Academic literature on the topic 'Lorentzian function'

The formula of the pseudo-Voigt function expressed by a weighted sum of Gaussian and Lorentzian functions is extended by adding two other types of peak functions in order to improve the accuracy when approximating the Voigt profile. The full width at half-maximum (FWHM) values and mixing parameters of the Gaussian, the Lorentzian and the other two component functions in the extended formula can be approximated by polynomials of a parameter ρ = ΓL/(ΓG+ ΓL), where ΓGand ΓLare the FWHM values of the deconvoluted Gaussian and Lorentzian functions, respectively. The maximum deviation of the extended pseudo-Voigt function from the Voigt profile is within 0.12% relative to the peak height when sixth-order polynomial expansions are used. The systematic errors of the integrated intensity ΓGand ΓL, estimated by fitting the extended formula to Voigt profiles, are typically less than 1/10 of the errors arising from the application of the original formula of the pseudo-Voigt approximation proposed by Thompsonet al.[J. Appl. Cryst.(1987),20, 79–83], while the time required for computation of the extended formula is only about 2.5 relative to the computation time required for the original formula.

Faddeeva–Voigt broadening (FVB) couples the physical characteristics of both Lorentzian and Gaussian profiles as a combined analytic function shaping the dielectric response. Accurate extraction of the Gaussian and Lorentzian broadening contents in line-shape analysis is essential for reliable optical characterization of semiconductors and dielectrics. By adding the Gaussian-broadening width to each Lorentzian width, we investigate how FVB affects critical-point (CP) analysis. We revisit a selection of earlier work based on classical Lorentz broadening in modulation spectroscopy and spectral ellipsometry. To generalize CP analysis, we derive the FVB’s analytical representation in terms of fractional derivatives of the Faddeeva function and apply the twenty-pole Martin–Donoso–Zamudio approximation for its precise and efficient computation of the FVB of model dielectric functions and derivatives. We investigate the FVB of the electroreflectance line shape of HgCdTe for three-dimensional M0 transitions and of the photoreflectance line shape of InP excitonic E0 transitions. Furthermore, we explore how FVB affects the dielectric functions of three-dimensional excitonic and two-dimensional M0 transitions vs Tanguy’s analytical two-dimensional exciton E1 and E1+Δ1 fits of GaAs to the second-order derivatives. We use the Akaike information criterion to quantitatively estimate the goodness of fit that statistically penalizes overfitting due to extraneous parameters. By consolidating both Gaussian and Lorentzian broadenings, the FVB significantly affects the CP analysis of modulation-spectroscopy line shapes and second-order derivatives of the dielectric function.

Connes' functional formula of the Riemannian distance is generalized to the Lorentzian case using the so-called Lorentzian distance, the d'Alembert operator and the causal functions of a globally-hyperbolic spacetime. As a step of the presented machinery, a proof of the almost-everywhere smoothness of the Lorentzian distance considered as a function of one of the two arguments is given. Afterwards, using a C*-algebra approach, the spacetime causal structure and the Lorentzian distance are generalized into noncommutative structures giving rise to a Lorentzian version of part of Connes' noncommutative geometry. The generalized noncommutative spacetime consists of a direct set of Hilbert spaces and a related class of C*-algebras of operators. In each algebra a convex cone made of self-adjoint elements is selected which generalizes the class of causal functions. The generalized events, called loci, are realized as the elements of the inductive limit of the spaces of the algebraic states on the C*-algebras. A partial-ordering relation between pairs of loci generalizes the causal order relation in spacetime. A generalized Lorentz distance of loci is defined by means of a class of densely-defined operators which play the role of a Lorentzian metric. Specializing back the formalism to the usual globally-hyperbolic spacetime, it is found that compactly-supported probability measures give rise to a non-pointwise extension of the concept of events.

In this note we are interested in isotropic totally real Lorentzian submanifolds of indefinite complex space forms. We show that such submanifolds are always H-umbilical warped product immersions and we determine also the warping function.

We study the problem of the cosmological constant in the context of the multiverse in Lorentzian space–time, and show that the cosmological constant will vanish in the future. This sort of argument was started by Sidney Coleman in 1989, and he argued that the Euclidean wormholes make the multiverse partition function a superposition of various values of the cosmological constant Λ, which has a sharp peak at Λ = 0. However, the implication of the Euclidean analysis to our Lorentzian space–time is unclear. With this motivation, we analyze the quantum state of the multiverse in Lorentzian space–time by the WKB method, and calculate the density matrix of our universe by tracing out the other universes. Our result predicts vanishing cosmological constant. While Coleman obtained the enhancement at Λ = 0 through the action itself, in our Lorentzian analysis the similar enhancement arises from the front factor of eiS in the universe wave function, which is in the next leading order in the WKB approximation.

From the assumption that the slow-roll parameter [Formula: see text] has a Lorentzian form as a function of the e-folds number [Formula: see text], a successful model of a quintessential inflation is obtained. The form corresponds to the vacuum energy both in the inflationary and in the dark energy epochs. The form satisfies the condition to climb from small values of [Formula: see text] to [Formula: see text] at the end of the inflationary epoch. At the late universe, [Formula: see text] becomes small again and this leads to the dark energy epoch. The observables that the models predict fits with the latest Planck data: [Formula: see text]. Naturally, a large dimensionless factor that exponentially amplifies the inflationary scale and exponentially suppresses the dark energy scale appearance, producing a sort of cosmological seesaw mechanism. We find the corresponding scalar Quintessential Inflationary potential with two flat regions — one inflationary and one as a dark energy with slow-roll behavior.

The accepted classical model for derivation of the frequency spectrum from the time-dependence of oscillator activity is considered. The exponential decay function giving rise to a Lorentzian band is modified to allow for a finite rate of growth of the oscillator, the resulting frequency spectrum is obtained, and the form normalized to unit band half-width is compared to the equivalent Lorentzian and Gaussian bands. It is shown that as the rate of oscillator growth decreases from infinity to one approximating the rate of decay, the resulting band contour changes from Lorentzian to near-Gaussian. At sufficiently fast growth-rates the band closely approximates a linear combination of Lorentzian and Gaussian.

Nesta dissertação apresentamos um método de quantização matemática e conceitualmente rigoroso para o campo escalar livre de interações. Trazemos de início alguns aspéctos importantes da Teoria de Distribuições e colocamos alguns pontos de geometria Lorentziana. O restante do trabalho é dividido em duas partes: na primeira, estudamos equações de onda em variedades Lorentzianas globalmente hiperbólicas e apresentamos o conceito de soluções fundamentais no contexto de equações locais. Em seguida, progressivamente construímos soluções fundamentais para o operador de onda a partir da distribuição de Riesz. Uma vez estabelecida uma solução para a equação de onda em uma vizinhança de um ponto da variedade, tratamos de construir uma solução global a partir da extensão do problema de Cauchy a toda a variedade, donde as soluções fundamentais dão lugar aos operadores de Green a partir da introdução de uma condição de contorno. Na última parte do trabalho, apresentamos um mínimo da Teoria de Categorias e Funtores para utilizar esse formalismo na contrução de um funtor de segunda quantização entre a categoria de variedades Lorentzianas globalmente hiperbólicas e a categoria de redes de álgebras C* satisfazendo os axiomas de Haag-Kastler. Ao fim, retomamos o caso particular do campo escalar quântico livre.<br>In this thesis we present a both mathematical and conceptually rigorous quantization method for the neutral scalar field free of interactions. Initially, we introduce some aspects of the Theory of Distributions and we establish some points of Lorentzian geometry. The rest of the work is divided in two parts: in the first one, we study wave equations on globally hyperbolic Lorentzian manifolds, hence presenting the concept of fundamental solutions within the context of locally defined wave equations. Next, we progressively construct fundamental solutions for the wave operator from the Riesz distribution. Once established a solution to the wave equation in a neighbourhood of a point of the manifold, we move forward to produce a global solution from the extension of the Cauchy problem to the whole manifold. At this stage, fundamental solutions are replaced by Green\'s operators by the imposition of appropriate boundary conditions. In the last part, we present a minimum on the Theory of Categories and Functors. This is followed by the use of this formalism in the development of a second-quantization functor between the category of Lorentzian globally hyperbolic manifolds and the category of nets of C*-algebras obeying Haag-Kastler axioms. Finally, we turn our attention to the particular case of the quantum free scalar field.

Cette thèse comporte deux parties, axées sur des aspects différents de la géométrie lorentzienne. La première partie porte sur les groupes d'isométries de surfaces lorentziennes globalement hyperboliques spatialement compactes, particulièrement lorsque le groupe exhibe une dynamique non triviale (action non propre). Le groupe d'isométries agit naturellement sur le cercle par difféomorphismes, et les résultats principaux portent sur la classification de ces représentations. Sous une hypothèse sur le bord conforme, on obtient une conjugaison par homéomorphisme avec l'action projective d'un sous-groupe de PSL(2,R) ou de l'un de ses revêtements finis. La différentiabilité de la conjuguante est étudiée, avec des résultats qui garantissent une conjugaison dans le groupe de difféomorphismes du cercle dans certains cas. On donne également des contre-exemples à l'existence d'une conjugaison différentiable, y compris pour des groupes ayant une dynamique riche. Ces constructions s'appuient sur l'étude de flots hyperboliques en dimension trois. Sans l'hypothèse sur le bord conforme, on obtient une semi conjugaison et un isomorphisme de groupes. On construit également des exemples pour lesquels il n'existe pas de conjugaison topologique. La seconde partie de cette thèse étudie un espace-temps vu comme un système dynamique multi-valuée : à un point on associe sont futur causal. Cette approche, déjà présente dans les travaux de Fathi et Siconolfi, permet de concrétiser le lien entre fonctions de Lyapunov en systèmes dynamiques et fonctions temps. Le résultat principal est une version lorentzienne du Théorème de Conley : on peut définir l'ensemble récurrent par chaînes d'un espace-temps, et il existe une fonction continue croissante le long de toute courbe causale orientée vers le futur, strictement croissante si le point de départ de la courbe n'est pas dans l'ensemble récurrent par chaînes. Ces techniques s'adaptent aussi dans un espace-temps stablement causal, ce qui permet de donner une nouvelle preuve d'une partie du Théorème d'Hawking.

Η αντιμετώπιση της περιορισμένης χωρικής ανάλυσης των εικόνων, η οποία οφείλεται στους φυσικούς περιορισμούς που εμφανίζουν οι αισθητήρες σύλληψης εικόνας, αποτελεί το αντικείμενο μελέτης της παρούσας διδακτορικής διατριβής. Στη διατριβή αυτή αρχικά γίνεται προσπάθεια μοντελοποίησης της λειτουργίας του ψηφιοποιητή εικόνας κατά τη δημιουργία αντίγραφου ενός εγγράφου μέσω απλών μοντέλων. Στην εξομοίωση της λειτουργίας του ψηφιοποιητή, το προτεινόμενο μοντέλο θα πρέπει να προτιμηθεί έναντι των μοντέλων Gaussian και Cauchy, που συναντώνται στη βιβλιογραφία, καθώς είναι ισοδύναμο στην απόδοση, απλούστερο στην υλοποίηση και δεν παρουσιάζει εξάρτηση από συγκεκριμένα χαρακτηριστικά λειτουργίας του ψηφιοποιητή. Έπειτα, μορφοποιούνται νέες μέθοδοι για τη βελτίωση της χωρικής ανάλυσης σε εικόνες. Προτείνεται μέθοδος μη ομοιόμορφης παρεμβολής για ανακατασκευή εικόνας Super-Resolution (SR). Αποδεικνύεται πειραματικά πως η προτεινόμενη μέθοδος η οποία χρησιμοποιεί την παρεμβολή Kriging υπερτερεί της μεθόδου η οποία δημιουργεί το πλέγμα υψηλής ανάλυσης μέσω της σταθμισμένης παρεμβολής κοντινότερου γείτονα που αποτελεί συμβατική τεχνική. Επίσης, παρουσιάζονται τρεις νέες μέθοδοι για στοχαστική ανακατασκευή εικόνας SR regularized. Ο εκτιμητής Tukey σε συνδυασμό με το Bilateral Total Variation (BTV) regularization, ο εκτιμητής Lorentzian σε συνδυασμό με το BTV regularization και ο εκτιμητής Huber συνδυασμένος με το BTV regularization είναι οι τρεις μέθοδοι που προτείνονται. Μία πρόσθετη καινοτομία αποτελεί η απευθείας σύγκριση των τριών εκτιμητών Tukey, Lorentzian και Huber στην ανακατασκευή εικόνας super-resolution, άρα στην απόρριψη outliers. Η απόδοση των προτεινόμενων μεθόδων συγκρίνεται απευθείας με εκείνη μίας τεχνικής SR regularized που υπάρχει στη βιβλιογραφία, η οποία αποδεικνύεται κατώτερη. Σημειώνεται πως τα πειραματικά αποτελέσματα οδηγούν σε επαλήθευση της θεωρίας εύρωστης στατιστικής συμπεριφοράς. Επίσης, εκπονείται μία πρωτότυπη μελέτη σχετικά με την επίδραση που έχει κάθε ένας από τους όρους έκφρασης πιστότητας στα δεδομένα και regularization στη διαμόρφωση του αποτελέσματος της ανακατασκευής εικόνας SR. Τα συμπεράσματα που προκύπτουν βοηθούν στην επιλογή μίας αποτελεσματικής μεθόδου για ανακατασκευή εικόνας SR ανάμεσα σε διάφορες υποψήφιες μεθόδους για κάποια δεδομένη ακολουθία εικόνων χαμηλής ανάλυσης. Τέλος, προτείνεται μία μέθοδος παρεμβολής σε εικόνα μέσω νευρωνικού δικτύου. Χάρη στην προτεινόμενη τεχνική εκπαίδευσης το νευρωνικό δίκτυο μαθαίνει το point spread function του ψηφιοποιητή εικόνας. Τα πειραματικά αποτελέσματα αποδεικνύουν πως η προτεινόμενη μέθοδος υπερτερεί σε σχέση με τους κλασικούς αλγόριθμους δικυβικής παρεμβολής και παρεμβολής spline. Η τεχνική που προτείνεται εξετάζει για πρώτη φορά το ζήτημα της σειράς της παρουσίασης των δεδομένων εκπαίδευσης στην είσοδο του νευρωνικού δικτύου.<br>Coping with the limited spatial resolution of images, which is caused by the physical limitations of image sensors, is the objective of this thesis. Initially, an effort to model the scanner function when generating a document copy by means of simple models is made. In a task of scanner function simulation the proposed model should be preferred over the Gaussian and Cauchy models met in bibliography as it is equivalent in performance, simpler in implementation and does not present any dependence on certain scanner characteristics. Afterwards, new methods for improving images spatial resolution are formulated. A nonuniform interpolation method for Super-Resolution (SR) image reconstruction is proposed. Experimentation proves that the proposed method employing Kriging interpolation predominates over the method which creates the high-resolution grid by means of the weighted nearest neighbor interpolation that is a conventional interpolation technique. Also, three new methods for stochastic regularized SR image reconstruction are presented. The Tukey error norm in combination with the Bilateral Total Variation (BTV) regularization, the Lorentzian error norm in combination with the BTV regularization and the Huber error norm combined with the BTV regularization are the three proposed methods. An additional novelty is the direct comparison of the three estimators Tukey, Lorentzian and Huber in the task of super-resolution image reconstruction, thus in rejecting outliers. The performance of the proposed methods proves superior to that of a regularized SR technique met in bibliography. Experimental results verify the robust statistics theory. Moreover, a novel study which considers the effect of each one of the data-fidelity and regularization terms on the SR image reconstruction result is carried out. The conclusions reached help to select an effective SR image reconstruction method, among several potential ones, for a given low-resolution sequence of frames. Finally, an image interpolation method employing a neural network is proposed. The presented training procedure results in the network learning the scanner point spread function. Experimental results prove that the proposed technique predominates over the classical algorithms of bicubic and spline interpolation. The proposed method is novel as it treats, for the first time, the issue of the training data presentation order to the neural network input.

Wydział Fizyki.<br>W pracy przedstawiono opis sektora pól cechowania dla elektrodynamiki kwantowej z prądami elektrycznymi i magnetycznymi. Wykorzystano model Zwanzigera zawierający dwa niezależne potencjały cechowania i . Założono, że prądy elektryczne i magnetyczne mają takie same własności transformacyjne, co prowadzi do złamania symetrii i , a jedynie złożenie transformacji lub ogólniej jest symetrią układu. Dla pól cechowania, badano własności 2-punktowej funkcji Wightmana, czyli próżniowej wartości oczekiwanej dla nieuporządkowanego iloczynu dwóch operatorów pola cechowania. Pokazano ogólną postać diagonalnych funkcji Wightmana, i dla różnych warunków cechowania, która ma postać symetryczną i zawiera część niezmienniczą proporcjonalną do . Ponadto znaleziono równanie różniczkowe na mieszane funkcje Wightmana i wykazano, że nie ma ono lorentzowsko-niezmienniczego rozwiązania. Znaleziono sferycznie symetryczne rozwiązanie, które odpowiada cechowaniu Coulomba, co zostało udowodnione na drodze kanonicznego kwantowania metodą Diraca. Dla innych warunków cechowania: planarnym, cechowaniu stożka świetlnego w kierunku oraz , została przeprowadzona procedura kwantowania kanonicznego metodą Faddeeva-Jackiwa. Wyznaczone funkcje Wightmana w tych cechowaniach są zgodne z ogólnymi wzorami, wyprowadzonymi wcześniej i znaleziono postać członów zależnych od cechowania.<br>This dissertation analyses the gauge field sector for the quantum electrodynamics with electric and magnetic currents. It uses the Zwanziger model with two independent gauge field potentials and . The assumption that the electric and magnetic currents have the same transformation properties, effectively leads to the and symmetry breaking, while the composed transformation or generally recovers the symmetry of the system. The properties of two-point Wightman functions, i.e. the vacuum expectations values of the unordered product two gauge field operators, were intensively studied. The general form of the diagonal Wightman functions and is found for different choices of gauge fixing conditions - it has a symmetrical form and includes the invariant part proportional to . Furthermore, the differential equation has been found for the mixed Wightman functions . It is shown that this equation has no Lorentz covariant solution. But the spherically symmetric solution has been found and it corresponds to the Coulomb gauge condition - this is proven by the canonical quantization procedure for systems with constraints - the Dirac method. For other gauge conditions: planar, light cone gauge in end direction, the canonical quantization procedure is carried out within the simplified procedure - the Faddeev-Jackiw method. The Wightman functions for the gauge field potentials are found and they are consistent with general formulas derived before. Also the gauge dependent parts are explicitly given for different gauge fixing conditions.

The thesis presents a study of bound states of heavy quarks in heavy-flavor QCD (only quarks b and c, without light quarks). The theory is written in the front form of Hamiltonian dynamics, with the canonical Hamiltonian as a starting point that needs regularization and renormalization. It is renormalized using renormalization group procedure for effective particles (RGPEP). The RGPEP operates with the concept of effective particles whose size is a scale parameter of renormalization group transformation. The renormalized Hamiltonian, written in terms of creation and annihilation operators for effective particles, is finite when the regularization cutoff on transverse momenta is sent to infinity. Asymptotic freedom follows and allows one to compute the renormalized Hamiltonian in perturbation theory at hadronic scales. I limit the calculation to the terms that are of second order in the effective coupling constant. The renormalized Hamiltonian eigenvalue equation, written in the whole Fock space of effective particles, is replaced by the eigenvalue equation written in terms of only two sectors of the Fock space – one containing the minimal quark content (QQ¯ for mesons and QQQ for baryons, where Q = b, c), and the other one that in addition to the quarks contains also just one effective gluon. The replacement is made plausible by the additional step, which is the assumption that in the reduced eigenvalue equation the gluon has mass, which is allowed to be a function of the gluon momentum relative to the quarks. In contrast to sectors with gluons, sectors with extra quark-antiquark pairs may be safely omitted because the renormalized Hamiltonian does not allow interactions to change invariant masses of states by much more than the RGPEP scale parameter. The sector with one massive gluon is then eliminated using the perturbative procedure of Bloch. The resulting effective Hamiltonians for QQ¯ and QQQ states contain the RGPEP form factors in interaction vertices and can be approximated by the form they take in the nonrelativistic limit. Finally, the effective approximate Hamiltonians turn out to contain the Coulomb potentials and harmonic oscillator potentials between quarks. Using the effective Hamiltonians, approximate masses and wave functions of heavy hadrons are found. I derived also formulas for their electromagnetic form factors in elastic scattering and their structure functions in the electron deep inelastic scattering off them. To calculate the mass spectra of mesons and baryons, the masses of quarks at the corresponding RGPEP scale parameters were fitted to the masses of the three lightest spin-1 states (excluding 1+− states) in the meson families of bottomonia and in the meson family of charmonia. and charmonia. The coupling constant was evaluated using the leading asymptotic freedom formula for two flavors. The results for masses of ground states of bbb and ccc baryons contain no free parameters and they agree well with predictions of other theoretical approaches. The excitation spectrum is in a qualitative agreement with lattice QCD studies. Therefore, these results constitute a firm foothold for future studies of bound states in QCD with one flavor of heavy quarks using our method. The results for ccb and bbc baryons, as well as c ¯b and bc¯ mesons, bear some theoretical uncertainty due to the universal mass shift that appears because of significant scale difference between bottom and charm quark masses. Removal of this uncertainty requires studies in the fourth order of perturbation theory in the coupling constant. Present calculations predict that in ccb the two c quarks form a diquark, while in bbc the b quarks are bound with each other more loosely than with the quark c. I calculated the form factors of the hadrons I found and I used them to compute the radii of these hadrons. My results for the radii agree well with results found in literature and especially good agreement is obtained with the lattice calculations wherever comparison is possible. I also calculated magnetic moments of baryons and vector quarkonia. Comparison with literature shows that for systems that involve only one flavor of quarks my results agree with other theoretical predictions, and for systems that involve both charm and bottom quarks the moments are larger than those predicted using other methods. The hadron structure functions I computed in a simplified way. I neglected the huge difference between the scale of quark binding and the scale of the virtual photon in deep inelastic scattering (formally infinite). My calculations show interesting features, such as dependence of the structure function shape on the wave function nodes. However, my calculations of the structure functions can be considered merely a demonstration of potential utility of the method, because I approximated the transformation that connects effective particles at two different RGPEP scales, that of DIS and that of bound-state formation, by identity. Corrections due to scale evolution are not included.<br>Dysertacja przedstawia dociekania dotyczące stanów związanych w QCD ciężkich kwarków (tylko kwarki b i c, bez uwzględnienia lekkich kwarków). Teoria ta jest napisana we frontowej formie dynamiki Hamiltonowskiej z kanonicznym Hamiltonianem jako punktem startowym, który wymaga regularyzacji i renormalizacji. Renormalizacja jest przeprowadzona przy użyciu procedury grupy renormalizacji dla cząstek efektywnych (w skrócie RGPEP od ang. renormalization group procedure for effective particles). RGPEP używa koncepcji cząstek efektywnych, których rozmiar jest parametrem skali w procedurze renormalizacji. Zrenormalizowany Hamiltonian, zapisany przy użyciu operatorów kreacji i anihilacji cząstek efektywnych, jest skończony w granicy usuwania obcięcia na poprzeczne pędy do nieskończoności. Asymptotyczna swoboda charakteryzuje zrenormalizowany Hamiltonian i pozwala na obliczenie go w rachunku zaburzeń dla skali hadronowej. Moje rachunki ograniczam do wyrazów drugiego rzędu w efektywnej stałej sprzężenia. Równanie własne zrenormalizowanego Hamiltonianu, zapisane w całej przestrzeni Focka cząstek efektywnych, jest zastąpione równaniem własnym zapisanym w dwóch sektorach przestrzeni Focka – jednym, zawierającym minimalny zestaw kwarków (QQ¯ dla mezonów i QQQ dla barionów, gdzie Q = b, c), oraz drugim, który w dodatku do kwarków zawiera jeszcze tylko jeden efektywny gluon. Żeby uwiarygodnić to zastąpienie, poczynione jest dodatkowe założenie, że w zredukowanym problemie własnym gluon ma masę, która może być funkcją pędu gluonu względem kwarków. W przeciwieństwie do sektorów z gluonami, sektory z dodatkowymi parami kwark-antykwark mogą być bezpiecznie pominięte, ponieważ zrenormalizowany Hamiltonian nie dopuszcza, żeby oddziaływania zmieniały masę inwariantną o dużo więcej niż parametr skali RGPEP. Sektor z jednym masywnym gluonem jest później wyeliminowany przy użyciu zaburzeniowej metody Blocha. W wyniku otrzymuje się efektywne Hamiltoniany dla stanów QQ¯ oraz QQQ zawierające czynniki kształtu RGPEP w wierzchołkach oddziaływania i można je przybliżyć ich postacią jaką przyjmują w granicy nierelatywistycznej. Efektywne i przybliżone Hamiltoniany zawierają potencjały Coulomba oraz potencjały oscylatora harmonicznego pomiędzy kwarkami. Korzystając z efektywnych Hamiltonianów, przybliżone masy i funkcje falowe ciężkich hadronów zostały znalezione. Wyprowadziłem także wzory na elektromagnetyczne czynniki kształtu w zderzeniach elastycznych z elektronami oraz na funkcje struktury w głęboko nieelastycznym rozpraszaniu elektronów na tych hadronach. Do obliczenia widm masowych mezonów i barionów, masy kwarków dla odpowiednich parametrów skali RGPEP zostały dopasowane do mas trzech najlżejszych cząstek o spinie jeden (pomijając stany 1+−) z rodziny bottomonium (b ¯b) oraz z rodziny czarmonium (cc¯). Stała sprzężenia została obliczona, korzystając z wiodącego przybliżenia asymptotycznej swobody dla dwóch zapachów kwarków. Wyniki dla mas stanów podstawowych barionów złożonych z trzech kwarków b lub trzech kwarków c, czyli bbb oraz ccc nie zawierają żadnych swobodnych parametrów i zgadzają się z przewidywaniami innych teoretycznych metod ich liczenia. Widma masowe stanów wzbudzonych są w jakościowej zgodności z obliczeniami na sieci (Lattice QCD). Dlatego wyniki te stanowią mocne oparcie dla przyszłych badań stanów związanych w QCD z jednym zapachem kwarków przy użyciu naszej metody. Wyniki dla barionów ccb i bbc oraz dla mezonów c ¯b i bc¯ obarczone są teoretyczną niepewnością związaną z uniwersalnym przesunięciem mas, które pojawia się z powodu znacznej różnicy skal masowych kwarków b i c. Usunięcie tej niepewności wymaga rozszerzenia rachunku do czwartego rzędu w stałej sprzężenia. Obecne wyniki sugerują, że w ccb dwa kwarki c tworzą dikwark, podczas gdy w bbc kwarki b są słabiej związane ze sobą niż z kwarkiem c. Obliczyłem czynniki kształtu hadronów, które rozważałem, i użyłem ich do obliczenia promieni tych hadronów. Moje wyniki na promienie zgadzają się z wynikami znalezionymi w literaturze, przy czym zgodność z wynikami Lattice QCD jest szczególnie dobra (w tych przypadkach gdzie porównanie jest możliwe). Obliczyłem także momenty magnetyczne barionów i wektorowych kwarkoniów. Porównanie z literaturą przedstawia zgodność z innymi metodami rachunkowymi dla cząstek złożonych z jednego rodzaju kwarków. Dla cząstek zawierających dwa rodzaje kwarków, momenty, które znalazłem, są większe niż momenty uzyskane przy użyciu innych metod. Funkcje struktury obliczyłem w uproszczony sposób, pomijając wielką różnicę pomiędzy skalą wiązania kwarków i skalą wirtualnego fotonu w rozpraszaniu głęboko nieelastycznym (która jest formalnie nieskończona). Moje rachunki ujawniają ciekawe cechy funkcji struktury, jak zależność ich kształtu od zer funkcji falowej. Jednakże, moje rachunki funkcji struktury można uznać jedynie za demonstrację potencjalnej użyteczności metody, ponieważ transformację, która łączy cząstki efektywne dla dwóch różnych skal RGPEP, przybliżyłem identycznością. Poprawki związane z ewolucją ze skalą nie są uwzględnione.

Line shapes describe how absorption and emission are spectrally distributed around the line positions formed by rotational, vibrational, and electronic transitions. Line shapes arise from the different processes that spectrally broaden the absorption and emission of radiation. Optical thickness and equivalent width are shown to be fundamentally related to line shape. The fundamental line shape functions for atmospheres including the Gaussian line shape due to molecular motion and the Lorentzian line shape from lifetime broadening, including collision (pressure) broadening are described. Their convolution, the Voigt line shape, which is important in some atmospheric conditions is also described. The standard HITRAN database of spectroscopic parameters of molecules for use in calculation of radiative transfer in planetary atmospheres, from radiofrequencies to the near ultraviolet, is introduced.

“Spontaneous emission of radiation” calculates the rate of spontaneous electric-dipole emission of a photon by an excited atom or molecule. The calculation proceeds by using basic quantum mechanics (i.e. not using the short cuts of Chapter 19); it uses quantum electrodynamics but is not, on that account, particularly difficult. A 2p–1s transition in hydrogen is used as exemplar; the radiation is elliptically polarized. The spectral line profile (lineshape function) is approximately Lorentzian, but has a high-frequency cut-off, needed to prevent the power radiated from diverging. A radiation-induced frequency shift is negligible. The width of the line profile agrees with the Einstein A-coefficient. A high-frequency cut-off is shown to apply similarly in the derivation of Golden Rule Number Two.

Different plasma diagnostic methods are briefly discussed, and the framework of a test charge technique is effectively used as diagnostic tool for investigating interaction potentials in Lorentzian plasma, whose constituents are the superthermal electrons and ions with negatively charged dust grains. Applying the space-time Fourier transformations to the linearized coupled Vlasov-Poisson equations, a test charge potential is derived with a modified response function due to energetic ions and electrons. For a test charge moving much slower than the dust-thermal speed, there appears a short-range Debye-Hückel (DH) potential decaying exponentially with distance and a long-range far-field (FF) potential as the inverse cube of the distance from test charge. The FF potentials exhibit more localized shielding curves for low-Kappas, and smaller effective shielding length is observed in dusty plasma compared to electron-ion plasma. However, a wakefield (WF) potential is formed behind the test charge when it resonates with dust-acoustic oscillations, whereas a fast moving test charge leads to the Coulomb potential having no shielding around. It is revealed that superthermality and plasma parameters significantly alter the DH, FF, and WF potentials in space plasmas of Saturn’s E-ring, where power-law distributions can be used for energetic electrons and ions in contrast to Maxwellian dust grains.

Abstract Begins by discussing the fundamentals of absorption and amplification in two-state optical systems, including energy levels, Planck’s radiation law, the Einstein A and B coefficients, the Lorentzian lineshape function, population inversion, amplification, optical resonators, and the conditions for oscillation and lasing. Introduces the main laser types, including solid state, gas discharge, dye, gas-dynamic, excimer, chemical, fibre, and semiconductor lasers. Details the main variants of semiconductor lasers, including the Fabry-Perot and distributed Bragg reflector lasers, lasers based on quantum wells, wires, dots and more general bandgap engineering, the laser array, and vertical cavity surface emitting lasers (VCSELs). Introduces the idea of transverse and axial modes, and presents a number of methods for their control, including Q-switching, cavity dumping and mode locking. Covers a number of related topics including parametric oscillators, optical fibre amplifiers, masers, and noise. Concludes with a discussion of applications, including optical sensors, optical communications, holography and Raman scattering.

Hyperbolic rotation is commonly used to effectively model knowledge graphs and their inherent hierarchies. However, existing hyperbolic rotation models rely on logarithmic and exponential mappings for feature transformation. These models only project data features into hyperbolic space for rotation, limiting their ability to fully exploit the hyperbolic space. To address this problem, we propose a novel fully hyperbolic model designed for knowledge graph embedding. Instead of feature mappings, we define the model directly in hyperbolic space with the Lorentz model. Our model considers each relation in knowledge graphs as a Lorentz rotation from the head entity to the tail entity. We adopt the Lorentzian version distance as the scoring function for measuring the plausibility of triplets. Extensive results on standard knowledge graph completion benchmarks demonstrated that our model achieves competitive results with fewer parameters. In addition, our model get the state-of-the-art performance on datasets of CoDEx-s and CoDEx-m, which are more diverse and challenging than before. Our code is available at https://github.com/llqy123/FHRE.

Fluorescence excitation spectroscopy of single dye molecules in crystals offers the possibility to determine molecular paramaters of single absorbers and their distribution instead of ensemble averages [1,2]. Informations about single molecules can be derived from the optical absorption spectra and from the time distribution of the fluorescence photons. The inhomogeneous broadening of an electronic transition reflects the variety of local environments accessible to the guest molecules and is therefore a measure for the degree of disorder or the defect concentration in the host crystal [3,4]. For pentacene molecules in p-terphenyl crystals (c ≈ 10−8mole/mole) we observed different types of inhomogeneous broadening: Some crystals show a narrow (FWHM ≈ 1 GHz) Lorentzian line with a satellite structure at short wavelengths due to 13C substitutional isomers of pentacene. The inhomogeneous line of other crystals is remarkable broadened and neither Lorentzian nor Gaussian. The intersystem crossing (ISC) rates k23 and k31 can be determined by measuring the intensity autocorrelation function of the fluorescence light [5]. In a crystal with low degree of disorder (narrow Lorentzian inhomogeneous broadening) the ISC rates of the molecules show a smaller distribution than in a crystal with higher degree of disorder (broadened line). The fully saturated fluorescence emission rates as calculated using the ISC rates are consistent with the experimentally measured photocount rates.