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Simonas Vainbergas, a teacher of Jewish primary schools in prewar Lithuania (1894?), was one of few his colleagues that actively collaborated with Lithuanian pedagogical journals. He published in them 11 papers in didactics of mathematics about teaching mathematics at primary schools. The said papers helped young teachers of primary schools to improve their qualification. There is a great probability that S. Vainbergas was a victim of holocaust.

One of the ways to improve the accuracy of probability density estimation is multi-mode density treating as the mixture of single-mode one. In this paper we offer to use data clustering in the first place and to estimate density in every cluster separately. To objectively compare the performance, Monte Carlo approximation is used. While using various methods to evaluate the accuracy of probability density estimations we tried to use clustered and not clustered data. In this paper we also tried to reveal the usefulness of using clustering for data generated by single-mode and multi-mode distributions.

BDI logic is widely used to describe agent based systems, since it can express a lot of different real world domains with three main operators: belief, desire and intention. There are lots of works where BDI logic is used as descriptive language, but authors do not talk about implementation issues [5,2,4,8,1]. There is even known general agent implementation scheme, which do not describe howto use formal logic for such scheme implementation [12]. The main aim of this paper is to show the bridge, which connects formal and practical parts of logic. We describe how sequent calculus can be used for implementing some parts of general scheme. In this paper, we show how flexible constraints can be implemented. Sequent calculus usage is illustrated with simple examples.

The paper describes shortly the main principles of representative examples development, provides the essence of the methodology to develop representative examples and an example of its application. The main contribution of the paper is the methodology to develop representative examples that could be used to evaluate the sufficiency of specification language for specification of software system requirements.

In this paper we present separate cases of Sturm–Liouville problem with nonlocal two-point boundary condition. The existence of positive eigenvalues and eigenfunctions is analysed. Obtained results we compare with results that are described in paper [1] and denote that in some cases they don’t coincide.

In this paper authors research the problem of traceability of assumptions in logical derivation. The essence of this task is to trace which assumptions from the available knowledge base of assumptions are necessary to derive a certain conclusion. The paper presents a new derivation procedure for propositional logic, which ensures traceability feature. For the derivable conclusion formula derivation procedure also returns the smallest set of assumptions those are enough to get derivation of the conclusion formula. Verification of the procedure were performed using authors implementation.

The article presents several observations about the examination papers of the state-level Maturity Examinations in Mathematics in the years 2014–2021 in Lithuania. Some inaccuracies were observed in the appendix of the paper titled ``Mathematical formulas'', in the wordings of the problems. Also the issue of proof problems in the examination is discussed. The purpose of these observations is to draw attention to areas for improvement in the examination system. The article presents the subjective opinion of the authors.

The technology for automatization of lecturer’sworkplace is presented in the paper, enhancing and simplifying the preparation of assignments for practical works and their inspection. The main focus of paper is placed on the inspection of students’works. The informational system created and exploited at the department of EngineeringMechanics, Vilnius Gediminas TechnicalUniversity, is described. Similar principles may be employed also for analogical systems of generating of assigments for practical works as well as inspection of them for other subjects (mathematics, informatics, etc).

The essential feature of Internet GIS is the interactivity between the users and the spatial data. Interactive GIS can offer interactive maps rather than static map images on the Web. Users can work with the maps interactively by performing conventionalGIS functions such as zoom, pan, search, etc. However, the interactivity of many InternetGIS is quite limited so far. In this paper, we propose to use AJAX to improve the interactivity of Internet GIS. AJAX design patterns that can be used to this aim are described in the paper. We present the architecture of the Internet GIS with AJAX-based user interface and integration with GIS web services.

In this paper, authors emphasise on the correctness of the transformation. Therefore, types of correctness are analysed here. As a context of transformation, the transformation of PAL constraints in SQL rules is chosen for more details. Formal rules for the transformation of PAL constraints in SQL rules are presented in this paper also. And finally, the analysis of the proposed transformation is checked.

Multiple Criteria Decision Making (MCDM) methods are effectively used in decision making tasks. The weights of criteria are an integral part of MCDM methods. The paper proposes the Bayesian approach to recalculate the weights of the criteria, when the decision-maker takes into account the opinions of other expert groups. Recalculation is relevant when the selection is individualized by the opinion of separate expert group. In this paper the distance learning course was chosen by separate group of experts, using SAW and TOPSIS methods and recounted criteria weights by Bayesian method.

In this research paper, a comprehensive analysis of particulate matter (PM10) and nitrogen dioxide (NO2) pollution concentrations in six different Lithuanian regions is presented. The analysis employs data smoothing, principal component analysis (PCA), exploratory data analysis, hypothesis testing, and time series analysis to provide a thorough examination. Functional data analysis approaches were used to find the origins and effects of these air pollutants by revealing their data patterns. The functional data analysis techniques demonstrate their effectiveness in revealing deep links within large datasets, assisting in the control of air quality problems. This research provides valuable insights into air quality challenges in Lithuanian regions. The study, aimed at comparing air quality across different regions, indicates that there are no significant differences in PM10 and NO2 between the two groups. Notably, reliable forecasts for 2023 data are attainable for PM10 in regions such as Vilnius Old Town, Vilnius Lazdynai, Šiauliai, and Klaipėda. For NO2, successful forecasting can be applied to Vilnius Old Town, Vilnius Lazdynai, and Šiauliai.

Usually, proofs of mathematical statements involve both algebraic rearrangements and logical reasoning. But there are mathematical statements whose truth is obvious at first glance when there is a diagram illustrating that proof. Although the proofs based on the drawing are not necessarily full and complete, but the drawing helps to notice facts that are then easily supported by algebra and logic. The paper presents proofs of mathematical propositions where, upon careful study of the drawing, the main idea of the proof can be seen from the drawing, and the proof itself becomes beautiful and clear.

In this paper we consider a modification of the attack on the classic RSA cryptosystem aimed at factoring the public modulus n, which is a product of three primes. To improve the performance of the modified attack we introduce additional parameters. We present the theoretical upper bound on the search range parameter and define a shifting parameter based on the empirical results. Since these changes make our attack probabilistic, we investigate the dependence of the success on the values of the newly defined parameters.

In this paper the error estimation for n-th order Filon quadrature formula is discussed.

Poisson approximation for the sum of independent random variables is investigates in this paper.

In this paper the local limit theorem for density of k-th maxima of independent identically distributed random variables is proved.

The paper reviews the visualisation problems of genetic sequences and of their analysis results as well as other related problems. A possible formulation of the problem based on similarity, complexity measure (distance) between DNA sequences is proposed. It is solved by making use of multidimensional scaling (MDS) of principal component analysis (PCA) methods.

The analysis of machine scheduling problem and possible algorithms for solving it are presented in this paper.

The analysis of heuristic algorithm for solving two dimensional cutting stock problem is presented in this paper.

This paper considered deterministic and stochastic delayed linear growth laws for the modeling of the process of growth. For the estimation of parameters the L1 distance procedure was proposed. As an illustrative experience was used a real data set from repeated measurements on permanent sample plots of pine stands in Lithuania.

Multiple criteria decision-making methods enable to choose the optimal decision, if his criterions is contradictorys. In herein paper is describe MAUT – multiattribute utility theory. This method is reasoned in theory. MAUT up for indeterminations situations, but can to use for determinations situations. In this method multiple criteria problem solvable with the help of unitary utility functions.

The positioning problem considered in this paper is an attempt to find system’s of a few points position using accelerations. Today’s industry provides with various types of accelerometers suitable for acceleration measure. That data may be used to trace object movement. Final results showed that such movement trace is possible and may be applicable. The results also showed origins for errors which according to the results, are quite high globally. The final conclusion follows: the model created must be analized further in a deeper level, and error origins must be eliminated or at least minimized by either chosing different solving or modeling methods, either by revising used ones.

This paper deals with an analysis of the electrochemical biosensors and their response to multi-component mixtures. The main task is to build a mathematical model for estimation the concentration of each mixture component from the biosensor response data. Two different types of biosensors: amperometric and potenciometric are analysed. Due to high dimensionality of biosensor output data the principal component analysis is applied. Additional multivariate analysis of variance is used to analyze the response sensitivity of each biosensor type. Finally a concentration estimation model based on ensemble of neural networks is presented.

The Baltic States equity market is a challenge for investors and financial analysts. Unfortunately strong assivity is observed in ``young'' markets, therefore any (Gaussian, α-stable etc) distribution fitting tests (Anderson–Darling, Kolmogorov–Smirnov, etc.) are poorly applicable. Improvement based on mixed distributions is proposed and its adequacy in the Baltic States market is tested. In this paper we use Koutrouvelis goodness-of-fit test and modified χ2 test.

The analysis of ACO algorithm for transmitters locating problem is presented in this paper.

The subject of this work is the analysis of hazard distribution in the network systems. In the age when transport flows in the railway and road systems, information flows in the internet channels, etc. intensify, hazard distribution in the network systems becomes a more relevant and important issue. In the paper various theoretical hazard type distributions in the network systems are analysed. Mathematical model of hazard distribution and accumulation in the nodes of the network was created.

In this paper we present estimated generalized least squares (EGLS) estimator for the coefficient vector β in the linear regression model y = βX + ε, where disturbance term can be heteroskedastic. For the heteroskedasticity of the changed segment type, using Monte-Carlo method, we investigate empirical properties of the proposed and ordinary least squares (OLS) estimators. The results show that the empirical covariance of the EGLS estimators is smaller than that of OLS estimators.

Large-scale aggregation and its inverse, disaggregation, problems are important in many fields of studies like macroeconomics, astronomy, hydrology and sociology. It was shown in Granger (1980) that a certain aggregation of random coefficient AR(1) models can lead to long memory output. Dacunha-Castelle and Oppenheim (2001) explored the topic further, answering when and if a predefined long memory process could be obtained as the result of aggregation of a specific class of individual processes. In this paper, the disaggregation scheme of Leipus et al. (2006) is briefly discussed. Then disaggregation into AR(1) is analyzed further, resulting in a theorem that helps, under corresponding assumptions, to construct a mixture density for a given aggregated by AR(1) scheme process. Finally the theorem is illustrated by FARUMA mixture densityÆs example.

In this paper DNA sequents are modelled as discrete-state Markov chains. Statistical data is presented in contingency tables form. The generalized logit model is used to test the first-order Markov property for all coding and non-coding subsequences of DNA.

The reflective study models are quite frequent and well analyzed in the scientific literature. Still their realization premises being different in many countries keep forming a scientific issue which needs a special attention, especially in learning mathematics. In this context there arises a problem question: what are the realization limitations of the traditional and reflective mathematics learning models? The research paper consists of two parts. The first part clears a problems of the realization a traditional study model. The second part clears out the possible realization limitations of the above mentioned model in Lithuanian context.

In this paper the discrete logarithm problem in matrix in finite fields is formulated, possible ways of solution are given.

In this paper enrolment and study problems in the faculty of Mathematics and informatics of Vilnius Pedagogical university are discussed.

The paper is devoted to multivariate statistical analysis of academic performance data. The phenomina of general grade inflation and the effect of the first passing of an egzam on grades of the egzam are investigated by making use of general linear (mixed) model and logistic regresion.

A hyperbolic system of first order partial differential equations with small parameter and periodical initial conditions is obtained. The uniformly valid in a long time interval asymptotical solution can be constructed using the averaging along characteristics of the system. The asymptotical estimation of the approximation is made in the paper. Some aspects of practical realization of the method are discussed in the work too.

The paper aims to present a methodology (i.e. model and method) to evaluate suitability, acceptance and use of personalised learning scenarios. High-quality learning scenarios should consist of the learning components (i.e. learning objects, learning activities, and learning environment) optimised to particular students according to their personal needs, e.g. learning styles. In the paper, optimised learning scenarios mean learning scenarios composed of the components having the highest probabilistic suitability indexes to particular students according to Felder–Silverman learning styles model. Personalised learning scenarios evaluation methodology presented in the paper is based on (1) probabilistic suitability indexes to identify learning components suitability to particular students needs according to their learning styles, and (2) Educational Technology Acceptance & Satisfaction Model (ETAS-M) based on well-known Unified Theory on Acceptance and Use of Technology (UTAUT) model.

Brodsky’s coding method for propositional logic is considered in the paper. Based on the sequent calculus, the method allows us to determine whether an arbitrary sequent is derivable in the calculus without constructing proof-search trees. The coding method, presented in the paper, can be used as a decision procedure for propositional logic.

The paper is aimed to analyse the problem of personalisation of Virtual Reality/Augmented Reality/Mixed Reality (VR/AR/MR) based learning systems. Research results are two-fold: first, the results of systematic literature review are presented, and, second, VR/AR/MR-based learning systems personalisation framework is proposed. First of all, systematic literature review on research topic was conducted in Thomson Reuters Web of Science database and applying Semantic Scholar search tool. The review revealed that strides are being made in education using VR/AR/MR, although much needs to be done. The possibilities of VR/AR/MR application in education seem to be endless and bring many advantages to students of all ages. Few are creating content that may be used for educational purposes, with most advances being made in the entertainment industry, but many understand and realise the future and importance of education applying VR/AR/MR. Many studies argue that new VR/AR/MR-based learning systems are more effective in comparison with traditional ones. Teachers and students like learning content and activities provided by VR/AR/MR technologies. On the other hand, although the concept of VR/AR/MR has already been proposed more than 20 years ago, most applications are still limited to simple visualisation of virtual objects onto spatially limited scenes, and the developed systems did not pass the barrier of demonstration prototypes. Many authors agree that personalisation of VR/AR/MR-based learning platforms should be further analysed. Original personalisation framework of VR/AR/MR-based learning systems is also presented in the paper. According to the framework, personalisation of VR/AR/MR learning systems should be based on applying learners models and intelligent technologies e.g. expert evaluation, ontologies, recommender systems, software agents etc. This pedagogically sound personalisation framework is aimed to improve learning quality and effectiveness.

Aim of multiple criteria decision-aid (MCDA) methods is to find the best alternative among the ones that are available or to rank alternatives in the order of preference. There are the following core pillars of the methods: the set of criteria and matrix with values of criteria that characterise the evaluated alternatives (decision matrix); and vector of weights that reflect relative importance of criteria. Usually, two types of criteria are used by researchers. Maximising criteria (e.g. profits) reflect a better situation whenever the larger value has been attained. While in case a criterion is minimising (e.g. costs), the better situation is reflected when its value is smaller. Such situations, when the best value of a criterion has a certain value, which differs from the maximal or the minimal, are usually not considered. This paper aims to fill this gap. Such criteria will be named as criteria with a-priori chosen optimal values. The aim of the paper is to propose appropriate types of transformation for criteria with a-priori chosen optimal values. Such transformations appear to be general and can be used with all three types of criteria.

In the paper, learning scenarios (units) quality evaluation and optimisation problems are analysed. Learning scenarios optimisation is referred here as its personalisation according to learners needs. In the paper, comparative analysis of two popular optimisation methods based on Fuzzy numbers theory and Analytic Hierarchy Process is performed, aiming to measure what method is the most suitable to evaluate the quality of personalised learning scenarios. Learning scenarios quality is referred here as its suitability to learners needs. Research results show that Fuzzy numbers theorybased methods are more suitable to evaluate the quality of personalised learning scenarios.

The paper aims to present an original probabilistic method to creating personalised learning activities and units. The method is based on evaluating learning components suitability to students needs according to their learning styles. In the paper, Felder–Silverman Learning Styles Model and Inquiry Based Learning method suitable to learn Math are used. Expert evaluation method based on Trapezoidal Fuzzy Numbers theory is applied in the research to obtain numerical values of suitability of IBL activities to students learning styles. Personalised learning units should consist of learning components (i.e. learning activities) that are optimal (i.e. the most suitable) to particular students according to their learning styles. Optimal means having the highest suitability index. In the paper, this original probabilistic method is applied to establish probabilistic suitability of IBL activities to particular students.

In Plato’s later ethics, the key challenge to one’s personal progress is unorderly thoughts and motions of soul. This disorder should be corrected by recognising and repeating the orderly thoughts and motions of cosmic gods. This paper examines the theoretical assumptions of Plato’s later ethics. A closer analysis of the Timaeus reveals a peculiar conception of mathematics, according to which numbers and their relations not only reveal facts about the reality, but also express values and ideals. By learning mathematics human beings come to know the everyday ethical life of cosmic gods and become like them.

The present paper deals with efficiency improvement of backward proof-search of sequents of propositional linear temporal logic, using a loop-type sequent calculus. The improvement is achieved by syntactic transformation of sequents into equivalent to them simpler ones. It is proved that some formulas can be removed from sequents with no impact on their derivability.

When using Spatial Correlation Rule with Distance (SCRD) the selection of the neighborhood scheme influences classification accuracy. Spatial dependency in different situations remains at various distances, so, according to this, in applications it is important to choose a suitable neighborhood scheme. In the earlier papers of the authors, the nearest neighbor scheme was used. In this paper, several different neighborhood schemes are examined by large experiment.

In this paper the remote sensed image classification example using spacial classification rule with distance (SCRD) is examined. This supervised classification method was first presented in paper [11]. This method is improved version of earlier method PBDF [4, 10, 9], during the classification it incorporates more spatial information. The advantage of this method is its ability to classify data which is corrupted by Gaussian random field and it is typical to remotely sensed images classified in this letter which are corrupted by clouds. Classification accuracy is compared with earlier method and with other commonly used supervised classification methods.

In this paper hyperbolic system of the first order gas dynamics PDE with initial and boundary conditions is studied. The aim of the paper is to construct the averaged system of differential equations in order to find the uniformly valid in a large domain asymptotical solution. The averaged system is a new object of asymptotical analysis.

This paper discusses the concept of three-dimensional (3D) fractal interpolation and the possibility to use it in modeling 3D surfaces. It is important to notice that this paper treats fractal interpolation only as a numerical tool and not as a model. The purpose of the research is to create a methodology for obtaining models for the given 3D surface and making them similar to it to a certain degree. The set of models then can be investigated as required (3D graphical representation, simulation of particular technological process, quality assessment for bonded surfaces, etc.). Measuring a particular 3D surface and making a set of models is far more cost efficient than performing the measurements many times.

In this paper supervised classification method is proposed. It is based on Bayes discriminant functions (BDF) and it deals with the problem of optimal classification for images, which are corrupted by natural phenomenon such as cloud, smoke or fog. Solving such a problem is very important when we have remotely sensed information, which very often is corrupted by clouds. For example, the remotely sensed images from the territory of Lithuania are very often corrupted by clouds. The idea of classification, using BDF with incorporated spatial dependency between the observation to be classified and the training sample is presented in earlier works of the author. The novelty of this paper is the method how to use these methods for the real situation, i.e. for the remotely sensed image which is naturally covered by clouds. Visual and numerical results are presented in this paper, which show the advantage of this method against BDF ignoring spatial dependency between training sample and observation to be classified and against the method using grey level cooccurrence matrices.

In this paper the stationary Navier–Stokes system with non-homogeneous boundary condition is studied in domain which consists of two connected layers. The extension of the boundary value, which reduces the non-homogeneous boundary problem to the homogeneous one, is constructed in this paper.

The paper deals with the evaluation of the criteria impacting distance learning course content quality. The goal of the paper is to introduce the improved AHP method with Fuzzy triangle numbers used to determine the quality of distance learning courses. In the paper, the interval and discrete methods for weights determination are compared.