arXiv subjects

Piotr Nowakowski

Publications and source records attributed to Piotr Nowakowski.

At least 37 records · Page 2Linked to original sources

arXiv2023

Conditions for the difference set of a central Cantor set to be a Cantorval. Part II

Let C(a) be the central Cantor set generated by a sequence a with terms in (0,1). It is known that the difference set C(a)-C(a) of C(a) can has one of three possible forms: a finite union of closed intervals, a Cantor set, or a Cantorval. In the previous paper there was proved a sufficient condition for the sequence a which implies that C(a) - C(a) is a Cantorval. In this paper we give different conditions for a sequence a which guarantee the same assertion. We also prove a corollary, which provides infinitely many new examples of Cantorvals.

math.CA↗

arXiv2023

Serverless Approach to Sensitivity Analysis of Computational Models

Digital twins are virtual representations of physical objects or systems used for the purpose of analysis, most often via computer simulations, in many engineering and scientific disciplines. Recently, this approach has been introduced to computational medicine, within the concept of Digital Twin in Healthcare (DTH). Such research requires verification and validation of its models, as well as the corresponding sensitivity analysis and uncertainty quantification (VVUQ). From the computing perspective, VVUQ is a computationally intensive process, as it requires numerous runs with variations of input parameters. Researchers often use high-performance computing (HPC) solutions to run VVUQ studies where the number of parameter combinations can easily reach tens of thousands. However, there is a viable alternative to HPC for a substantial subset of computational models - serverless computing. In this paper we hypothesize that using the serverless computing model can be a practical and efficient approach to selected cases of running VVUQ calculations. We show this on the example of the EasyVVUQ library, which we extend by providing support for many serverless services. The resulting library - CloudVVUQ - is evaluated using two real-world applications from the computational medicine domain adapted for serverless execution. Our experiments demonstrate the scalability of the proposed approach.

cs.DC↗

arXiv2023

Characterization of the algebraic difference of special affine Cantor sets

We investigate some self-similar Cantor sets $C(l,r,p)$, which we call S-Cantor sets, generated by numbers $l,r,p \in \mathbb{N}$, $l+r<p$. We give a full characterization of the set $C(l_1,r_1,p)-C(l_2,r_2,p)$ which can take one of the form: the interval $[-1,1]$, a Cantor set, an L-Cantorval, an R-Cantorval or an M-Cantorval. As corollaries we give examples of Cantor sets and Cantorvals, which can be easily described using some positional numeral systems.

math.CA↗

arXiv2023

When the algebraic difference of two central Cantor sets is an interval?

Let $C(a ),C(b)\subset \lbrack 0,1]$ be the central Cantor sets generated by sequences $ a,b \in (0,1)^{\mathbb{N}}$. The first main result of the paper gives a necessary and a sufficient condition for sequences $a$ and $b$ which inform when $C(a )-C(b)$ is equal to $[-1,1]$ or is a finite union of closed intervals. One of the corollaries following from this results shows that the product of thicknesses of two central Cantor sets which algebraic difference is an interval may be arbitrarily small. We also show that there are sets $C(a)$ and $C(b)$ with the Hausdorff dimension equal to $0$ such that their algebraic difference is an interval. Finally, we give a full characterization of the case, when $C(a )-C(b)$ is equal to $[-1,1]$ or is a finite union of closed intervals.

math.CA↗

arXiv2022

FieldView: An interactive software tool for exploration of three-dimensional vector fields

Vector fields are one of the fundamental parts of mathematics which are key for modelling many physics phenomena such as electromagnetic fields or fluid and gas flows. Fields found in nature often exhibit complex structures which can be difficult to analyze in full detail without a graphical representation. In this paper we introduce FieldView -- an application targeted at academics and industry professionals wanting to visualize vector fields using various rendering techniques. FieldView uses a novel way of evaluation of the field and construction of the displayed elements directly on the GPU which is based on the recently introduced mesh shader graphics pipeline by NVIDIA. Research done for FieldView development has found practical application in the form of magnetic field visualisation integrated in the data processing framework developed by ALICE experiment at CERN.

physics.ed-ph↗

arXiv2022

How large is the space of almost convergent sequences?

We consider the subspaces $c$, $\widehat{c}$, $S$ of $\ell^\infty$, where $\widehat{c}$ consists of almost convergent sequences, and $S$ consists of sequences whose arithmetic means of consecutive terms are convergent. We know that $c\subset \widehat{c} \subset S$. We examine the largeness of $c$ in $\widehat{c}$, $\widehat{c}$ in $S$ and $S$ in $\ell^\infty$. We will do it from the viewpoints of porosity, algebrability and measure.

math.FA↗

arXiv2022

Protein-lipid domains in heterogeneous membranes beyond spontaneous curvature effects

We study a model of a lipid bilayer membrane described by two order parameters: the chemical composition described using the Gaussian model and the spatial configuration described with the elastic deformation model of a membrane with a finite thickness, or equivalently, for an adherent membrane. We assume and explain on physical grounds the linear coupling between the two order parameters. Using the exact solution, we calculate the correlation functions and order parameters profiles. We also study the domains that form around inclusions on the membrane. We propose and compare six distinct ways to quantify the size of such domains. Despite of its simplicity, the model has many interesting features like Fisher-Widom line or two distinct critical regions.

cond-mat.soft↗

arXiv2022

Homogeneity of abstract linear spaces

We discuss homogeneity and universality issues in the theory of abstract linear spaces, namely, structures with points and lines satisfying natural axioms, as in Euclidean or projective geometry. We show that the two smallest projective planes (including the Fano plane) are homogeneous and, assuming the continuum hypothesis, there exists a universal projective plane of cardinality $\aleph_1$ that is homogeneous with respect to its countable and finite projective subplanes. We also show that the existence of a generic countable linear space is equivalent to an old conjecture asserting that every finite linear space embeds into a finite projective plane.

math.LO↗

arXiv2021

Distributed Simulation and Visualization of The ALICE Detector Magnetic Field

The ALICE detector at CERN uses properties of the magnetic field acting on charged particles as part of the particle tracking and identification system -- via measuring the strength of bending of charged particles in a magnetic field supplied by electromagnets. During the calibration of the detector the magnetic field was measured by the scientific team. The measurement points were then fitted using Chebyshev polynomials to create a field model. That model is used extensively in particle trajectory reconstruction, but for maximum compatibility it is handled by CPU-only software. We propose multiple approaches to the implementation of the model in a shader language, which allows it to run in a graphical GPU environment. This makes the model run more efficiently, as well as enables it to be used for field visualisation, which was not previously done for ALICE.

physics.ins-det↗

arXiv2021

Protein induced lipid demixing in homogeneous membranes

Specific lipid environments are necessary for the establishment of protein signalling platforms in membranes, yet their origin has been highly debated. We present a continuum, exactly solvable model of protein induced local demixing of lipid membranes. The coupling between a local composition and a local thickness of the membrane induces lipid domains around inclusions with hydrophobic mismatch, even for temperatures above the miscibility critical point of the membrane. The model qualitatively explains the experimentally observed formation of lipid domains induced by anchoring of reconstituted actin in flat supported lipid bilayers.

physics.bio-ph↗

arXiv2021

Cutting sets of continuous functions on the unit interval

For a function $f\colon [0,1]\to\mathbb R$, we consider the set $E(f)$ of points at which $f$ cuts the real axis. Given $f\colon [0,1]\to\mathbb R$ and a Cantor set $D\subset [0,1]$ with $\{0,1\}\subset D$, we obtain conditions equivalent to the conjunction $f\in C[0,1]$ (or $f\in C^\infty [0,1]$) and $D\subset E(f)$. This generalizes some ideas of Zabeti. We observe that, if $f$ is continuous, then $E(f)$ is a closed nowhere dense subset of $f^{-1}[\{ 0\}]$ where each $x\in \{0,1\}\cap E(f)$ is an accumulation point of $E(f)$. Our main result states that, for a closed nowhere dense set $F\subset [0,1]$ with each $x\in \{0,1\}\cap E(f)$ being an accumulation point of $F$, there exists $f\in C^\infty [0,1]$ such that $F=E(f)$.

math.CA↗

arXiv2021

Mechanical pressure and work cycle of confined active Brownian particles

We derive an analytic expression for the mechanical pressure of a generic one-dimensional model of confined active Brownian particles (ABPs) that is valid for all values of Peclet number Pe and all confining scenarios. Our model reproduces the known scaling of bulk pressure with Pe^2 while in strong confinement pressure scales with Pe. Our analytic results are very well reproduced by simulations of ABPs in 2D. We use the pressure formula to calculate both the work performed by an active engine and its efficiency. In particular, efficiency is maximized for work cycles with finite period and not in the limit of infinitely slow cycles as in thermodynamic engines.

cond-mat.soft↗

arXiv2020

On functions preserving products of certain classes of semimetric spaces

In the paper we continue the research of Borsík and Doboš on functions which allow us to introduce a metric to the product of metric spaces. In this paper we extend their scope on broader class of spaces which usually fail to satisfy the triangle inequality, albeit they tend to satisfy some weaker form of this axiom. In particular, we examine the behavior of functions preserving $b$-metric inequality. We provide analogues of the results of Borsík and Doboš, adjusted to the new, broader setting. The results we obtained are illustrated with multitude of examples. Furthermore, the connections of newly introduced families of functions with the ones already known from the literature are investigated.

math.MG↗

arXiv2020

New developments for ALICE MasterClasses and the new Particle Therapy MasterClass

International MasterClasses (IMC), an outreach activity of the International Particle Physics Outreach Group (IPPOG), has been bringing cutting-edge particle physics research to schoolchildren for over 15 years now. All four LHC experiments participate in the event, including ALICE, the experiment optimised for the study of heavy-ion collisions. Heavy-ion physics is actively contributing to IMC with new developments including experimental measurements but also applications for society, such as treatment of cancer with ions. In particular, ALICE provides three MC measurements related to the main observables used to characterize the properties of the produced Quark-Gluon Plasma. Historically, those MC measurements were developed independently, inheriting from the first one, by several ALICE groups. Since all of them are based on the ROOT EVE package, a project to integrate them into a common framework was undertaken. ALICE delivers now a single and easy-to-use application, compiled under Linux, MacOS, and, for the first time, Windows. Then, in line with current IPPOG goals to increase the global reach and scope of the IMC programme a newly developed measurement on medical applications of particle physics, the Particle Therapy MasterClass (PTMC) was introduced in the IMC2020 programme. It is a simplified version of matRad, a MATLAB-based toolkit for calculation of dose deposition in the body and allows for planning of radiotherapy using different modalities and highlighting the benefits of treatment with ions.

physics.ed-ph↗

arXiv2020

Conditions for the difference set of a central Cantor set to be a Cantorval

Let $C(λ)\subset \lbrack 0,1]$ denote the central Cantor set generated by a sequence $ λ= \left( λ_{n} \right) \in \left( 0,\frac{1}{2} \right) ^{\mathbb{N}}$. By the known trichotomy, the difference set $ C(λ)-C(λ)$ of $C(λ)$ is one of three possible sets: a finite union of closed intervals, a Cantor set, and a Cantorval. Our main result describes effective conditions for $(λ_{n})$ which guarantee that $C(λ)-C(λ)$ is a Cantorval. We show that these conditions can be expressed in several equivalent forms. Under additional assumptions, the measure of the Cantorval $C(λ)-C(λ)$ is established. We give an application of the proved theorems for the achievement sets of some fast convergent series.

math.CA↗

arXiv2020

Effective Pair Interaction of Patchy Particles in Critical Fluids

We study the critical Casimir interaction between two spherical colloids immersed in a binary liquid mixture close to its critical demixing point. The surface of each colloid prefers one species of the mixture with the exception of a circular patch of arbitrary size, where the other species is preferred. For such objects we calculate, within the Derjaguin approximation, the scaling function describing the critical Casimir potential, and we use it to derive the scaling functions for all components of the forces and torques acting on both colloids. The results are compared with available experimental data. Moreover, the general relation between the scaling function for the potential and the scaling functions for the force and the torque is derived.

cond-mat.stat-mech↗

arXiv2016

Lateral critical Casimir force in two-dimensional inhomogeneous Ising strip. Exact results

We consider two-dimensional Ising strip bounded by two planar, inhomogeneous walls. The inhomogeneity of each wall is modeled by a magnetic field acting on surface spins. It is equal to $+h_1$ except for a group of $N_1$ sites where it is equal to $-h_1$. The inhomogeneities of the upper and lower wall are shifted with respect to each other by a lateral distance $L$. Using exact diagonalization of the transfer matrix, we study both the lateral and normal critical Casimir forces as well as magnetization profiles for a wide range of temperatures and system parameters. The lateral critical Casimir force tends to reduce the shift between the inhomogeneities, and the excess normal force is attractive. Upon increasing the shift $L$ we observe, depending on the temperature, three different scenarios of breaking of the capillary bridge of negative magnetization connecting the inhomogeneities of the walls across the strip. As long as there exists a capillary bridge in the system, the magnitude of the excess total critical Casimir force is almost constant, with its direction depending on $L$. By investigating the bridge morphologies we have found a relation between the point at which the bridge breaks and the inflection point of the force. We provide a simple argument that some of the properties reported here should also hold for a whole range of different models of the strip with the same type of inhomogeneity.

cond-mat.stat-mech↗

arXiv2016

Critical Casimir forces between defects in the 2D Ising model

An exact statistical mechanical derivation is given of the critical Casimir interactions between two defects in a planar lattice-gas Ising model. Each defect is a group of nearest-neighbor spins with modified coupling constants. Such a system can be regarded as a model of a binary liquid mixture with the molecules confined to a membrane and the defects mimicking protein inclusions embedded into the membrane. As suggested by recent experiments, certain cellular membranes appear to be tuned to the proximity of a critical demixing point belonging to the two-dimensional Ising universality class. Therefore one can expect the emergence of critical Casimir forces between membrane inclusions. These forces are governed by universal scaling functions, which we derive for simple defects. We prove that the scaling law appearing at criticality is the same for all types of defects considered here.

cond-mat.stat-mech↗