Discrete Data Research Articles

Estimating vegetation structure and aboveground carbon storage in Western Australia using GEDI LiDAR, Landsat and Sentinel data

Abstract Worsening climate change impacts are amplifying the need for accurate estimates of vegetation structure and aboveground biomass density (AGBD) to assess changes in biodiversity and carbon storage. In Australia, increasing wildfire frequency and interest in the role of forests in the carbon cycle necessitates biomass mapping across large geographic extents to monitor forest change. The availability of spaceborne Light Detection and Ranging (LiDAR) optimised for vegetation structure mapping through the Global Ecosystem Dynamics Investigation (GEDI) provides an opportunity for large-scale forest AGBD estimates of higher accuracy. This study assessed the use of the GEDI canopy height product to predict woody AGBD across five vegetation types in Western Australia: tall eucalypt forests, eucalypt open‒woodlands, low-lying heathland, tropical eucalypt savannas, and tussock and hummock grasslands. Canopy height models were developed using random forest regressions trained on GEDI canopy height discrete point data. Predictor variables included spectral bands and vegetation indices derived from synthetic aperture radar (SAR) Sentinel‒1 data, and multispectral Landsat and Sentinel‒2 data. AGBD was subsequently estimated using power-law models derived by relating the predicted canopy heights to field AGBD plots. Mapping was conducted for 2020 and 2021. The accuracy of canopy height predictions varied with height quantiles; models underestimated the height of taller trees and overestimated the height of smaller trees. A similar underestimation and overestimation trend was observed for the AGBD estimates. The mean carbon stock was estimated at 69.0 ±12.0 MgCha-1 in the tall eucalypt forests of the Warren region; 33.8 ± 5.0 MgCha-1 for the open eucalypt woodlands in the South Jarrah region; 7.1 ± 1.4 MgCha-1 for the heathland and shrublands in the Geraldton Sandplains region; 43.9 ± 4.9 MgCha-1 for the Kimberley eucalypt savanna; and 3.9 ± 1.0 MgCha-1 for the Kimberley savanna grasslands. This approach provides a useful framework for the future development of this process for fire management, and habitat health monitoring.

Bounding the beta invariant of 3-connected graphs

The beta invariant is closely related to the Chromatic and Tutte Polynomials and has been extensively studied, see Brylawski [A combinatorial model for series-parallel networks, Trans. Amer. Math. Soc. 154 (1971) 1–22], Crapo [A higher invariant for matroids, J. Combin. Theory 2 (1967) 406–417], Lee and Wu [Bounding the beta invariant of 3-connected matroids, Discrete Math. 354 (2022) 1–11], Oxley [On Crapo’s beta invariant for matroids, Stud. Appl. Math. 66(3) (1982) 267–277], and others. Lee and Wu [Bounding the beta invariant of 3-connected matroids, Discrete Math. 354 (2022) 1–11] established that a [Formula: see text]-connected graph with [Formula: see text] vertices possesses a beta invariant of at least [Formula: see text], reaching equality only when the graph is a wheel or the Prism. Additionally, Oxley [On Crapo’s beta invariant for matroids, Stud. Appl. Math. 66(3) (1982) 267–277] provided characterizations for matroids with beta invariant values of two, three, and four. This paper extends the findings of Lee and Wu by offering a comprehensive characterization of all [Formula: see text]-connected graphs with [Formula: see text] vertices that have a beta invariant of either [Formula: see text] or [Formula: see text]. As a consequence, we prove that any [Formula: see text]-connected graph with at least [Formula: see text] vertices other than a Wheel has a beta invariant of at least [Formula: see text].

Metaphorical meaning dynamics: Identifying patterns in the metaphorical evolution of English words using mathematical modeling techniques

Abstract Conceptual metaphor theory has been criticized due to its emphasis on concepts instead of words and its top-down direction of analysis. In response to these criticisms, this paper employs a new strategy, utilizing established mathematical modeling methods to allow a systematic, quantitative analysis of the entire dataset produced by the Mapping Metaphor project at the University of Glasgow. This dataset consists of 9609 words performing 18916 metaphorical mappings between 414 domains. The data is represented as a network consisting of 414 nodes, the domains, connected by shared words. Words are represented by groups of directed mappings between all domains in which they occur. This is made possible by the use of a directed hypergraph representation, a tool commonly used in discrete mathematics and various areas of computer science but not previously applied to the metaphorical meanings of words. Examining the dataset as a whole, rather than focusing on individual words or metaphors, allows global patterns of behavior to emerge from the data without pre-filtering or selection by the authors. Outcomes of the analysis relating to the distributions of source and target domains within the network, the growth mechanisms at work in the spread of metaphorical meanings and how these relate to existing concepts in CMT are discussed.

The Biomechanical differences in maximum snatch weight between elite and sub-elite weightlifters: A one-dimensional statistical parameter mapping study

Background: Current research primarily relies on discrete data collected at specific time points to analyze the weightlifting process, often overlooking the impact of continuous temporal changes on athletic performance. Purpose: This study aimed to quantitatively analyze the one-dimensional kinematic patterns of maximum snatch weight actions in elite and sub-elite weightlifters using statistical parameter mapping (SPM). It explores kinematic differences in snatch actions between elite and sub-elite weightlifters, which assists in summarizing the technical characteristics of elite athletes. Methods: Two cameras recorded three successful maximum snatch attempts of 10 elite and 10 sub-elite weightlifters at the World Weightlifting Championships and Chinese Olympic selection competitions. Simi Motion 10.2 was used for kinematic analysis. SPM was employed for comparative analysis of snatch kinematics among different levels of weightlifters, and independent sample t-tests were used for phase duration proportions. Results: Elite weightlifters showed significant advantages in multiple movement phases: smaller knee joint angles in M1 phase (p < 0.001, 35.82%–78.45%) and M3 phases (p = 0.047, 17.47%–24.74%; p = 0.036, 30.92%–49.38%; p = 0.040, 50.93%–65.85%); lower vertical body center of gravity (COG) height in M1 phase (p = 0.019, 0.00%–51.08%), M3 phase (p = 0.046, 48.27%–100.00%), and M5 phase (p = 0.045, 38.49%–59.54%); closer displacement between barbell COG and body COG in M1 phase (p = 0.006, 0%–38.18%; p = 0.048, 43.91%–48.94%) and M2 phase (p = 0.026, 0%–100%); greater barbell acceleration in M5 phase (p < 0.001, 0%–94.61%); and slower barbell descent speed in M6 phase (p = 0.001, 0%–30.58%; p = 0.046, 44.57%–51.37%; p = 0.048, 67.16%–72.41%). Moreover, elite weightlifters exhibited significantly higher phase duration proportions in M1 phase (p = 0.034, d = 1.03) than sub-elite weightlifters. Conclusion: Elite weightlifters demonstrated longer distance and time exerting work on the barbell in M1 phase, less energy consumption in barbell ascent in M2 phase, and greater upward power gain for the barbell in M1 and M3 phases. They showed faster squat-to-barbell catching speed in M4 phase and excellent braking and precise squat-to-barbell catching capabilities in M5 and M6 phases.

Application of the Gray Wolf Optimization Algorithm and Neural Networks for Solving Discrete Problems

Relevance. In recent decades, metaheuristic optimization methods have become popular for solving complex problems that require searching for global extrema. Algorithms such as genetic algorithm (GA), ant colony optimization (ACO), particle swarm optimization (PSO), as well as more modern approaches such as cat pack optimization (CSO) and gray wolf pack optimization (GWO) demonstrate high efficiency, but their application is often limited by the conditions of continuity and differentiability of the objective functions. This is a challenge when solving problems with discrete data, where such requirements are not met. In this context, the search for methods that allow adapting metaheuristic algorithms to work with discrete functions is of particular relevance.Aim. The study is aimed at testing the hypothesis about the possibility of using a neural network trained on a limited set of discrete data as an approximation of a function sufficient for the correct execution of the GWO algorithm when searching for a global minimum. The implementation of this hypothesis can significantly expand the scope of GWO, making it available for a wider range of problems where functions are defined on discrete sets.Methods. The study is based on the analysis of existing approaches and experimental verification of the hypothesis on two test functions: a linear function and a Booth function, which are widely used as standards for evaluating the performance of optimization algorithms. Numerical experiments were conducted using neural networks as an approximating model to obtain the results. Solution. During the experiments, an analysis of the applicability of neural networks for approximating discrete functions was carried out, which showed the success of this approach. It was found that neural networks can approximate discrete functions with high accuracy, creating conditions for a successful search for a global minimum using the GWO algorithm.Novelty. For the first time, a hypothesis was proposed and tested on the use of neural networks for approximating objective functions in metaheuristic optimization problems on discrete data. This direction has not previously received due coverage in the scientific literature, which adds significance to the obtained results and confirms the effectiveness of the proposed approach.Practical significance. The results of the study open up new prospects for the application of algorithms such as GWO in optimization problems based on discrete data, expanding the capabilities of metaheuristic methods and facilitating their implementation in a wider class of applied problems, including problems where the use of other methods is limited.

Congratulations to our Editorial Board Members Prof. Taekyun Kim, Dr. Mladen Vassilev-Missana and Prof. Krassimir Atanassov for their anniversaries in 2024!

[EDITORIAL] Journal “Notes on Number Theory and Discrete Mathematics” congratulates its Editorial Board Members Prof. Taekyn Kim and Dr. Mladen Vassilev-Missana, as well as its Editor-in-Chief Prof. Krassimir Atanassov on the occasion of their anniversaries in 2024!

Adversarial Attacks on Pre-trained Deep Learning Models for Encrypted Traffic Analysis

For web security, it’s essential to accurately classify traffic across various web applications to detect malicious activities lurking within network traffic. However, the encryption protocols for privacy protection, such as TLS 1.3 and IPSec, make it difficult to apply traditional traffic classification methods like deep packet inspection (DPI). Recently, the advent of deep learning has significantly advanced the field of encrypted traffic analysis (ETA), outperforming traditional traffic analysis approaches. Notably, pre-trained deep learning based ETA models have demonstrated superior analytical capabilities. However, the security aspects of these deep learning models are often overlooked during the design and development process. In this paper, we conducted adversarial attacks to evaluate the security of pre-trained ETA models. We targeted ET-BERT, a state-of-the-art model demonstrating superior performance, to generate adversarial traffic examples. To carry out the adversarial example generation, we drew inspiration from adversarial attacks on discrete data, such as natural language, defining fluency from a network traffic perspective and proposing a new attack algorithm that can preserve this fluency. Finally, in our experiments, we showed our target model is vulnerable to the proposed adversarial attacks.

Development and Validation of Concept of Innovative Method of Computer-Aided Monitoring and Diagnostics of Machine Components

The monitoring and diagnostic system has been suggested as a non-destructive diagnostic method. The structure and operation of the suggested system can be described by the concept of digital shadow (DS). One of the main DS subsystems is a set of sensors properly placed on the monitored object and coupled with a discrete data processing model created in Matlab/Simulink. The discrete model, as another important DS subsystem of the monitored facility, transfers information about its technical condition to the operator based on data recorded by the sensor system. The digital monitoring model processes the recorded data in the form of the object’s response to actions caused by its operating conditions. This work formalized a mathematical model determining the coupling of the digital model with the sensors placed on the monitored object. The formulated method using DS, due to its sensitivity, enables the detection of the damage in the object at an early stage. The tests allowed for detecting the regularities enabling the determination of the area of damage to the shaft and its size.

Walking biomechanics in women with patellofemoral osteoarthritis differ compared to men with and women without patellofemoral osteoarthritis

BackgroundDifferences in walking biomechanics between women and men with patellofemoral joint (PF) osteoarthritis (OA) may contribute to the development or progression of persistent symptoms in people with PFJ OA. ObjectiveEvaluate how walking biomechanics of women with PF OA differ from: (i) men with PFJ OA; and (ii) women without PF OA. Second, explore the relationship between knee-related symptoms/function and walking biomechanics in individuals with PF OA, and whether these are modified by sex. MethodsSixty-seven individuals with PF OA (43 women) and 14 women without PF OA were included. Biomechanics data were recorded during walking. Patient-reported symptoms and function were obtained using the Knee injury and Osteoarthritis Outcome Score. Differences in continuous biomechanical data were assessed using statistical parametric mapping, with discrete data and relationships evaluated using linear models. ResultsWomen with PF OA walked with a greater hip adduction angle throughout stance (t > 2.757) and lower impulses for the hip flexion, knee flexion, and ankle dorsiflexion moments (adjusted mean differences [95 % CI]:3.3 × 10–2 [-4.9 × 10–2, -1.6 × 10–2], -2.9 × 10–2 [-5.3 × 10–2, -0.4 × 10–2], -5.1 × 10–2 [-8.2 × 10–2, -2.0 × 10–2] Nms/kg, respectively) compared to men with PF OA. Compared to their asymptomatic peers, women with PF OA displayed a 5° offset towards greater hip flexion. Higher knee adduction moment impulse correlated with worse KOOS-ADL scores in men, not women. ConclusionObserved biomechanical differences were small in nature with moderate to weak relationship observed with the KOOS. Findings were not limited to the knee, indicating that women with PF OA display unique biomechanical features across the kinetic-chain.

Discrete Joint Random Variables in Fréchet-Weibull Distribution: A Comprehensive Mathematical Framework with Simulations, Goodness-of-Fit Analysis, and Informed Decision-Making

This paper introduces a novel four-parameter discrete bivariate distribution, termed the bivariate discretized Fréchet–Weibull distribution (BDFWD), with marginals derived from the discretized Fréchet–Weibull distribution. Several statistical and reliability properties are thoroughly examined, including the joint cumulative distribution function, joint probability mass function, joint survival function, bivariate hazard rate function, and bivariate reversed hazard rate function, all presented in straightforward forms. Additionally, properties such as moments and their related concepts, the stress–strength model, total positivity of order 2, positive quadrant dependence, and the median are examined. The BDFWD is capable of modeling asymmetric dispersion data across various forms of hazard rate shapes and kurtosis. Following the introduction of the mathematical and statistical frameworks of the BDFWD, the maximum likelihood estimation approach is employed to estimate the model parameters. A simulation study is also conducted to investigate the behavior of the generated estimators. To demonstrate the capability and flexibility of the BDFWD, three distinct datasets are analyzed from various fields, including football score records, recurrence times to infection for kidney dialysis patients, and student marks from two internal examination statistical papers. The study confirms that the BDFWD outperforms competitive distributions in terms of efficiency across various discrete data applications.

Aurally enhanced media promotion: Music and sound in Greek television commercials

The sonic dimensions of modern media—including speech, sound effects, and music—play a significant role in communicating signs, meanings, and messages through multimodal channels of audiovisual connotations. Each feature can be regarded as a discrete unit; however, the overall framework is established through an orchestrated “interaction” of these components. Receiving information through televised advertisements is a cognitive process that requires critical analysis. This study examines the functions of music and sound in television advertising, emphasizing the collaboration between visual and audio material. Our objective is to equally investigate optical and musical conventions in selected TV commercials. We will also analyze how meaning emerges through visuals, sounds, and music in correlation to the product, branding, and the purpose of advertisements as short, concise audiovisual narratives. The data for this study were collected from a series of food and drink ads shown on Greek television between 2015 and 2020. The research draws upon contemporary culturally informed, musicological, semiotic, and interpretive methodologies, applying both textual and contextual analyses to examine how these techniques captivate audiences and effectively communicate messages. Understanding music’s psychological and pragmatic effects in advertising is crucial for recognizing its role as a powerful tool that influences consumer behavior and perception.

Mathematical Modeling of the Dynamics of Trust in Automation

Trust in automation (TiA) is crucial for human-automation interactions, but traditionally measured statically via discrete survey data points. We developed a mathematical model capturing the dynamic nature of TiA, incorporating measurable components across different timescales, and validated it through experimental data. Our findings show that: (1) trust timescales are important for predicting automation usage, emphasizing trust’s key role in decision-making; (2) short-timescale trust significantly impacts overall trust dynamics, highlighting the importance of recent interactions; (3) trust levels are generally higher in highly reliable automation scenarios, though this pattern emerges after an initial evaluation period; and (4) automation type, implying varying workloads, does not significantly impact trust dynamics in highly reliable systems but fosters increased trust in lower-reliability systems. This study advances our understanding of trust dynamics in automation, contributing to more intuitive and trustworthy technology development.

Spectrum Broadening Due to Nonselective Linear Absorption

The position and linewidth of an emission spectrum reflect the physical properties of the luminophor. So, keeping the spectrum from distortion is very important in its measurement. However, we find that the spectrum linewidth will be broadened when the near-infrared radiation from a sodium lamp passes through a nonselective linear absorbing filter. This counterintuitive linewidth-broadening phenomenon is obvious when the residual light power after the filter is low enough, typically lower than 2.48×10−4 μW. This novel linewidth-broadening effect is different from the well-known Lorentzian, Doppler, and Voigt broadening, and is likely to be more independent evidence of the discrete wavelet structure of classical plane light waves. The effect is significant in high-sensitivity spectroscopy measurements, for example streak camera spectroscopy and Raman spectroscopy experiments. In addition, this effect may also be significant for cosmological research.

ValveVision AI - a multimodal language model for qualitative reporting of the aortic valve in echocardiography

Abstract Background The precise assessment of the aortic valve via echocardiography is critical for early detection and management of aortic valve diseases. Until recently, previous studies have examined machine learning models to estimate individual measurements and severity of aortic stenosis (AS) from echocardiographic images. These image processing algorithms, while precise in their narrow focus, fall short in mirroring the holistic and interconnected clinical judgment typical of human echocardiographers in producing a qualitative report. Large language models (LLMs), particularly image-to-text multi-modal LLMs, are a fundamental advance in the field of deep learning with implications for a host of applications in medical imaging. They promise to encapsulate not just discrete data points typical in traditional machine learning, but also the complex contextual interrelations in clinical diagnosis. Methods In this study, a large-scale heterogeneous database of echocardiographic images containing over 90,681 studies with textual descriptors of the aortic valve was used to train a single, image-to-text multimodal LLM called ValveVision AI. The ground truth textual summaries were drafted by level III echocardiographers in a clinical setting between 2015-2020. BLEU and ROUGE score was calculated. The models were retrospectively assessed on a holdout dataset. Reviewing physicians compared the generated summary to the ground truth and binarily agreed or rejected it. Receiver Operating Characteristics (ROC) for distinct pathologies were also assessed (Figure I). Results ValveVision AI performed with a BLEU score of 0.45 and a ROUGE score of 0.49. The performance of the model in reporting on classification of moderate/severe vs none/mild AS in concordance with the validation protocol described above achieved a specificity of 91.98% and a sensitivity of 83.89%, along with more precise qualitative description. Qualitatively, the model exhibited the capability of zero-shot learning in certain instances, however, this result remains an area of exploration. Conclusion This study represents to our knowledge, the first attempt at an image-representation to text-tokenizer deep learning model architecture to mimic the thought and subtlety of echocardiographic qualitative analysis of the aortic valve. The results suggest that this multimodal LLM has sufficient accuracy to create a preliminary textual summary of the aortic valve that, if paired with a point of care ultrasound (POCUS) device in a primary care setting, may facilitate case triage, increase efficiency, and determine a more precise care pathway for patients.

Using Rigidity and Conjugation of Subunits to Modulate Supramolecular Topologies Constructed by Half-Sandwich Fragments.

The synthesis of supramolecular compounds with a high degree of controllability and the targeted modulation of their topological transitions pose significant challenges in situ. In this study, we have successfully constructed an array of discrete structures based on a series of bidentate pyridyl ligands (L1, L2, and L3), which were subsequently ligated with half-sandwiched (Cp*Ir fragments) building blocks. Our further investigations elucidate a strategy for coordinating the relative lengths of the bidentate ligands with the building blocks, achieving specific concentrations that drive the transformation of tetranuclear metal macrocycles into Borromean rings. Notably, the distinct characteristics of the three pyridyl ligands markedly influence the efficiency of synthesis and the topological conversion of the supramolecular macrocycles. Detailed structural analyses reveal that π-π stacking interactions, the electron-donating capabilities of the ligands, and hydrogen-bonding interactions are pivotal in stabilizing these molecular macrocycles and in facilitating their transformation to Borromean rings. The analyses underscore the importance of the electron-rich effect induced by the sulfur atoms in the ligands and the regulation and modulation of the pyridine functional group in contributing to the structural stability and altered characteristics of the macrocycles.

Computing MEMs and Relatives on Repetitive Text Collections

We consider the problem of computing the Maximal Exact Matches (MEMs) of a given pattern \(P[1\mathinner{.. }m]\) on a large repetitive text collection \(T[1\mathinner{.. }n]\) over an alphabet of size \(\sigma\) , which is represented as a (hopefully much smaller) run-length context-free grammar of size \(g_{rl}\) . We show that the problem can be solved in time \(O(m^{2}\log^{\epsilon}n)\) , for any constant \(\epsilon\gt0\) , on a data structure of size \(O(g_{rl})\) . Further, on a locally consistent grammar of size \(O(\delta\log\frac{n\log\sigma}{\delta\log n})\) , the time decreases to \(O(m\log m(\log m+\log^{\epsilon}n))\) . The value \(\delta\) is a function of the substring complexity of \(T\) and \(\Omega(\delta\log\frac{n\log\sigma}{\delta\log n})\) is a tight lower bound on the compressibility of repetitive texts \(T\) , so our structure has optimal size in terms of \(n\) , \(\sigma\) , and \(\delta\) . We extend our results to several related problems, such as finding \(k\) -MEMs, MUMs, rare MEMs, and applications. Categories and Subject Descriptors: E.1 [Data structures] ; E.2 [Data storage representations] ; E.4 [Coding and information theory]: Data compaction and compression; F.2.2 [Analysis of algorithms and problem complexity] : Nonnumerical algorithms and problems— Pattern matching, Computations on discrete structures, Sorting and searching

An Investigation into the Behavior of Cathode and Anode Spots in a Welding Discharge

The effective development of modern welding technologies and the improvement of equipment and materials inevitably require deep theoretical knowledge about the physical phenomena occurring in the electric arc column and in the near-electrode region. However, there is still no convincing theoretical description of an arc discharge. This article demonstrates, through the generalization of known experimental facts and studies using a high-speed camera, that the conductive channel of an electric arc has a discrete structure, consisting of a set of thin channels through which the main discharge current passes. The cathode spot of an arc discharge is a highly heated and brightly glowing area on the cathode’s surface. Electron emission occurs from this region, which supports the discharge as well as the removal of the cathode material. We propose a new technique to study the reverse side of the cathode spot, revealing a structure consisting of individual cells or fragments of the cathode spot. For the first time, we present the anode spots captured by a high-speed camera. We carry out an analysis of the spots’ structure. We determine the parameters affecting the mobility of cathode and anode spots. We propose a hypothesis based on the obtained experimental facts about the heterogeneous structure of cathode and anode spots in an arc discharge and the existence of current filaments that affect the mobility of spots during arc combustion.

Controllable Human Trajectory Generation Using Profile-Guided Latent Diffusion

Trajectory generation is a vital element in AI applications. Firstly, it enables simulation such as traffic simulation and epidemic spreading modeling. Secondly, it can provide synthetic privacy-preserving data for training AI models. Notably, trajectory generation featuring controllable user profiles holds substantial value in generating customized mobility trajectories tailored to diverse requirements. However, relevant work is still lacking. On the one hand, traditional deep generative models fall short in guiding controllable trajectory generation due to the statistical nature of human mobility patterns and the corresponding insufficient control mechanisms. On the other hand, though the diffusion model has demonstrated strong generative capabilities in many fields, to achieve controllable generation on discrete trajectory data, we still need to redesign the structure of the continuous diffusion model. In this paper, we introduce a controllable trajectory generation framework that leverages a continuous diffusion model and classifier guidance for more robust condition control. Our proposed framework comprises two modules: a latent trajectory diffusion model and a trajectory classifier for profile guidance. Experiments on two real-world mobility datasets consistently demonstrate its capability of generating trajectories matching given user profiles and conforming to human mobility patterns. Our source code and trained models are released at https://github.com/tsinghua-fib-lab/User-Profile-Guided-Latent-Diffusion .

Transverse compressive behavior of plain and polypropylene fiber-reinforced concrete infilled steel tubes as hybrid steel-concrete nodes for discrete reticulated shell structures

This study investigates using concrete infilled steel tubes as hybrid nodes for discrete reticulated shell structures under compressive loading transverse to the tube axis. Three configurations were tested: empty, plain concrete infilled and polypropylene fiber-reinforced concrete infilled steel tubes. Displacement-controlled loading was applied through welded steel plates. Simplified beam theory models and nonlinear finite element analysis were used to evaluate the structural behavior. The concrete infilled steel tube specimens had higher stiffness and load-bearing capacity than the empty ones. The structural behavior of the concrete infilled specimens could be split into pre- and post-split-tensile cracking phases. No significant difference was found between the plain and fiber-reinforced concrete specimens under loads up to 1500 kN, which was the load limit of the testing machine. An additional fiber-reinforced concrete specimen, tested on a higher capacity machine, had a load-bearing capacity of 2400 kN and exhibited ductile failure.

Topology for discrete mathematics

A series of results is presented on the problem of the existence and the uniqueness of extensions of continuous mappings on topological spaces.