arXiv subjects
Publications and source records attributed to Piotr Nowakowski.
We study purely atomic measures whose range is a central Cantor set and characterize those central Cantor sets that are the range of exactly one such measure. Next, we extend the characterization to the case of symmetric Cantor sets using a different method of proof. Finally, we make a few initial observations and remarks on recovering a measure whose range is a Cantorval.
The coupled dynamics of local fields exert a drastic influence on the light-activated self-propulsion of a Janus particle in a binary solvent under spatial confinement. In this work, we investigate this problem using numerical simulations that account for local phase separation and wetting phenomena, as well as hydrodynamic effects. We find that confining the binary solvent within a channel results in a reduction of the active particle's propulsion speed and an extension of the duration of its directed motion. Furthermore, the orientational dynamics of this self-propelled particle are not restricted to two dimensions, unlike the phenomenon known as "orientational quenching". Increasing the light intensity leads to strong fluctuations in the local fields and, consequently, in the particle's speed. In this context, the significance of key physical parameters governing the efficiency of particle motion control is elucidated.
The continuous regulation of transport properties through softness remains a longstanding challenge in active matter. Here, we show that encasing a programmable active particle within a deformable membrane naturally gives rise to intermittent stop-and-go dynamics, with ballistic motion at short times crossing over to diffusion at long times. Crucially, membrane softness acts as a single control parameter that continuously tunes persistence, intermittency, and long-time transport, linking the internal driving to the emergent locomotion of the synthetic cell. Combining experiments, simulations, and a run-and-tumble theoretical framework, we identify the minimal physical ingredients underlying this behavior and establish design principles for programmable soft active transport, opening new avenues at the interface of active matter physics and synthetic robotics.
Let $S$ be a family of sequences of positive numbers that decrease to 0, let $X$ be a metric space and $A \subset X$. $A$ is said to be $S$-dominated if, for every $s\in S$, a countable cover $\{E_n\}$ of $E$ can be found such that $diam E_n < s_n$ for all $n$. We examine the family of all $S$-dominated sets, denoted by $\mathcal{D}(S)$. In particular, we examine the connections between $\mathcal{D}(S)$ and families of sets with zero Hausdorff measure for some gauges.
Given a nonincreasing sequence of positive numbers $(a_n)$ such that the series $\sum a_n$ is convergent, by $E(a_n)$ we denote the set of all subsums of the series $\sum a_n$ and call it the achievement set of $(a_n)$. It is well known that such a set can be a finite union of closed intervals, a Cantor set or a Cantorval. We give a new condition implying that the last possibility occurs. We also show how we can use this condition to produce new achievable Cantorvals. In particular, we prove that Kakeya conditions cannot tell us more about the form of the achievement set than it was proved by Kakeya.
Let $(X,+,d)$ be an Abelian metric group and $A\subset X$. We investigate the spectre of a set $A$, defined as the set of all elements $z\in X$ such that for every $x\in A$ either $x+z \in A$ or $x-z \in A$. We consider the corresponding to this notion operator $S$ acting on the hyperspace of compact sets and examine its properties. Furthermore, we study the families of achievement sets and sets of $P$-sums in this hyperspace, as well as prove some properties of achievement sets in the plane.
Cell migration is a fundamental process underlying the survival and function of both unicellular and multicellular organisms. Crawling motility in eukaryotic cells arises from cyclic protrusion and retraction driven by the cytoskeleton, whose organization is regulated by reaction-diffusion (RD) dynamics of Rho GTPases between the cytosol and the cortex. These dynamics generate spatial membrane patterning and establish front-rear polarity through the coupling of biochemical signalling and mechanical feedback. We develop a cross-scale mean-field framework that integrates RD signalling with cytosolic and cortical hydrodynamics to capture emergent cellular locomotion. Our model reproduces diverse experimentally observed shape and motility phenotypes with small parameter changes, indicating that these behaviours correspond to self-organized limit cycles. Phase-space analysis reveals that coupling to both cytosolic flow and spatially varying surface tension is essential to recover the full spectrum of motility modes, providing a theoretical foundation for understanding amoeboid migration.
We prove that the boundary of every multigeometric Cantorval is a null set, and extend this result to a larger class of standard achievable Cantorvals. In addition, we discuss the sets of uniqueness of achievement sets and show that they always belong to the Borel class $\mathcal{G}_δ$.
We employ a procedure that enables us to calculate the excess free energies for a finite Ising cylinder with domain walls analytically. This procedure transparently covers all possible configurations of the domain walls under given boundary conditions and allows for a physical interpretation in terms of coarse-grained quantities such as surface and point tensions. The resulting integrals contain all the information about finite-size effects; we extract them by careful asymptotic analysis using the steepest descent method. To this end, we exactly determine the steepest descent path and analyse its features. For the general class of integrals, which are usually found in the study of systems with inclined domain walls, knowledge of the steepest descent path is necessary to detect possible intersections with poles of the integrand in the complex plane.
Let $\mathcal{C}\subseteq[0,1]$ be a Cantor set. In the classical $\mathcal{C}\pm\mathcal{C}$ problems, modifying the ``size'' of $\mathcal{C}$ has a magnified effect on $\mathcal{C}\pm\mathcal{C}$. However, any gain in $\mathcal{C}$ necessarily results in a loss in $\mathcal{C}^c$, and vice versa. This interplay between $\mathcal{C}$ and its complement $\mathcal{C}^c$ raises interesting questions about the delicate balance between the two, particularly in how it influences the ``size'' of $\mathcal{C}^c-\mathcal{C}$. One of our main results indicates that the Lebesgue measure of $\mathcal{C}^c-\mathcal{C}$ has a greatest lower bound of $\frac{3}{2}$.
Critical Casimir forces emerge among particles or surfaces immersed in a near-critical fluid, with the sign of the force determined by surface properties and with its strength tunable by minute temperature changes. Here, we show how such forces can be used to trap a colloidal particle and levitate it above a substrate with a bull's-eye pattern consisting of a ring with surface properties opposite to the rest of the substrate. Using the Derjaguin approximation and mean-field calculations, we find a rich behavior of spherical colloids at such a patterned surface, including sedimentation towards the ring and levitation above the ring (ring levitation) or above the bull's-eye's center (point levitation). Within the Derjaguin approximation, we calculate a levitation diagram for point levitation showing the depth of the trapping potential and the height at which the colloid levitates, both depending on the pattern properties, the colloid size, and the solution temperature. Our calculations reveal that the parameter space associated with point levitation shrinks if the system is driven away from a critical point, while, surprisingly, the trapping force becomes stronger. We discuss the application of critical Casimir levitation for sorting colloids by size and for determining the thermodynamic distance to criticality. Our results show that critical Casimir forces provide rich opportunities for controlling the behavior of colloidal particles at patterned surfaces.
In modern democracies, the outcome of elections and referendums is often remarkably tight. The repetition of these divisive events are the hallmark of a split society; to the physicist, however, it is an astonishing feat for such large collections of diverse individuals. Many sociophysics models reproduce the emergence of collective human behavior with interacting agents, which respond to their environment according to simple rules, modulated by random fluctuations. A paragon of this class is the Ising model which, when interactions are strong, predicts that order can emerge from a chaotic initial state. In contrast with many elections, however, this model favors a strong majority. Here, we introduce a new element to this classical theory, which accounts for the influence of opinion polls on the electorate. This brings about a new phase in which two groups divide the opinion equally. These political camps are spatially segregated, and the sharp boundary that separates them makes the system size-dependent, even in the limit of a large electorate. Election data show that, over the last 30 years, countries with more than about a million voters often found themselves in this state, whereas elections in smaller countries yielded more consensual results. We suggest that this transition hinges on the electorate's awareness of the general opinion.
The manipulation of microscopic objects requires precise and controllable forces and torques. Recent advances have led to the use of critical Casimir forces as a powerful tool, which can be finely tuned through the temperature of the environment and the chemical properties of the involved objects. For example, these forces have been used to self-organize ensembles of particles and to counteract stiction caused by Casimir-Liftshitz forces. However, until now, the potential of critical Casimir torques has been largely unexplored. Here, we demonstrate that critical Casimir torques can efficiently control the alignment of microscopic objects on nanopatterned substrates. We show experimentally and corroborate with theoretical calculations and Monte Carlo simulations that circular patterns on a substrate can stabilize the position and orientation of microscopic disks. By making the patterns elliptical, such microdisks can be subject to a torque which flips them upright while simultaneously allowing for more accurate control of the microdisk position. More complex patterns can selectively trap 2D-chiral particles and generate particle motion similar to non-equilibrium Brownian ratchets. These findings provide new opportunities for nanotechnological applications requiring precise positioning and orientation of microscopic objects.
We examine properties of achievement sets of series in $\mathbb{R}^2$. We show several examples of unusual sets of subsums on the plane. We prove that we can obtain any set of P-sums as a cut of an achievement set in $\mathbb{R}^2.$ We introduce a notion of the spectre of a set in an Abelian group, which generalizes the notion of the center of distances. We examine properties of the spectre and we use it, for example, to show that the Sierpi\'nski carpet is not an achievement set of any series.
The ALICE Collaboration at CERN developed a 3D visualisation tool capable of displaying a representation of collected collision data (particle trajectories, clusters and calorimeter towers) called the Event Display. The Event Display is constantly running in the ALICE Run Control Center as part of the Quality Assurance system, providing the monitoring personnel with visual cues about possible problems of both hardware and software components during periods of data gathering. In the software, particle trajectories (which are curved due to presence of magnetic field inside the detector) are generated from physical parameters of detected particles, such as electrical charge and momentum. Previously this process in the Event Display used a uniform, constant magnetic field for these calculations, which differs from the spatial variations of the real magnetic field and does not model one of the two magnets used in the detector. Recently, a detailed model of ALICE magnetic field was made available as a shader program for execution on the GPU. In this work we attempt to implement the reconstruction algorithm in a shader form as well, allowing us to combine it with the detailed model to create a full solution for rendering trajectories from collision event data directly on the GPU. This approach has several possible advantages, such as better performance and the ability to alter the magnetic field properties in real-time. This was not previously done for ALICE and as such could be used in the future to upgrade the Event Display.
In this paper we look at the topological type of algebraic sum of achievement sets. We show that there is a Cantorval such that the algebraic sum of its $k$ copies is still a Cantorval for any $k \in \mathbb{N}$. We also prove that for any $m,p \in (\mathbb{N}\setminus \{1\}) \cup \{\infty\}$, $p \geq m$, the algebraic sum of $k$ copies of a Cantor set can transit from a Cantor set to a Cantorval for $k=m$ and then to an interval for $k=p$. These two main results are based on a new characterization of sequences whose achievement sets are Cantorvals. We also define a new family of achievable Cantorvals which are not generated by multigeometric series. In the final section we discuss various decompositions of sequences related to the topological typology of achievement sets.
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.
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.