2021 - Repozytorium publikacji - Politechnika Gdańska

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Publikacje z roku 2021

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A Wideband Channel Model for Body Area Networks in Circular Metallic Indoor Environments
- Filipe D. Cardoso
- Manuel M. Ferreira
- Sławomir Ambroziak
- Luis M. Correia
2021 Pełny tekst IEEE Access
In this paper, the wideband characterization of the propagation channel in circular metallic indoor environments is addressed, regarding Body Area Networks and 5G small cells, an analytical model for the dependence of the mean delay and the average delay spread on the circle radius, the working frequency and the distance between the transmitter and the receiver being proposed. The derivation of the model is initially done analytically, based on optical physics, after which simulation results allow to obtain the values of the coefficients. The simulator was previously assessed with measurements at 2.45 GHz in a passenger ferry room with a circular shape. For a random positioning of the transmitter and the receiver, and a given distance between them, it is observed that the mean delay and the delay spread increase linearly with the radius; furthermore, the mean delay increases quadratically with the distance, while the delay spread has a concave parabolic behavior, with the maximum being at a distance equal to the radius. In a practical case, regarding the positioning of an Access Point inside the room, it is recommended that it is done at the circle center, in order to reduce delay spread.
A wireless network for off-shore marine research
- Krzysztof Gierłowski
- Michał Hoeft
- Marcin Wichorowski
2021 Bollettino di Geofisica Teorica ed Applicata
This paper presents functionality, the most important technology developments and potential use cases of digital communication system, developed to facilitate selected tasks commonly required during off-shore operations and marine research, including: broadband information sharing between platforms isolated from an on-shore communication infrastructure, acquisition of large data-sets from variety of off-shore data sources (e.g. fleet of drones) as well as remote access and control to the research infrastructure exploited in environment with limited communication between neighboring devices.
Abilities, Motivations, and Opportunities of Furloughed Employees in the Context of Covid-19: Preliminary Evidence From the UK
- Joanna Szulc
- Rachael Smith
2021 Pełny tekst Frontiers in Psychology
The Covid-19 global pandemic is a crisis like no other, forcing governments to implement prolonged national lockdowns in an effort to limit the spread of the disease. As organizations aim to adapt and remain operational, employers can suspend or reduce work activity for events related to Covid-19 and claim government support to subsidize employee wages. In this way, some employees are placed on furlough (i.e., temporary unemployment) as opposed to being made redundant. While the impact of such schemes on global economy attracted much attention, their micro-level impact on individual employees is still unknown. Building on the ability-motivation-opportunity (AMO) framework, this pilot study explores how employees' perceptions of abilities, motivation, and opportunities are affected as a result of furlough. Rapid ethnography including interviews, observations, and document analysis in a British organization provided insights into the perceptions and experiences of employees put on furlough and highlighted that all three elements of AMO are affected by the current situation, either positively or negatively. We identify theoretical contributions and suggest a number of AMO enhancing practices in the context of furlough.
Accelerated Gradient-Based Optimization of Antenna Structures Using Multi-Fidelity Simulations and Convergence-Based Model Management Scheme
- Sławomir Kozieł
- Anna Pietrenko-Dąbrowska
2021 Pełny tekst IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION
The importance of numerical optimization has been steadily growing in the design of contemporary antenna structures. The primary reason is the increasing complexity of antenna topologies, [ a typically large number of adjustable parameters that have to be simultaneously tuned. Design closure is no longer possible using traditional methods, including theoretical models or supervised parameter sweeping. To ensure reliability, optimization is normally carried out at the level of full-wave electromagnetic (EM) simulations, which incurs major computational expenses. The issue can be alleviated using a variety of methods such as the incorporation of adjoint sensitivities, sparse sensitivity updates (for local optimization), or the employment of surrogate modeling methods (in the context of global search). Another possibility is utilization of variable-fidelity simulation models, which, in practice, is most often restricted to two levels (coarse/fine or low-/high-fidelity models), and accompanied by appropriate low-fidelity model correction (e.g., space mapping). This paper proposes an accelerated version of a trust-region gradient-based procedure, which involves simulation model management by continuous adjustment of EM analysis fidelity throughout the optimization process. Decision making process is based on the convergence status of the algorithm. The initial stages of the optimization run are executed using the coarsest discretization of the structure at hand with the model being gradually refined towards the end of the process. This enables considerable computational savings without degrading the quality of the final design. The presented approach has been comprehensively validated using a benchmark set of four broadband antennas and compared to the reference trust-region procedure and two state-of-the-art accelerated algorithms. The average computational savings are almost sixty percent as compared to the reference.
Accelerated large scale test set-up design in natural corrosion marine environment
- Krzysztof Wołoszyk
- Yordan Garbatov
2021
The standards for conducting small-scale specimen tests are well developed, but there is a lack of direct guidelines for conducting corrosion tests for large-scale specimens. The objective here is to develop a methodology which may be used in designing an accelerated corrosion test of large-scale structural components subjected to a natural corrosion marine environment. Different factors influencing corrosion degradation of steel structures are analysed, including salinity and pH of the water, water velocity, oxygen content etc. The most important governing factors of corrosion degradation are identified, and the process of corrosion testing is defined in order to produce accelerated corrosion degradation in relatively short interval of time, developing a natural corrosion degradation mechanism, leading to irregular random corroded surfaces and material properties changes. Existing commercial testing equipment, that can be employed in creating accelerated corrosion conditions for testing large scale specimens, are investigated and the most efficient salt corrosion testing tank is identified. Several features of the corrosion test set-up are discussed.
Accumulation of volatile constituents in agar and bioreactor shoot cultures of Verbena officinalis L.
- Adam Kokotkiewicz
- Bożena Zabiegała
- Pawel Kubica
- Agnieszka Szopa
- Adam Bucinski
- Halina Ekiert
- Maria Luczkiewicz
2021 PLANT CELL TISSUE AND ORGAN CULTURE
In vitro shoot cultures of common vervain (Verbena officinalis L., Verbenaceae) were established and evaluated for essential oil content. The shoots were maintained as stationary microshoot culture, grown on Schenk–Hildebrandt medium supplemented with 9.84 µM of 2-isopentenyladenine and 1.0 µM of thidiazuron. The shoots were further adapted to grow in two temporary immersion bioreactors (RITA and Plantform), using Schenk–Hildebrandt medium supplemented with 4.92 µM indole-3-butyric acid. The best growth, in terms of dry weight production, was recorded for agar culture (0.73 g l−1 day−1), followed by 35 days (day) Plantform (0.53 g l−1 day−1), 35 days RITA (0.42 g l−1 day−1) and 56 days Plantform culture (0.29 g l−1 day−1). The volatile fractions were obtained by simultaneous hydrodistillation-extraction of the plant materials using Likens–Nickerson apparatus. Aerial parts of V. officinalis intact plant and 56 d Plantform culture had the highest essential oil content (8.1 g kg−1) whereas the stationary agar culture excelled in essential oil productivity (5.0 mg l−1 day−1). The volatile oils isolated from in vitro shoots contained 1-octen-3-ol, benzyl alcohol, phenethyl alcohol, coumaran and hexadecenoic acid as major constituents. 1-Octen-3-ol was the predominant (> 50%) constituent of 35 day shoot cultures maintained in temporary immersion bioreactors whereas coumaran (19%) and hexadecanoid acid (21%) prevailed in stationary microshoot culture. The volatile fraction obtained from aerial parts of V. officinalis intact plant was composed mainly of hexadecanoic acid (> 20%), linalool (> 8%), anethole (> 5%) and carvone (> 3%).
Accuracy of Pretreatment Ultrasonography Assessment of Intra-Abdominal Spread in Epithelial Ovarian Cancer: A Prospective Study
- Agnieszka Tomasińska
- Maciej Stukan
- Michał Badocha
- Aleksandra Myszewska
2021 Pełny tekst Diagnostics
The aim of this study was to test the accuracy of ultrasonography performed by gynecological oncologists for the preoperative assessment of epithelial ovarian cancer (EOC) spread in the pelvis and abdominal cavity. A prospective, observational cohort study was performed at a single tertiary cancer care unit. Patients with suspected EOC were recruited and underwent comprehensive transvaginal and abdominal ultrasonography performed by a gynecological oncologist. Sixteen intra-abdominal localizations and parameters were assessed using ultrasonography and compared with surgical-pathological status (reference standard). Sensitivity, specificity, positive and negative predictive values, and overall accuracy were calculated. Differences were analyzed using Fisher’s exact and chi-square tests. Ultimately, we included 132 patients (median age 62 years), of whom 67% were in stage IIIC–IVB and 72% had serous cancer. Overall prediction accuracies for the involvement of the omentum, small bowel mesentery root, and frozen pelvis, and detecting ascites were >90%. Detecting the involvement of the pelvis peritoneum, liver and spleen hilum, and rectosigmoid colon, and predictions of disease stage and residual disease had overall accuracies of 80–90%. The lowest accuracy was for involvement of the abdominal peritoneum (69%) and diaphragm peritoneum (right 71%; left 75%) and surgical complexity prediction (77%). Stratification of results by presence or absence of ascites revealed significantly higher specificity of ultrasonography (clinically meaningful) for assessments of the abdominal/pelvic peritoneum, spleen hilum, and rectum wall, if there were ascites. A gynecological oncologist, experienced in surgery and sonology, performing comprehensive ultrasonography on patients with EOC can accurately detect intraperitoneal lesions and recognize critical disease manifestations and predict stage, surgical complexity, and residual disease, which allow accurate qualification of patients for primary debulking surgery or neoadjuvant chemotherapy.
Accurate Modeling of Frequency Selective Surfaces Using Fully-Connected Regression Model with Automated Architecture Determination and Parameter Selection Based on Bayesian Optimization
- Nurullah Calik
- Mehmet Belen
- Peyman Mahouti
- Sławomir Kozieł
2021 Pełny tekst IEEE Access
Surrogate modeling has become an important tool in the design of high-frequency structures. Although full-wave electromagnetic (EM) simulation tools provide an accurate account for the circuit characteristics and performance, they entail considerable computational expenditures. Replacing EM analysis by fast surrogates provides a way to accelerate the design procedures. Unfortunately, modeling of microwave passives is a challenging task due to their highly-nonlinear outputs. Frequency selective surfaces (FSSs) constitute a representative example with their multi-resonant reflection and transmission responses that need to be represented over broad frequency ranges. Deep neural networks (DNNs) seem to be the promising techniques for handling such cases. However, a serious practical issue associated with their employment is an appropriate selection of the model parameters, including its architecture. A common practice is experience-driven setup, heavily based on trial and error, which does not guarantee the optimum model determination and may lead to multiple problems such as poor generalization or high variance of the model predictive power with respect to the training data set selection. This paper proposes a novel modeling framework, referred to as a fully-connected regression model (FCRM), where the crucial role is played by Bayesian Optimization (BO), incorporated to determine the DNN-based model setup, including both its architecture and the hyperparameter values, in a fully automated manner. For validation, FCRM is applied to construct the model of a Minkowski Fractal-Based FSS. The efficacy of the methodology is demonstrated through comparisons with several benchmark techniques, including the DNN surrogates established using the traditional methods as well as conventional regression models. The numerical results indicate that FCRM exhibits considerably improved prediction power and reduced sensitivity to the training sample assignment.
Achieving Efficient and Stable Deammonification at Low Temperatures—Experimental and Modeling Studies
- Hussein Al-Hazmi
- Xi Lu
- Dominika Grubba
- Joanna Majtacz
- Przemysław Kowal
- Jacek Mąkinia
2021 Pełny tekst ENERGIES
The short-term effects of temperature on deammonification sludge were evaluated in a laboratory-scale sequencing batch reactor (SBR). Mathematical modeling was used for further evaluations of different intermittent aeration strategies for achieving high and stable deammonification performance at decreasing temperatures. As for the biomass cultivated at high temperatures (e.g., 30 C), a higher temperature dependency (the adjusted Arrhenius coefficient for 11–17 C = 1.71 vs. for 17–30 C = 1.12) on the specific anammox growth rates was found at lower temperatures (11–17 C) in comparison with higher temperatures (17–30 C). Further evaluations of recovering the nitrogen removal efficiency at decreasing temperatures with the mathematical model by modifying the intermittent aeration strategies (aeration frequency (F) and the ratio (R) between non-aerated (non-aer) phase and aerated (aer) phase durations) indicated that intermittent aeration with a prolonged non-aerated phase (e.g., R 4 regardless of F value) would help to maintain high and stable deammonification performance (~80%) at decreasing temperatures (14–22 C). Extending the non-aerated phases (increasing R) and reducing the frequency (F) of off/on phase changes have a positive effect on increasing energy savings, leading to increasing interest in this method.
Acid Rider. Idea „komunizmu kwasowego” Marka Fishera a doświadczenie zewnętrzności
- Andrzej Karalus
2021 Pełny tekst Praktyka Teoretyczna
Acid communism pozostaje intrygującym wstępem Marka Fishera do planowanej przezeń historycznej rekonstrukcji radykalnych projektów politycznych wyrosłych na gruncie kultury psychodelicznej. Idea ta nie narodziła się w próżni, lecz stanowi konsekwencję jego poszukiwań teoretycznych. Artykuł próbuje osadzić koncepcję „komunizmu kwasowego” w szerszym kontekście, skupiając się na refleksji poświęconej doświadczeniu psychodelicznemu jako kontakcie z Zewnętrzem. Twierdzę, że istnieje zbieżność między koncepcją komunizmu kwasowego a zagadnieniami poruszanym przez Fishera w książce The Weird And The Eerie, gdzie analizuje potencjalnie transformujący szok zetknięcia z tym, co radykalnie Inne. Omówię przy tym krótko dwa fenomeny muzyczno-kulturowe, które zaistniały już po epoce psychodelii lat sześćdziesiątych. Posłużą one za egzemplifikacje trybów weird oraz eerie, obietnic przyszłości pokapitalistycznej, które nie znalazły politycznej reprezentacji i osunęły się na poziom widma niezrealizowanej przyszłości modernizmu. W końcowej części artykułu spróbuję pokazać jak zagadnienie komunizmu kwasowego wiąże się z kwestią pragnienia postkapitalistycznego oraz z Fisherowską wersją futuro-akceleracjonizmu.
Acoustic Detector of Road Vehicles Based on Sound Intensity
- Grzegorz Szwoch
- Józef Kotus
2021 Pełny tekst SENSORS
A method of detecting and counting road vehicles using an acoustic sensor placed by the road is presented. The sensor measures sound intensity in two directions: parallel and perpendicular to the road. The sound intensity analysis performs acoustic event detection. A normalized position of the sound source is tracked and used to determine if the detected event is related to a moving vehicle and to establish the direction of movement. The algorithm was tested on a continuous 24-h recording made in real-world conditions. The overall results were: recall 0.95, precision 0.95, F-score 0.95. In the analysis of one-hour slots, the worst results obtained in dense traffic were: recall 0.9, precision 0.93, F-score 0.91. The proposed method is intended for application in a network of traffic monitoring sensors, such as a smart city system. Its advantages include using a small, low cost and passive sensor, low algorithm complexity, and satisfactory detection accuracy
Acoustic Sensing Analytics Applied to Speech in Reverberation Conditions
- Piotr Odya
- Józef Kotus
- Adam Kurowski
- Bożena Kostek
2021 Pełny tekst SENSORS
The paper aims to discuss a case study of sensing analytics and technology in acoustics when applied to reverberation conditions. Reverberation is one of the issues that makes speech in indoor spaces challenging to understand. This problem is particularly critical in large spaces with few absorbing or diffusing surfaces. One of the natural remedies to improve speech intelligibility in such conditions may be achieved through speaking slowly. It is possible to use algorithms that reduce the rate of speech (RoS) in real time. Therefore, the study aims to find recommended values of RoS in the context of STI (speech transmission index) in different acoustic environments. In the experiments, speech intelligibility for six impulse responses recorded in spaces with different STIs is investigated using a sentence test (for the Polish language). Fifteen subjects with normal hearing participated in these tests. The results of the analytical analysis enabled us to propose a curve specifying the maximum RoS values translating into understandable speech under given acoustic conditions. This curve can be used in speech processing control technology as well as compressive reverse acoustic sensing.
Action Plan “Women”
- Marzena Grzesiak
- Anita Richert-Kaźmierska
2021
Employees of a company are not the first alternative coming into one’s mind when searching suitable successor for the business. Those belonging to the company’s management group are in many cases asked for their interest in being a successor, but the lower level an employee is in the hierarchy, the less probably he/she is inquired of his/her interest in becoming a successor. The contemporary entrepreneur may have had conflicts with employees, he may underestimate their capacities or there might be suspects concerning their loyalty to the company. On the other hand, it can be also difficult for an employee to express his interests in acquiring the business. The social pressure against the colleague wishing to start a trip across the classes, i.e. from worker to capitalist, may be, epending to the country, area and culture, crushing. An employee’s own family may be very critic against plans like that too. Nevertheless, many aspects are in favor of employees as considerable successors. In this chapter, these will be further discussed.
Activation of endogenous regenerative potential in mammals using epigenetic inhibitor zebularine
- Piotr Sass
2021
In this work, a nucleoside inhibitor of DNA methyltransferases, zebularine, was investigated as a wound healing and regeneration promoting agent. It was found that a high dose of intraperitoneally delivered zebularine improved punch wound closure in the ear pinna in mice. Both cytidine, a zebularine analogue, and uridine, a zebularine metabolite, did not promote ear pinna hole closure. The effect of zebularine on ear pinna healing was dose-dependent. Further experiments showed that zebularine administration should be carried out shortly after the injury as several-day delays resulted in decreased effectiveness. Transcriptional activity of genes related to stem cell development and maintenance and cell cycle progression regulation was evaluated in the ear pinna tissues regenerating in response to zebularine treatment. Two of the crucial cell pluripotency genes, Nanog and Sox2, were remarkably induced in zebularine-treated ear pinnae on day 7 after injury. Also, the Cdkn2a gene encoding an inhibitor of a cell cycle progression was significantly upregulated in zebularine-treated pinnae. These findings suggest that zebularine enhances wound repair by activating genes that are silenced in adult tissues in normal conditions. Preliminary trials to translate the zebularine regenerative therapy to other injury models were performed. Topically delivered zebularine moderately accelerated dorsal skin wound healing in mice and positively affected the tissue architecture in the restored skin. This observation suggests that zebularine might drive skin repair into a regenerative response. Besides, a pilot experiment was carried out to evaluate the zebularine treatment effect on sciatic nerve healing following a crush-injury. To my knowledge, the zebularine-induced ear pinna regeneration is the first example of regenerating a complex tissue by use of an epigenetic inhibitor in vivo. These findings indicate that epigenetic modulation might become an essential step in regenerative therapies.
Active Annotation in Evaluating the Credibility of Web-Based Medical Information: Guidelines for Creating Training Data Sets for Machine Learning
- Aleksandra Nabożny
- Bartłomiej Balcerzak
- Adam Wierzbicki
- Mikołaj Morzy
- Małgorzata Chlabicz
2021 Pełny tekst JMIR Medical Informatics
Methods Results Discussion References Abbreviations Copyright Abstract Background: The spread of false medical information on the web is rapidly accelerating. Establishing the credibility of web-based medical information has become a pressing necessity. Machine learning offers a solution that, when properly deployed, can be an effective tool in fighting medical misinformation on the web. Objective: The aim of this study is to present a comprehensive framework for designing and curating machine learning training data sets for web-based medical information credibility assessment. We show how to construct the annotation process. Our main objective is to support researchers from the medical and computer science communities. We offer guidelines on the preparation of data sets for machine learning models that can fight medical misinformation. Methods: We begin by providing the annotation protocol for medical experts involved in medical sentence credibility evaluation. The protocol is based on a qualitative study of our experimental data. To address the problem of insufficient initial labels, we propose a preprocessing pipeline for the batch of sentences to be assessed. It consists of representation learning, clustering, and reranking. We call this process active annotation. Results: We collected more than 10,000 annotations of statements related to selected medical subjects (psychiatry, cholesterol, autism, antibiotics, vaccines, steroids, birth methods, and food allergy testing) for less than US $7000 by employing 9 highly qualified annotators (certified medical professionals), and we release this data set to the general public. We developed an active annotation framework for more efficient annotation of noncredible medical statements. The application of qualitative analysis resulted in a better annotation protocol for our future efforts in data set creation. Conclusions: The results of the qualitative analysis support our claims of the efficacy of the presented method.
Active Suppression of Nonstationary Narrowband Acoustic Disturbances
- Maciej Niedźwiecki
- Michał Meller
2021 Pełny tekst
In this chapter, a new approach to active narrowband noise control is presented. Narrowband acoustic noise may be generated, among others, by rotating parts of electro-mechanical devices, such as motors, turbines, compressors, or fans. Active noise control involves the generation of “antinoise”, i.e., the generation of a sound that has the same amplitude, but the opposite phase, as the unwanted noise, which causes them to interfere destructively, rather than constructively. In the range of low frequencies (below 1 kHz), the active approach is more effective than passive methods that employ dampers, barriers, absorbers, and other forms of acoustic isolation.
Ad hoc prayer spaces – a challenge of the architecture of the 21th century
- Agnieszka Szuta
2021 Pełny tekst Studia oecumenica
This article considers the problem of arranging places of prayer in the vicinity of universities and dormitories. A survey was conducted among students living in the dormitories of the Gdańsk University of Technology in order to determine if there was a real need to arrange such spaces. Subsequently, examples of temporary chapels designed in the last 20 years in Western European countries were presented. Analysis of the selected facilities and discussions based on literature studies made it possible to draw conclusions on the possibilities and ways of shaping temporary prayer spaces in the often historic and crowded university environments. The research included in situ investigations.
Adaptation of Architectural Heritage in the Era of Climate Change. Torpedo Platform in Babie Doły
- Agnieszka Nyka
- Jakub Szczepański
2021
The paper provides original insight into the problem of adap¬tation of historical objects in the era of climate change. It conveys a message that in times of climate crisis, there is a need to go beyond an anthropocentric perspective, specifically, that architec¬ture should not only be designed for people, but should also provide a framework for nature. As well as shaping spaces for human beings, the architect is also responsible for creating conditions for ecosys¬tem recovery and species protection. To illustrate this problem and as a critical reflection, the stunning example of the architecture of destruction was chosen – that is, the torpedo platform in Babie Doły built during WWII on the waters of Gdańsk Bay. The present¬ed project is a discussion on how to change this place of destruction into a place of both social and ecological regeneration, a laboratory that demonstrates, from a broader perspective, the opportunity of renaturalisation of urban areas through architecture.
Adaptive Method for Modeling of Temporal Dependencies between Fields of Vision in Multi-Camera Surveillance Systems
- Karol Lisowski
- Andrzej Czyżewski
2021 Pełny tekst Electronics
A method of modeling the time of object transition between given pairs of cameras based on the Gaussian Mixture Model (GMM) is proposed in this article. Temporal dependencies modeling is a part of object re-identification based on the multi-camera experimental framework. The previously utilized Expectation-Maximization (EM) approach, requiring setting the number of mixtures arbitrarily as an input parameter, was extended with the algorithm that automatically adapts the model to statistical data. The probabilistic model was obtained by matching to the histogram of transition times between a particular pair of cameras. The proposed matching procedure uses a modified particle swarm optimization (mPSO). A way of using models of transition time in object re-identification is also presented. Experiments with the proposed method of modeling the transition time were carried out, and a comparison between previous and novel approach results are also presented, revealing that added swarms approximate normalized histograms very effectively. Moreover, the proposed swarm-based algorithm allows for modelling the same statistical data with a lower number of summands in GMM.
ADAPTIVE METHOD FOR THE SOLUTION OF 1D AND 2D ADVECTION-DIFFUSION EQUATIONS USED IN ENVIRONMENTAL ENGINEERING
- Romuald Szymkiewicz
- Dariusz Gąsiorowski
2021 Pełny tekst JOURNAL OF HYDROINFORMATICS
The paper concerns the numerical solution of one-dimensional (1D) and two-dimensional (2D) advection-diffusion equations. For the numerical solution of the 1D advection-diffusion equation a method, originally proposed for solution of the 1D pure advection equation, has been developed. A modified equation analysis carried out for the proposed method allowed increasing of the resulting solution accuracy and consequently, to reduce the numerical dissipation and dispersion. This is achieved by proper choice of the involved weighting parameter being a function of the Courant number and the diffusive number. The method is adaptive because for uniform grid point and for uniform flow velocity the weighting parameter takes constant value whereas, for non-uniform grid and for varying flow velocity its value varies in the region of solution. For solution of the 2D transport equation the dimensional decomposition in the form of Strang splitting technique is used. Consequently, this equation is reduced to a series of the 1D equations with regard to x and y directions which next are solved using afore mentioned method. The results of computational experiments compared with the exact solutions confirmed that the proposed approaches ensure high solution accuracy of the transport equations.