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This research explores the application of quadratic polynomials in Python for advanced data analysis. The study demonstrates how quadratic models can effectively capture nonlinear relationships in complex datasets by leveraging Python libraries such as NumPy, Matplotlib, scikit-learn, and Pandas. The methodology involves fitting quadratic polynomials to the data using least-squares regression and evaluating the model fit using the coefficient of determination (R-squared). The results highlight the strong performance of the quadratic polynomial fit, as evidenced by high R-squared values, indicating the model's ability to explain a substantial proportion of the data variability. Comparisons with linear and cubic models further underscore the quadratic model's balance between simplicity and precision for many practical applications. The study also acknowledges the limitations of quadratic polynomials and proposes future research directions to enhance their accuracy and efficiency for diverse data analysis tasks. This research bridges the gap between theoretical concepts and practical implementation, providing an accessible Python-based tool for leveraging quadratic polynomials in data analysis.

In this paper, we introduce a new nonlinear optical channel equalizer based on Transformers. By leveraging parallel computation and attending directly to the memory across a sequence of symbols, we show that Transformers can be used effectively for nonlinear compensation (NLC) in coherent long-haul transmission systems. For this application, we present an implementation of the encoder part of the Transformer and analyze its performance over a wide range of different hyper-parameters. It is shown that by proper embeddings and processing blocks of symbols at each iteration and also carefully selecting subsets of the encoder's output to be processed together, an efficient nonlinear equalization can be achieved for different complexity constraints. To reduce the computational complexity of the attention mechanism, we further propose the use of a physic-informed mask inspired by nonlinear perturbation theory. We also compare the Transformer-NLC with digital back-propagation (DBP) under different transmission scenarios in order to demonstrate the flexibility and generalizability of the proposed data-driven solution.

In distributed computing by mobile robots, robots are deployed over a region, continuous or discrete, operating through a sequence of \textit{look-compute-move} cycles. An extensive study has been carried out to understand the computational powers of different robot models. The models vary on the ability to 1)~remember constant size information and 2)~communicate constant size message. Depending on the abilities the different models are 1)~$\mathcal{OBLOT}$ (robots are oblivious and silent), 2)~$\mathcal{FSTA}$ (robots have finite states but silent), 3)~$\mathcal{FCOM}$ (robots are oblivious but can communicate constant size information) and, 4)~$\mathcal{LUMI}$ (robots have finite states and can communicate constant size information). Another factor that affects computational ability is the scheduler that decides the activation time of the robots. The main three schedulers are \textit{fully-synchronous}, \textit{semi-synchronous} and \textit{asynchronous}. Combining the models ($M$) with schedulers ($K$), we have twelve combinations $M^K$. In the euclidean domain, the comparisons between these twelve variants have been done in different works for transparent robots, opaque robots, and robots with limited visibility. There is a vacant space for similar works when robots are operating on discrete regions like networks. It demands separate research attention because there have been a series of works where robots operate on different networks, and there is a fundamental difference when robots are operating on a continuous domain versus a discrete domain in terms of robots' movement. This work contributes to filling the space by giving a full comparison table for all models with two synchronous schedulers: fully-synchronous and semi-synchronous.

For unforeseen emergencies, such as natural disasters and pandemic events, it is highly demanded to cope with the explosive growth of mobile data traffic in extremely critical environments. An Unmanned aerial vehicle (UAV) fleet is an effective way to facilitate the Emergency wireless COmmunication NETwork (EcoNet). In this article, a MUlti-tier Heterogeneous UAV Network (MuHun), which is with different UAV fleets in different altitudes, is proposed to flexibly serve various emergencies. We refresh the key performance indicators of full coverage, network capacity, low latency, and energy efficiency in harsh environments. Then, we present the special challenges regarding shadowing-dominated complex channel model, energy supply limited short-endurance, various communication mechanisms coexistence, and communication island for underground users in UAV-based EcoNet, followed by the MuHun-based EcoNet architecture and its advantages. Furthermore, some potential solutions such as the new hybrid-channel adapted resource allocation, reconfigurable intelligent surface assisted UAV communications, competitive heterogenous-networks, and magnetic induction based air-to-ground/underground communications are discussed to effectively achieve full coverage, high capacity, high energy efficiency, and diverse qualities of services for EcoNets in harsh environments.

Quantum computing holds immense potential for solving classically intractable problems by leveraging the unique properties of quantum mechanics. The scalability of quantum architectures remains a significant challenge. Multi-core quantum architectures are proposed to solve the scalability problem, arising a new set of challenges in hardware, communications and compilation, among others. One of these challenges is to adapt a quantum algorithm to fit within the different cores of the quantum computer. This paper presents a novel approach for circuit partitioning using Deep Reinforcement Learning, contributing to the advancement of both quantum computing and graph partitioning. This work is the first step in integrating Deep Reinforcement Learning techniques into Quantum Circuit Mapping, opening the door to a new paradigm of solutions to such problems.

Regular transition systems (RTS) are a popular formalism for modeling infinite-state systems in general, and parameterised systems in particular. In a CONCUR 22 paper, Esparza et al. introduce a novel approach to the verification of RTS, based on inductive invariants. The approach computes the intersection of all inductive invariants of a given RTS that can be expressed as CNF formulas with a bounded number of clauses, and uses it to construct an automaton recognising an overapproximation of the reachable configurations. The paper shows that the problem of deciding if the language of this automaton intersects a given regular set of unsafe configurations is in $\textsf{EXPSPACE}$ and $\textsf{PSPACE}$-hard. We introduce $\textit{regular abstraction frameworks}$, a generalisation of the approach of Esparza et al., very similar to the regular abstractions of Hong and Lin. A framework consists of a regular language of $\textit{constraints}$, and a transducer, called the $\textit{interpretation}$, that assigns to each constraint the set of configurations of the RTS satisfying it. Examples of regular abstraction frameworks include the formulas of Esparza et al., octagons, bounded difference matrices, and views. We show that the generalisation of the decision problem above to regular abstraction frameworks remains in $\textsf{EXPSPACE}$, and prove a matching (non-trivial) $\textsf{EXPSPACE}$-hardness bound. $\textsf{EXPSPACE}$-hardness implies that, in the worst case, the automaton recognising the overapproximation of the reachable configurations has a double-exponential number of states. We introduce a learning algorithm that computes this automaton in a lazy manner, stopping whenever the current hypothesis is already strong enough to prove safety. We report on an implementation and show that our experimental results improve on those of Esparza et al.

We contribute to a better understanding of the class of functions that can be represented by a neural network with ReLU activations and a given architecture. Using techniques from mixed-integer optimization, polyhedral theory, and tropical geometry, we provide a mathematical counterbalance to the universal approximation theorems which suggest that a single hidden layer is sufficient for learning any function. In particular, we investigate whether the class of exactly representable functions strictly increases by adding more layers (with no restrictions on size). As a by-product of our investigations, we settle an old conjecture about piecewise linear functions by Wang and Sun (2005) in the affirmative. We also present upper bounds on the sizes of neural networks required to represent functions with logarithmic depth.

The existence of representative datasets is a prerequisite of many successful artificial intelligence and machine learning models. However, the subsequent application of these models often involves scenarios that are inadequately represented in the data used for training. The reasons for this are manifold and range from time and cost constraints to ethical considerations. As a consequence, the reliable use of these models, especially in safety-critical applications, is a huge challenge. Leveraging additional, already existing sources of knowledge is key to overcome the limitations of purely data-driven approaches, and eventually to increase the generalization capability of these models. Furthermore, predictions that conform with knowledge are crucial for making trustworthy and safe decisions even in underrepresented scenarios. This work provides an overview of existing techniques and methods in the literature that combine data-based models with existing knowledge. The identified approaches are structured according to the categories integration, extraction and conformity. Special attention is given to applications in the field of autonomous driving.

As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.

Detecting carried objects is one of the requirements for developing systems to reason about activities involving people and objects. We present an approach to detect carried objects from a single video frame with a novel method that incorporates features from multiple scales. Initially, a foreground mask in a video frame is segmented into multi-scale superpixels. Then the human-like regions in the segmented area are identified by matching a set of extracted features from superpixels against learned features in a codebook. A carried object probability map is generated using the complement of the matching probabilities of superpixels to human-like regions and background information. A group of superpixels with high carried object probability and strong edge support is then merged to obtain the shape of the carried object. We applied our method to two challenging datasets, and results show that our method is competitive with or better than the state-of-the-art.