arXiv Math @arxiv_math@qoto.org

The high simulation cost has been a bottleneck of practical analog/mixed-signal design automation. Many learning-based algorithms require thousands of simulated data points, which is impractical for expensive to simulate circuits. We propose a learning-based algorithm that can be trained using a small amount of data and, therefore, scalable to tasks with expensive simulations. Our efficient algorithm solves the post-layout performance optimization problem where simulations are known to be expensive. Our comprehensive study also solves the schematic-level sizing problem. For efficient optimization, we utilize Bayesian Neural Networks as a regression model to approximate circuit performance. For layout-aware optimization, we handle the problem as a multi-fidelity optimization problem and improve efficiency by exploiting the correlations from cheaper evaluations. We present three test cases to demonstrate the efficiency of our algorithms. Our tests prove that the proposed approach is more efficient than conventional baselines and state-of-the-art algorithms.

We apply Game Theory to a mathematical representation of two competing teams of agents connected within a complex network, where the ability of each side to manoeuvre their resource and degrade that of the other depends on their ability to internally synchronise decision-making while out-pacing the other. Such a representation of an adversarial socio-physical system has application in a range of business, sporting, and military contexts. Specifically, we unite here two physics-based models, that of Kuramoto to represent decision-making cycles, and an adaptation of a multi-species Lotka-Volterra system for the resource competition. For complex networks we employ variations of the Barabási-Alberts scale-free graph, varying how resources are initially distributed between graph hub and periphery. We adapt as equilibrium solution Nash Dominant Game Pruning as a means of efficiently exploring the dynamical decision tree. Across various scenarios we find Nash solutions where the side initially concentrating resources in the periphery can sustain competition to achieve victory except when asymmetries exist between the two. When structural advantage is limited we find that agility in how the victor stays ahead of decision-state of the other becomes critical.

For any generic finite reductive group $\mathbb{G}$, integer $e > 0$, and $Φ_e$-cuspidal pair $(\mathbb{L}, λ)$, Broué-Malle-Michel conjectured that the endomorphism rings of the Deligne-Lusztig representations attached to $\mathbb{G}, (\mathbb{L}, λ)$ all come from the same generic cyclotomic Hecke algebra. We propose a new conjecture about the Harish-Chandra theory of such pairs, involving two integers $e$ and $m$: namely, that the intersection of an $Φ_e$-Harish-Chandra series and a $Φ_m$-Harish-Chandra series is parametrized by both a union of $Φ_m$-blocks of the $Φ_e$-Hecke algebra and a union of $Φ_e$-blocks of the $Φ_m$-Hecke algebra, in a way that matches blocks. We also conjecture that when blocks match, there is an equivalence of categories between their highest-weight covers. When $e = 1$, we provide evidence that our bijections are essentially realized by bimodules that Oblomkov-Yun construct from the cohomology of affine Springer fibers. This suggests a strange analogy: Roughly, homogeneous affine Springer fibers are to roots of unity as tensor products of Deligne-Lusztig representations are to prime powers. We predict the generic Hecke parameters for arbitrary $Φ$-cuspidal pairs of the groups $\mathbb{G}\mathbb{L}_n$ and $\mathbb{G}\mathbb{U}_n$, unifying the known cases. We prove that they would imply our conjectural bijections for these groups and coprime $e, m$. Then we show that the bijections for $\mathbb{G}\mathbb{L}_n$ are related by affine permutations to Uglov's bijections between bases of higher-level Fock spaces. This relates our conjectural equivalences of categories to those conjectured by Chuang-Miyachi, and proved by several authors, under the name of level-rank duality. Finally, for many cases in exceptional types, we verify that the parameters predicted by Broué-Malle are compatible with our conjectures.

A gravitational close encounter of a small body with a planet may produce a substantial change of its orbital parameters which can be studied using the circular restricted three-body problem. In this paper we provide parametric representations of the fast close encounters with the secondary body of the planar CRTBP as arcs of non-linear focus-focus dynamics. The result is the consequence of a remarkable factorization of the Birkhoff normal forms of the Hamiltonian of the problem represented with the Levi-Civita regularization. The parametrizations are computed using two different sequences of Birkhoff normalizations of given order N. For each value of N, the Birkhoff normalizations and the parameters of the focus-focus dynamics are represented by polynomials whose coefficients can be computed iteratively with a computer algebra system; no quadratures, such as those needed to compute action-angle variables of resonant normal forms, are needed. We also provide some numerical demonstrations of the method for values of the mass parameter representative of the Sun-Earth and the Sun-Jupiter cases.

In an acceptance monitoring system, acceptance sampling techniques are used to increase production, enhance control, and deliver higher-quality products at a lesser cost. It might not always be possible to define the acceptance sampling plan parameters as exact values, especially, when data has uncertainty. In this work, acceptance sampling plans for a large number of identical units with exponential lifetimes are obtained by treating acceptable quality life, rejectable quality life, consumer's risk, and producer's risk as fuzzy parameters. To obtain plan parameters of sequential sampling plans and repetitive group sampling plans, fuzzy hypothesis test is considered. To validate the sampling plans obtained in this work, some examples are presented. Our results are compared with existing results in the literature. Finally, to demonstrate the application of the resulting sampling plans, a real-life case study is presented.

A dynamical system is considered such that, in this system, particles move on a toroidal lattice of the dimension $N_1\times N_2$ according to a version of the rule of particle movement in Biham--Middleton--Levine traffic model. Particles of the first type move along rows, and the particles of the second type move along columns. A particle can change its type. The probability that, at a step, a particle changes the type equals $q.$ It has been proved that, if whether $q>0$ or $q=0$ and there are at least one particle of the first type and there is at least one particle of the second type, then a necessary condition for a state of free movement to exist is that the greatest divisor of $N_1$ and $N_2$ be not less than~3. The system is represented as an algebraic structure over a set of matrices corresponding to states of the system. The theorems are formulated in terms of the algebraic structures and terms of the dynamical systems. The model is also presented as a Buslaev net. The spectrum of the net $2\times 2$ has been found.

We study a weighted $\frac{N}{2}$ biharmonic equation involving a positive continuous potential in $\overline{B}$. The non-linearity is assumed to have critical exponential growth in view of logarithmic weighted Adams' type inequalities in the unit ball of $\mathbb{R}^{N}$. It is proved that there is a nontrivial weak solution to this problem by the mountain Pass Theorem. We avoid the loss of compactness by proving a concentration compactness result and by a suitable asymptotic condition.

Calabi-Yau four-folds may be constructed as hypersurfaces in weighted projective spaces of complex dimension 5 defined via weight systems of 6 weights. In this work, neural networks were implemented to learn the Calabi-Yau Hodge numbers from the weight systems, where gradient saliency and symbolic regression then inspired a truncation of the Landau-Ginzburg model formula for the Hodge numbers of any dimensional Calabi-Yau constructed in this way. The approximation always provides a tight lower bound, is shown to be dramatically quicker to compute (with compute times reduced by up to four orders of magnitude), and gives remarkably accurate results for systems with large weights. Additionally, complementary datasets of weight systems satisfying the necessary but insufficient conditions for transversality were constructed, including considerations of the IP, reflexivity, and intradivisibility properties. Overall producing a classification of this weight system landscape, further confirmed with machine learning methods. Using the knowledge of this classification, and the properties of the presented approximation, a novel dataset of transverse weight systems consisting of 7 weights was generated for a sum of weights $\leq 200$; producing a new database of Calabi-Yau five-folds, with their respective topological properties computed. Further to this an equivalent database of candidate Calabi-Yau six-folds was generated with approximated Hodge numbers.

In a dimensionless multi-scalar extension of the Standard Model, Gildener and Weinberg assumed that along a flat direction, there is only one classically massless scalar, known as the {\it scalon}, which acquires mass through radiative corrections à la Coleman-Weinberg, while all other scalars remain heavy. In this paper, by introducing a toy model with four scalar degrees of freedom, we demonstrate the existence of {\it two} scalons along a specific flat direction that we construct. We present the effective potential for the model and provide the masses of the heavy scalars and two radiatively light pseudo-Goldstone bosons.

Since the early years of General Relativity, understanding the long-time behavior of the cosmological solutions of Einstein's vacuum equations has been a fundamental yet challenging task. Solutions with global symmetries, or perturbations thereof, have been extensively studied and are reasonably understood. On the other hand, thanks to the work of Fischer-Moncrief and M. Anderson, it is known that there is a tight relation between the future evolution of solutions and the Thurston decomposition of the spatial 3-manifold. Consequently, cosmological spacetimes developing a future asymptotic symmetry should represent only a negligible part of a much larger yet unexplored solution landscape. In this work, we revisit a program initiated by the second named author, aimed at constructing a new type of cosmological solution first posed by M. Anderson, where (at the right scale) two hyperbolic manifolds with a cusp separate from each other through a thin torus neck. Specifically, we prove that the so-called double-cusp solution, which models the torus neck, is stable under $S^1 \times S^1$ - symmetry-preserving perturbations. The proof, which has interest on its own, reduces to proving the stability of a geodesic segment as a wave map into the hyperbolic plane and partially relates to the work of Sideris on wave maps and the work of Ringström on the future asymptotics of Gowdy spacetimes.

The growing penetration of distributed energy resources (DERs) in distribution networks (DNs) raises new operational challenges, particularly in terms of reliability and voltage regulation. In response to these challenges, we introduce an innovative DN operation framework with multi-objective optimization, leveraging community battery energy storage systems (C-BESS). The proposed framework targets two key operational objectives: first, to minimize voltage deviation, which is a concern for a distribution network service provider (DNSP), and second, to maximize the utilization of DERs on the demand side. Recognizing the conflicting nature of these objectives, we utilize C-BESS to enhance the system's adaptability to dynamically adjust DN operations. The multi-objective optimization problem is solved using the non-dominated sorting genetic algorithm-II (NSGA-II). Case studies using real-world data are conducted to validate the effectiveness of the proposed framework. The results show significant improvements in voltage regulation and DER utilization, demonstrating the potential of C-BESS in enabling more reliable DN operation. Our findings contribute to the ongoing discourse on the role of C-BESS in DN operation enhancement and DER integration.

Large antenna arrays can steer narrow beams towards a target area, and thus improve the communications capacity of wireless channels and the fidelity of radio sensing. Hardware that is capable of continuously-variable phase shifts is expensive, presenting scaling challenges. PIN diodes that apply only discrete phase shifts are promising and cost-effective; however, unlike continuous phase shifters, finding the best phase configuration across elements is an NP-hard optimization problem. Thus, the complexity of optimization becomes a new bottleneck for large-antenna arrays. To address this challenge, this paper suggests a procedure for converting the optimization objective function from a ratio of quadratic functions to a sequence of more easily solvable quadratic unconstrained binary optimization (QUBO) sub-problems. This conversion is an exact equivalence, and the resulting QUBO forms are standard input formats for various physics-inspired optimization methods. We demonstrate that a simulated annealing approach is very effective for solving these sub-problems, and we give performance metrics for several large array types optimized by this technique. Through numerical experiments, we report 3D beamforming performance for extra-large arrays with up to 10,000 elements.

In this study, we explore the application of Topological Data Analysis (TDA) and Lipschitz-Killing Curvatures (LKCs) as powerful tools for feature extraction and classification in the context of biomedical multiomics problems. TDA allows us to capture topological features and patterns within complex datasets, while LKCs provide essential geometric insights. We investigate the potential of combining both methods to improve classification accuracy. Using a dataset of biomedical images, we demonstrate that TDA and LKCs can effectively extract topological and geometrical features, respectively. The combination of these features results in enhanced classification performance compared to using each method individually. This approach offers promising results and has the potential to advance our understanding of complex biological processes in various biomedical applications. Our findings highlight the value of integrating topological and geometrical information in biomedical data analysis. As we continue to delve into the intricacies of multiomics problems, the fusion of these insights holds great promise for unraveling the underlying biological complexities.

In this work, we propose an end-to-end adaptive sampling neural network (MMPDE-Net) based on the moving mesh PDE method, which can adaptively generate new coordinates of sampling points by solving the moving mesh PDE. This model focuses on improving the efficiency of individual sampling points. Moreover, we have developed an iterative algorithm based on MMPDE-Net, which makes the sampling points more precise and controllable. Since MMPDE-Net is a framework independent of the deep learning solver, we combine it with PINN to propose MS-PINN and demonstrate its effectiveness by performing error analysis under the assumptions given in this paper. Meanwhile, we demonstrate the performance improvement of MS-PINN compared to PINN through numerical experiments on four typical examples to verify the effectiveness of our method.