomial number of parameters that covers all states in ${\cal G}_k$. Potential applications of this ansatz and its connections to the coupled-cluster wavefunction are discussed.

We set out the general theory of "Beck modules" in a variety of algebras and describe them as modules over suitable "universal enveloping" unital associative algebras. We pay particular attention to the somewhat nonstandard case of "alternative algebras," defined by a restricted associative law, and determine the Poincaré polynomial of the universal enveloping algebra in the homogenous case.

The restoration lemma is a classic result by Afek, Bremler-Barr, Kaplan, Cohen, and Merritt [PODC '01], which relates the structure of shortest paths in a graph $G$ before and after some edges in the graph fail. Their work shows that, after one edge failure, any replacement shortest path avoiding this failing edge can be partitioned into two pre-failure shortest paths. More generally, this implies an additive tradeoff between fault tolerance and subpath count: for any $f, k$, we can partition any $f$-edge-failure replacement shortest path into $k+1$ subpaths which are each an $(f-k)$-edge-failure replacement shortest path. This generalized result has found applications in routing, graph algorithms, fault tolerant network design, and more. Our main result improves this to a multiplicative tradeoff between fault tolerance and subpath count. We show that for all $f, k$, any $f$-edge-failure replacement path can be partitioned into $O(k)$ subpaths that are each an $(f/k)$-edge-failure replacement path. We also show an asymptotically matching lower bound. In particular, our results imply that the original restoration lemma is exactly tight in the case $k=1$, but can be significantly improved for larger $k$. We also show an extension of this result to weighted input graphs, and we give efficient algorithms that compute path decompositions satisfying our improved restoration lemmas.

Numerous accidents caused by parametric rolling have been reported on container ships and pure car carriers (PCCs). A number of theoretical studies have been performed to estimate the occurrence condition of parametric rolling in both regular and irregular seas. Some studies in random wave conditions have been the approximate extension of the occurrence conditions for regular waves (e.g. Maki et al). Furthermore, several researches have been based on the stochastic process in ocean engineering (Roberts and Dostal). This study tackled the parametric rolling in irregular seas from the stability of the system's origin. It provided a novel theoretical explanation of the instability mechanism for two cases: white noise parametric excitation and colored noise parametric excitation. The authors then confirmed the usefulness of the previously provided formulae by Roberts and Dostal through numerical examples.

The parabolic problem $u_t-Δu=\frac{λf(x)}{(1-u)^2}+P$ on a bounded domain $Ω$ of $R^n$ with Dirichlet boundary condition models the microelectromechanical systems(MEMS) device with an external pressure term. In this paper, we classify the behavior of the solution to this equation. We first show that under certain initial conditions, there exists critical constants $P^*$ and $λ_P^*$ such that when $0\leq P\leq P^*$, $0<λ\leq λ_P^*$, there exists a global solution, while for $0\leq P\leq P^*,λ>λ_P^*$ or $P>P^*$, the solution quenches in finite time. The estimate of voltage $λ_P^*$, quenching time $T$ and pressure term $P^*$ are investigated. The quenching set $Σ$ is proved to be a compact subset of $Ω$ with an additional condition, provided $Ω\subset R^n$ is a convex bounded set. In particular, if $Ω$ is radially symmetric, then the origin is the only quenching point. Furthermore, we not only derive the two-side bound estimate for the quenching solution, but also study the asymptotic behavior of the quenching solution in finite time.

In this paper, we study the convolution structure in the special affine Fourier transform domain to combine the advantages of the well known special affine Fourier and Stockwell transforms into a novel integral transform coined as special affine Stockwell transform and investigate the associated constant Q property in the joint time frequency domain. The preliminary analysis encompasses the derivation of the fundamental properties, Rayleighs energy theorem, inversion formula and range theorem. Besides, we also derive a direct relationship between the recently introduced special affine scaled Wigner distribution and the proposed SAST. Further, we establish Heisenbergs uncertainty principle, logarithmic uncertainty principle and Nazarovs uncertainty principle associated with the proposed SAST. Towards the culmination of this paper, some potential applications with simulation are presented.

Partial differential equations (PDEs) are used to describe a variety of physical phenomena. Often these equations do not have analytical solutions and numerical approximations are used instead. One of the common methods to solve PDEs is the finite element method. Computing derivative information of the solution with respect to the input parameters is important in many tasks in scientific computing. We extend JAX automatic differentiation library with an interface to Firedrake finite element library. High-level symbolic representation of PDEs allows bypassing differentiating through low-level possibly many iterations of the underlying nonlinear solvers. Differentiating through Firedrake solvers is done using tangent-linear and adjoint equations. This enables the efficient composition of finite element solvers with arbitrary differentiable programs. The code is available at github.com/IvanYashchuk/jax-firedrake.

Urban Air Mobility (UAM) offers a solution to current traffic congestion by providing on-demand air mobility in urban areas. Effective traffic management is crucial for efficient operation of UAM systems, especially for high-demand scenarios. In this paper, we present VertiSync, a centralized traffic management policy for on-demand UAM networks. VertiSync schedules the aircraft for either servicing trip requests or rebalancing in the network subject to aircraft safety margins and separation requirements during takeoff and landing. We characterize the system-level throughput of VertiSync, which determines the demand threshold at which travel times transition from being stabilized to being increasing over time. We show that the proposed policy is able to maximize the throughput for sufficiently large fleet sizes. We demonstrate the performance of VertiSync through a case study for the city of Los Angeles. We show that VertiSync significantly reduces travel times compared to a first-come first-serve scheduling policy.

The Sendov conjecture asserts that if $p(z) = \prod_{j=1}^{N}(z-z_j)$ is a polynomial with zeros $|z_j| \leq 1$, then each disk $|z-z_j| \leq 1$ contains a zero of $p'$. Our purpose is the following: Given a zero $z_j$ of order $n \geq 2$, determine whether there exists $ζ\not= z_j$ such that $p'(ζ) = 0$ and $|z_j - ζ| \leq 1$. In this paper we present some partial results on the problem.

In 1976, Donald Cartwright and John McMullen characterized axiomatically the Riemann-Liouvile fractional integral in a paper that was published in 1978. The motivation for their work was to answer affirmatively to a conjecture stated by J. S. Lew a few years before, in 1972. Essentially, their ``Cartwright-McMullen theorem in fractional calculus'' proved that the Riemann-Liouville fractional integral is the only continuous extension of the usual integral operator to positive real orders, in such a way that the Index Law holds. In this paper, we propose an analogous result for the uniqueness of the extension of the Stieltjes integral operator, in the case of a smooth integrator.

We study Clark measures on the unit polydisc, giving an overview of recent research and investigating the Clark measures of some new examples of multivariate inner functions. In particular, we study the relationship between Clark measures and multiplication; first by introducing compositions of inner functions and multiplicative embeddings, and then by studying products of one-variable inner functions.

Andrew Wiles' proof of Fermat's Last Theorem, with an assist from Richard Taylor, focused renewed attention on the foundational question of whether the use of Grothendieck's Universes in number theory entails that the results proved therewith make essential use of the large cardinal axiom that there is an uncountable strongly inaccessible cardinal, or more generally, that every cardinal is less than a strongly inaccessible cardinal. If one traces back through the references in Wiles' proof, one finds that the proof does depend upon explicit use of Grothendieck's Universes. Thus, prima facie, it appears that the proof of Fermat's Last Theorem depends upon a foundation that is strictly stronger than ZFC. Colin McLarty removes this appearance by demonstrating that all of Grothendieck's large tools, i.e., entities whose construction depended upon Grothendieck's Universes, can instead be founded on a fragment of ZFC with the logical strength of Finite-Order Arithmetic. The goal of this article is to present overviews both of the history of Fermat's Last Theorem and of McLarty's foundation for Grothendieck's large tools.

Line outage identification in distribution grids is essential for sustainable grid operation. In this work, we propose a practical yet robust detection approach that utilizes only readily available voltage magnitudes, eliminating the need for costly phase angles or power flow data. Given the sensor data, many existing detection methods based on change-point detection require prior knowledge of outage patterns, which are unknown for real-world outage scenarios. To remove this impractical requirement, we propose a data-driven method to learn the parameters of the post-outage distribution through gradient descent. However, directly using gradient descent presents feasibility issues. To address this, we modify our approach by adding a Bregman divergence constraint to control the trajectory of the parameter updates, which eliminates the feasibility problems. As timely operation is the key nowadays, we prove that the optimal parameters can be learned with convergence guarantees via leveraging the statistical and physical properties of voltage data. We evaluate our approach using many representative distribution grids and real load profiles with 17 outage configurations. The results show that we can detect and localize the outage in a timely manner with only voltage magnitudes and without assuming a prior knowledge of outage patterns.