Solutions of two-point boundary value problems via phase-plane analysis

We consider period annuli (continua of periodic solutions) in equations of the type $x''+g(x)=0$ and $x''+f(x) x'^2 + g(x)= 0,$ where $g$ and $f$ are polynomials. The conditions are provided for existence of multiple nontrivial (encircling more than one critical point) period annuli. The conditions are obtained (by phase-plane analysis of period annuli) for existence of families of solutions to the Neumann boundary value problems

Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth

We prove the existence and multiplicity of subharmonic solutions for Hamiltonian systems obtained as perturbations of N planar uncoupled systems which, e.g., model some type of asymmetric oscillators. The nonlinearities are assumed to satisfy Landesman\u2013Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is carried out by the use of a generalized version of the Poincar\ue9\u2013Birkhoff Theorem. Different situations, including Lotka\u2013Volterra systems, or systems with singularities, are also illustrated

Combinatorics of Pisot Substitutions

Siirretty Doriast

Expected sensitivity of the AugerPrime Radio Detector to the masses of ultra-high-energy cosmic rays using inclined air showers

Despite enormous efforts in the last several decades, the origin of ultra-high-energy cosmic rays (UHECRs) -- their acceleration sites and acceleration mechanism(s) -- remains unidentified and is subject of active research. The progress made during that time, in particular by the Pierre Auger Observatory, established that significant advances in our understanding of the nature of UHECRs are only achieved with a better knowledge of their mass composition, i.e., through more precise measurements. To this end, the Pierre Auger Observatory is upgrading its large-aperture Surface Detector (SD) to enhance its mass sensitivity for the detection of the highest-energy cosmic rays ($E \gtrsim 4 \times 10^{19}\,$eV). As part of this effort, the AugerPrime Radio Detector (RD) will consist of over 1600 dual-polarized radio antennas mounted on top of each of the SD\u27s water-Cherenkov detector (WCD) stations. The RD will be measuring the electromagnetic radiation in the 30$\,$MHz to 80$\,$MHz frequency band produced by highly inclined air showers with zenith angles $\gtrsim\,$65$^\circ$. Thus, the RD will allow us to determine the cosmic-ray energy by measuring the shower\u27s electromagnetic component, which is largely independent of the cosmic-ray mass. In contrast, since most particles in highly-inclined air showers are absorbed in the atmosphere and do not reach the ground, the WCDs will mainly record muons from the muonic shower component, which is highly correlated to the cosmic-ray mass. The combination of that complementary information allows us to infer the cosmic-ray mass with high precision. With this work, I have laid the foundation to process, reconstruct, and analyze data measured by the RD. To develop a signal and reconstruction model for the radio detection of inclined air showers, I have conducted comprehensive studies of the nature of the radio emission from inclined air showers by utilizing numerical CoREAS simulations. In particular, I have investigated the origin of the radio emission within the extensive particle cascades and studied the correlation between the emission strength and ambient conditions. Furthermore, I have identified and characterized a refractive displacement of the radio-emission footprints at the ground, caused by the propagation of the electromagnetic radiation through the Earth\u27s atmosphere. This causes the radio emission from an 85$^\circ$ air shower to be displaced by about 1.5$\,$km and thus has essential implications for the description of the radio-emission footprint and the interpretation of the reconstructed geometry for very inclined air showers with zenith angles above 80$^\circ$. With that at hand, I have developed a signal model of the 2-dimensional lateral distribution of the radio emission in the 30$\,$MHz to 80$\,$MHz frequency band. This model enables the reconstruction of the (electromagnetic) shower energy with sparse radio-antenna arrays and an intrinsic resolution of below 5\% without taking into account instrumental uncertainties. As the electromagnetic energy can be reconstructed without any dependency on the cosmic-ray mass, this model is suitable to perform precise studies of the mass(-composition) of UHECRs, for example, with RD-SD hybrid detections of the AugerPrime Observatory. In addition, I have evaluated the possibility of improving this mass sensitivity by measuring the slant depth of the shower maximum $X_\mathrm{max}$ with a newly-proposed interferometric reconstruction technique. I have worked out, that the RD does not meet the specifications for an accurate reconstruction of $X_\mathrm{max}$, and that a time synchronization between antenna stations of $\lesssim\,$1$\,$ns and a signal multiplicity of $\gtrsim\,20$ are required to achieve accurate results. With this theoretical framework, I have thoroughly studied the expected performance of the RD to detect and reconstruct inclined air showers and its potential to determine the mass(-composition) of UHECRs with RD-SD hybrid measurements. These studies utilize Monte-Carlo-generated air showers, perform end-to-end simulations of the RD instrumental response including measured noise, and a reconstruction of all relevant air shower observables with the here-developed signal model. I have found that the RD will be fully efficient to detect inclined air showers with zenith angles above 70$^\circ$ and energies above $6.3 \times 10^{18}\,$eV. For a 10-year operation period, the RD will collect over 3900 events with energies above $10^{19}\,$eV and around 570 events for energies above $4 \times 10^{19}\,$eV. An accurate reconstruction of the shower energy with the RD is already possible for air showers measured with 5 radio antennas and zenith angles above 68$^\circ$. For current assumptions on the instrumental response of the RD, I have obtained an expected energy resolution of well below 10\% for energies above $10^{19}\,$eV and find no bias in the reconstructed electromagnetic energy for air showers induced by different primary particles. This study is concluded with an assessment of possible systematic uncertainties. By combining the RD-reconstructed (electromagnetic) energy and the SD-reconstructed number of muons, I assessed the potential discrimination between different primary particle types and to measure the average mass composition of UHECRs. The separation for proton- and iron-induced air showers with zenith angles above 70$^\circ$ and electromagnetic energies above $10^{19}\,$eV is quantified with a figure of merit $\text{FOM} \approx 1.6$. The (simulated) measurements of the mean muon number with the RD and SD were found to reproduce the injected mass compositions. Hence, RD-SD hybrid measurements carry the potential to extend such measurements currently performed with the Fluorescence Detector and SD to higher energies, and thereby, to distinguish between different astrophysical scenarios that could explain the nature of UHECRs. With the reconstruction model and mass-composition analysis developed in this work, the Pierre Auger Observatory is well-prepared for the arrival of experimental data from AugerPrime of inclined air showers

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

Fraktalna analiza neomeđenih skupova u euklidskim prostorima i Lapidusove zeta funkcije

In this thesis , we consider relative fractal drums and their corresponding Lapidus fractal zeta functions, as well as a generalization of this notions to the case of unbounded sets at infinity. Relative fractal drums themselves are a generalization of the notion of a bounded subset in an Euclidean space. Here, we continue the ongoing research into their properties and the higher-dimensional theory of their fractal zeta functions and complex dimensions which started as a collaboration between M. L. Lapidus and D. Žubrinić in 2009 with the later addition of the author of this thesis. The theory of complex dimensions is already well developed for fractal strings; that is, for fractal subsets of the real line. The complex dimensions of a relative fractal drum are defined as poles of a meromorphic continuation of its corresponding distance or tube zeta function. Complex dimensions of a relative fractal drum generalize, in a way, the notion its box (or Minkowski) dimension. More precisely, under some mild conditions, the value of the box dimension of a relative fractal drum is a pole of its corresponding fractal zeta function with maximal real part. Moreover, the residue computed at this pole is closely related to its Minkowski content. Here we derive important results which further justify the notion of ‘complex dimensions’ and connect it to fractal properties of a given relative fractal drum. More precisely, we establish fractal tube formulas for a class of relative fractal drums which express their relative tube function; that is, the Lebesgue measure of their relative δ-neighborhood for small values of δ, as a sum over the residues of their fractal zeta function. These formulas are given with or without an error term and hold pointwise or distributionally depending on the growth properties of the corresponding fractal zeta function. The importance of these formulas is that they show how the complex dimensions are related to the asymptotic development of the relative tube function of a given relative fractal drum. As an application we derive a Minkowski measurability criterion for a large class of relative fractal drums. Furthermore, we also show that the complex dimensions of a relative fractal drum are, as expected, invariant to the dimension of the ambient space. We introduce a further generalization of the theory of complex dimensions to the context of unbounded sets at infinity which can be used as a new approach of applying fractal analysis to unbounded subsets in Euclidean spaces. This is done for unbounded sets of finite Lebesgue measure by introducing the notions of Minkowski content at infinity and Minkowski (or box) dimension at infinity which describe their fractal properties. Furthermore, we proceed by introducing an appropriate Lapidus (or distance) zeta function at infinity and show that it is well connected with the fractal properties of unbounded sets. We proceed by constructing interesting examples of quasiperiodic sets at infinity with arbitrary number (even infinite) of quasiperiods that exhibit complex fractal behavior. We also address the natural question which arises when dealing with unbounded sets and their fractal properties; that is, establish results about the fractal properties of their images under the one-point compactification and under the geometric inversion. Furthermore, we also investigate fractal properties of unbounded sets of infinite Lebesgue measure by introducing notions of the parametric φ-shell Minkowski content at infinity and the corresponding parametric φ-shell Minkowski (or box) dimension at infinity and we establish results connecting these notions with the distance zeta function at infinity. Finally we demonstrate how fractal analysis of unbounded sets via the geometric inversion may be applied to investigate bifurcations of dynamical systems occurring at infinity.U ovoj disertaciji bavimo se relativnim fraktalnim bubnjevima i njihovim fraktalnim zeta funkcijama Lapidusovog tipa, kao i generalizacijama ovih pojmova za slučaj neomeđenih skupova u beskonačnosti. Relativni fraktalni bubnjevi su sami po sebi generalizacija pojma omeđenog skupa u Euklidskom prostoru. Ovdje nastavljamo istraživanje njihovih svojstava i višedimenzionalne teorije njihovih fraktalnih zeta funkcija te pripadajućih kompleksnih dimenzija koje je započeto suradnjom M. L. Lapidusa i D. Žubrinića 2009. godine a kojoj se autor disertacije pridružio nešto kasnije. Teorija kompleksnih dimenzija već je vrlo dobro razvijena za slučaj fraktalnih struna, odnosno, fraktalnih podskupova realnog pravca. Kompleksne dimenzije relativnog fraktalnog bubnja definirane su kao polovi meromorfnog proširenja pripadajuće razdaljinske ili cijevne zeta funkcije. Na određeni način kompleksne dimenzije relativnog fraktalnog bubnja generaliziraju pojam njegove box dimenzije (ili dimenzije Minkowskog). Preciznije, uz neke blage uvjete, vrijednost box dimenzije relativnog fraktalnog bubnja jest pol njegove pripadajuće fraktalne zeta funkcije s maksimalnom vrijednošću realnog dijela. Štoviše, reziduum u tom polu usko je povezan sa sadržajem Minkowskog danog relativnog fraktalnog bubnja. U ovoj radnji izvodimo važne rezultate koji donose daljnje opravdanje pojma ‘kompleksnih dimenzija’ i povezuju ga s fraktalnim svojstvima danog relativnog fraktalnog bubnja. Preciznije, kao rezultat dobivamo fraktalne cijevne formule za klasu relativnih fraktalnih bubnjeva koje izražavaju njihovu relativnu cijevnu funkciju, odnosno, Lebesgueovu mjeru njihove relativne δ-okoline za male vrijednosti δ, kao sumu po reziduumima njihove fraktalne zeta funkcije. Te formule su dane s greškom ili bez greške i vrijede po točkama ili distribucijski ovisno svojstvima rasta pripadajuće fraktalne zeta funkcije. Važnost ovih formula je u tome što pokazuju kako su kompleksne dimenzije povezane s asimptotikom relativne cijevne funkcije danog relativnog fraktalnog bubnja. Kao primjenu izvodimo kriterij za Minkowskivljevu izmjerivost velike klase relativnih fraktalnih bubnjeva. Nadalje, očekivano, pokazujemo da su kompleksne dimenzije danog relativnog fraktalnog bubnja invarijantne u odnosu na dimenziju ambijentnog prostora. U nastavku radnje uvodimo generalizaciju teorije kompleksnih dimenzija u kontekstu neomeđenih skupova u beskonačnosti koja može poslužiti kao novi pristup primjeni fraktalne analize na neomeđene skupove u Euklidskim prostorima. U slučaju neomeđenih skup ova konačne Lebesgueove mjere, generalizaciju provodimo uvođenjem pojmova sadržaja Minkowskog u beskonačnosti i box dimenzije u beskonačnosti (ili dimenzije Minkowskog u beskonačnosti) koji opisuju njihova fraktalna svojstva. Nadalje, uvodimo i pripadajuću Lapidusovu (ili razdaljinsku) zeta funkciju u beskonačnosti te pokazujemo da je dobro povezana s fraktalnim svojstvima neomeđenih skupova. Nastavljamo s konstrukcijom zanimljivih primjera kvaziperiodičkih skupova u beskonačnosti s proizvoljnim brojem (moguće i beskonačnim) kvaziperioda koji posjeduju složena fraktalna svojstva. Također se bavimo i prirodnim pitanjem koje se postavlja prilikom istraživanja neomeđenih skupova i njihovih fraktalnih svojstava, u vidu pronalaženja rezultata koji ih povezuju s fraktalnim svojstvima njihovih slika po jednotočkovnoj kompaktifikaciji i po geometrijskoj inverziji. Nadalje, također istražujemo i fraktalna svojstva neomeđenih skupova beskonačne Lebesgueove mjere uvođenjem pojmova parametarskog φ-omotačkog sadržaja Minkowskog u beskonačnosti i pripadajuće parametarske φ-omotačke dimenzije Minkowskog u beskonačnosti (ili φ-omotačke box dimenzije u beskonačnosti) te izvodimo rezultate koji povezuju ove pojmove s razdaljinskom zeta funkcijom u beskonačnosti. Naposljetku, demonstriramo kako se fraktalna analiza neomeđenih skupova preko geometrijske inverzije može primijeniti u istraživanju bifurkacija dinamičkih sustava koje se događaju u beskonačnosti