arXiv subjects

Marcin Magdziarz

Publications and source records attributed to Marcin Magdziarz.

At least 19 recordsLinked to original sources

arXiv2026

Optimizing the Geometry of an L-Shaped Building to Enhance Energy Efficiency and Sustainability

The geometric form of a building strongly influences its material use, heat losses, and energy efficiency. This paper presents an analytical optimization of L-shaped residential buildings aimed at minimizing the external surface area for a prescribed volume. Both symmetric and asymmetric configurations are examined under realistic design constraints, including fixed or bounded wing aspect ratios and fixed building height. Using explicit optimization methods and Karush-Kuhn-Tucker conditions, closed-form expressions for the optimal geometric parameters and minimal envelope area are derived. The results show that unconstrained optimization leads to degenerate cuboid shapes, highlighting the importance of geometric constraints to preserve the L-shaped form. The obtained results provide practical design guidelines for architects and engineers, supporting informed early stage decisions that balance functional requirements, regulatory constraints, architectural intent, and energy performance. Case studies of existing houses demonstrate that the proposed approach can reduce external surface area or confirm near-optimality of practical designs, supporting energy-efficient early-stage architectural decisions.

math.OC↗

arXiv2026

From geometry to sustainability: Optimal shapes of hip roof houses

In this paper, we develop a rigorous mathematical framework for the optimization of hip roof house geometry, with the primary goal of minimizing the external surface of the building envelope for a given set of design constraints. Five optimization scenarios are systematically analyzed: fixed volume, fixed footprint ratio, fixed slenderness ratio, fixed floor area, and constrained height. For each case, explicit formulas for the optimal dimensions are derived, offering architects and engineers practical guidelines for improving material efficiency, reducing construction costs, and enhancing energy performance. To illustrate the practical relevance of the theoretical results, case studies of real-world hip roof houses are presented, revealing both inefficiencies in common practice and near-optimal examples. Furthermore, a freely available software application has been developed to support designers in applying the optimization methods directly to architectural projects. The findings confirm that square-based footprints combined with balanced slenderness ratios yield the most efficient forms, while deviations toward elongated or flattened proportions significantly increase energy and material demands. This work demonstrates how mathematical modeling and architectural design can be integrated to support sustainable architecture, providing both theoretical insight and practical tools for shaping energy-efficient, cost-effective, and aesthetically coherent residential buildings.

math.OC↗

arXiv2026

Designing sustainable barn-type houses: Optimal shapes for minimal envelope and energy use

Barn-type houses have become one of the most popular single-family housing typologies in Poland and across Europe due to their simplicity, functionality, and potential for energy efficiency. Despite their widespread use, systematic methods for optimizing their geometry in terms of envelope area and energy performance remain limited. This paper develops a rigorous mathematical framework for determining the optimal proportions of barn-type houses with respect to minimizing the external surface area while satisfying constraints of either fixed volume or fixed floor area. Closed-form solutions for the optimal width, length, and height are derived as explicit functions of the roof slope, together with formulas for the minimal achievable surface. A recently introduced dimensionless compactness measure is also calculated, allowing quantitative assessment of how far a given design deviates from the theoretical optimum. The methodology is applied to case studies of three existing houses, showing that while some designs deviate significantly from optimal compactness, others already closely approximate it. The results confirm that theoretical optimization can lead to meaningful reductions in construction costs and energy demand. To support practical implementation, two original freely available software tools were developed, enabling architects and engineers to perform optimization analyses.

math.OC↗

arXiv2025

Arcsine laws for Brownian motion with Poissonian resetting

We analyze the equivalents of the celebrated arcsine laws for Brownian motion undergoing Poissonian resetting. We obtain closed-form formulae for the probability density functions of the corresponding random variables in the cases of the first and second arcsine law. Furthermore, we obtain numerical results for the third law.

math.PR↗

arXiv2023

Stochastic representation of processes with resetting

In this paper we introduce a general stochastic representation for an important class of processes with resetting. It allows to describe any stochastic process intermittently terminated and restarted from a predefined random or non-random point. Our approach is based on stochastic differential equations called jump-diffusion models. It allows to analyze processes with resetting both, analytically and using Monte Carlo simulation methods. To depict the strength of our approach, we derive a number of fundamental properties of Brownian motion with Poissonian resetting, such as: the Itô lemma, the moment-generating function, the characteristic function, the explicit form of the probability density function, moments of all orders, various forms of the Fokker-Planck equation, infinitesimal generator of the process and its adjoint operator. Additionally, we extend the above results to the case of time-nonhomogeneous Poissonian resetting. This way we build a general framework for the analysis of any stochastic process with intermittent random resetting.

math.PR↗

arXiv2022

About subordinated generalizations of 3 classical models of option pricing

In this paper, we investigate the relation between Bachelier and Black-Scholes models driven by the infinitely divisible inverse subordinators. Such models, in contrast to their classical equivalents, can be used in markets where periods of stagnation are observed. We introduce the subordinated Cox-Ross-Rubinstein model and prove that the price of the underlying in that model converges in distribution and in Skorokhod space to the price of underlying in the subordinated Black-Scholes model defined in [31]. Motivated by this fact we price the selected option contracts using the binomial trees. The results are compared to other numerical methods.

math.NA↗

arXiv2022

A tempered subdiffusive Black-Scholes model

In this paper, we focus on the tempered subdiffusive Black-Scholes model. The main part of our work consists of the finite difference method as a numerical approach to the option pricing in the considered model. We derive the governing fractional differential equation and the related weighted numerical scheme. The proposed method has the $2-α$ order of accuracy with respect to time, where $α\in(0,1)$ is the subdiffusion parameter, and $2$ with respect to space. Furthermore, we provide the stability and convergence analysis. Finally, we present some numerical results.

math.NA↗

arXiv2020

A computational weighted finite difference method for American and barrier options in subdiffusive Black-Scholes model

Subdiffusion is a well established phenomenon in physics. In this paper we apply the subdiffusive dynamics to analyze financial markets. We focus on the financial aspect of time fractional diffusion model with moving boundary i.e. American and barrier option pricing in the subdiffusive Black-Scholes (B-S) model. Two computational methods for valuing American options in the considered model are proposed - the weighted finite difference (FD) and the Longstaff-Schwartz method. In the article it is also shown how to valuate numerically wide range of barrier options using the FD approach.

q-fin.CP↗

arXiv2020

A weighted finite difference method for subdiffusive Black Scholes Model

In this paper we focus on the subdiffusive Black Scholes model. The main part of our work consists of the finite difference method as a numerical approach to the option pricing in the considered model. We derive the governing fractional differential equation and the related weighted numerical scheme being a generalization of the classical Crank-Nicolson scheme. The proposed method has $2-α$ order of accuracy with respect to time where $α\in(0,1)$ is the subdiffusion parameter, and $2$ with respect to space. Further, we provide the stability and convergence analysis. Finally, we present some numerical results.

cs.CE↗

arXiv2019

First passage properties of asymmetric Lévy flights

Lévy Flights are paradigmatic generalised random walk processes, in which the independent stationary increments---the "jump lengths"---are drawn from an $α$-stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of Lévy Flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering Lévy Flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of Lévy Flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index $α$ and the skewness (asymmetry) parameter $β$. The other approach is based on the stochastic Langevin equation with $α$-stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times.

cond-mat.stat-mech↗

arXiv2019

Codifference can detect ergodicity breaking and non-Gaussianity

We show that the codifference is a useful tool in studying the ergodicity breaking and non-Gaussianity properties of stochastic time series. While the codifference is a measure of dependence that was previously studied mainly in the context of stable processes, we here extend its range of applicability to random-parameter and diffusing-diffusivity models which are important in contemporary physics, biology and financial engineering. We prove that the codifference detects forms of dependence and ergodicity breaking which are not visible from analysing the covariance and correlation functions. We also discuss a related measure of dispersion, which is a non-linear analogue of the mean squared displacement.

cond-mat.stat-mech↗

arXiv2018

Superstatistical generalised Langevin equation: non-Gaussian viscoelastic anomalous diffusion

Recent advances in single particle tracking and supercomputing techniques demonstrate the emergence of normal or anomalous, viscoelastic diffusion in conjunction with non-Gaussian distributions in soft, biological, and active matter systems. We here formulate a stochastic model based on a generalised Langevin equation in which non-Gaussian shapes of the probability density function and normal or anomalous diffusion have a common origin, namely a random parametrisation of the stochastic force. We perform a detailed analytical analysis demonstrating how various types of parameter distributions for the memory kernel result in the exponential, power law, or power-log law tails of the memory functions. The studied system is also shown to exhibit a further unusual property: the velocity has a Gaussian one point probability density but non-Gaussian joint distributions. This behaviour is reflected in relaxation from Gaussian to non-Gaussian distribution observed for the position variable. We show that our theoretical results are in excellent agreement with Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv2017

Limit theorems for continuous time random walks with continuous paths

The continuous time random walks (CTRWs) are typically defned in the way that their trajectories are discontinuous step fuctions. This may be a unwellcome feature from the point of view of application of theese processes to model certain physical phenomena. In this article we propose alternative definition of continuous time random walks with continuous trajectories. We also give the functional limit theorem for sequence of such random walks. This result requires the use of strong Skorohod M1 topology instead of Skorohod J1 topology, which is usually used in limit theorems for ordinary CTRW processes.

math.PR↗

arXiv2016

Method of calculating densities for isotropic Lévy Walks

We provide explicit formulas for asymptotic densities of $d$-dimensional isotropic Lévy walks, when $d>1$. The densities of multidimensional undershooting and overshooting Lévy walks are presented as well. Interestingly, when the number of dimensions is odd the densities of all these Lévy walks are given by elementary functions. When $d$ is even, we can express the densities as fractional derivatives of hypergeometric functions, which makes an efficient numerical evaluation possible.

math.PR↗

arXiv2015

Explicit Densities of Multidimensional Lévy Walks

We provide explicit formulas for asymptotic densities of the 2- and 3-dimensional ballistic Lévy walks. It turns out that in the 3D case the densities are given by elementary functions. The densities of the 2D Lévy walks are expressed in terms of hypergeometric functions and the right-side Riemann-Liouville fractional derivative which allows to efficiently evaluate them numerically. The theoretical results agree with Monte-Carlo simulations. The obtained functions solve certain differential equations with the fractional material derivative.

cond-mat.stat-mech↗

arXiv2015

Comment on "Fokker-Planck equations for nonlinear dynamical systems driven by non-Gaussian Lévy processes" [J. Math. Phys. 53, 072701 (2012)]

In an article [J. Math. Phys. 53, 072701 (2012)] X. Sun and J. Duan presented Fokker-Planck equations for nonlinear stochastic differential equations with non-Gaussian Lévy processes. In this comment we show a serious drawback in the derivation of their main result. In the proof of Theorem 1 in the aforementioned paper, a false assumption that each infinitely differentiable function with compact support is equal to its Taylor series, is used. We prove that although the derivation is incorrect, the result remains valid only if we add certain additional assumptions.

math-ph↗

arXiv2015

Fractional diffusion equation with distributed-order material derivative. Stochastic foundations

In this paper we present stochastic foundations of fractional dynamics driven by fractional material derivative of distributed order-type. Before stating our main result we present the stochastic scenario which underlies the dynamics given by fractional material derivative. Then we introduce a Levy walk process of distributed-order type to establish our main result, which is the scaling limit of the considered process. It appears that the probability density function of the scaling limit process fulfills, in a weak sense, the fractional diffusion equation with material derivative of distributed-order type.

math.PR↗

arXiv2015

Stochastic representation of fractional subdiffusion equation. The case of infinitely divisible waiting times, Levy noise and space-time-dependent coefficients

In this paper we analyze fractional Fokker-Planck equation describing subdiffusion in the general infinitely divisible (ID) setting. We show that in the case of space-time-dependent drift and diffusion and time-dependent jump coefficient, the corresponding stochastic process can be obtained by subordinating two-dimensional system of Langevin equations driven by appropriate Brownian and Levy noises. Our result solves the problem of stochastic representation of subdiffusive Fokker-Planck dynamics in full generality.

math.PR↗