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A class of subharmonic functions are proved to have the growth estimates $u(x)= o(x_n^{1-\frac{\alpha}{p}}|x|^{\frac{\gamma}{p}+\frac{n-1}{q}-n+\frac{\alpha}{p}})$ at infinity in the upper half space of ${\bf R}^{n}$, which generalizes the growth properties of analytic functions and harmonic functions.

Based on the dual representation in terms of the recently established eigenfunctions of the evolution kernel in heavy-quark effective theory, we investigate the description of the B-meson light-cone distribution amplitude (LCDA) beyond tree-level. In particular, in dual space, small and large momenta do not mix under renormalization, and therefore perturbative constraints from a short-distance expansion in the parton picture can be implemented independently from non-perturbative modelling of long-distance effects. It also allows to (locally) resum perturbative logarithms from large dual momenta at fixed values of the renormalization scale. We construct a generic procedure to combine perturbative and non-perturbative information on the B-meson LCDA and compare different model functions and the resulting logarithmic moments which are the relevant hadronic parameters in QCD factorization theorems for exclusive B-meson decays.

A three Dimensional (3D) space truss structural member is used to transfer the force in 3D manner. The space truss is unstable and brittle in nature. Due to over loading, the buckling of one member may cause the subsequent failure in other members. This may even leads to collapse the entire structure. The slab will reduce the buckling in the chord member and to increase the stiffness. In this study a composite space truss model was created in Hyper Mesh-a FEM software using P ROD and P SHELL elements and its deflection values were obtained. Various design parameters such as grade of concrete, slab thickness, steel module sizes for truss members were incorporated in the software and the deflection values were found. Finally the optimum slab thickness for different grades of concrete and module size of steel truss by weight optimization and deflection criteria were found.

We present an extension of our earlier work [Ritt operators and convergence in the method of alternating projections, J. Approx. Theory, 205:133-148, 2016] by proving a general asymptotic result for orbits of an operator acting on a reflexive Banach space. This result is obtained under a condition involving the growth of the resolvent, and we also discuss conditions involving the location and the geometry of the numerical range of the operator. We then apply the general results to some classes of iterative projection methods in approximation theory, such as the Douglas-Rachford splitting method and, under suitable geometric conditions either on the ambient Banach space or on the projection operators, the method of alternating projections.

This project aims to develop and test a theory of how electoral behaviour is determined by parties, individual voters, and their interaction: how do voters and party foot soldiers interact? And what impact does this interaction have on electoral outcomes, namely vote choice and turnout. The project uses a panel survey of households in Harare and Bulawayo to investigate these questions. We focus here on those living and/or working in urban informality. Given the poor economic conditions in the country, the vast majority of the urban population lives in poor conditions and works in the informal sector. The vast majority of those in the informal sector work as street vendors - buying small amounts of goods, which are sold on the streets. We focus on this population as it is a large segment of the urban population and is one that has frequently found itself in opposition to the ruling party. We seek to explain how the ruling party’s influence on informality shapes political behavior for those living in informality. This is also an important population to study given our focus on positive and negative inducements. The literature has found that positive inducements (e.g. vote buying) is primarily used among the poorer population and while it is still an open question as to whether or not negative inducements like violence are more common among the poor (Mares and Young 2016), there is evidence that this is the case in Zimbabwe (Young 2015).

A detailed analysis of the structure and gauge dependence of the bulk-to-bulk propagators for the higher spin gauge fields in $AdS$ space is performed. The possible freedom in the construction of the propagators is investigated and fixed by the correct boundary behaviour and correspondence to the representation theory results for the $AdS$ space isometry group. The classical origin of the Goldstone mode and its connection with the gauge fixing procedure is considered.

We characterise asymptotic behaviour of families of symmetric orthonormal polynomials whose recursion coefficients satisfy certain conditions, satisfied for example by the (normalised) Hermite polynomials. More generally, these conditions are satisfied by the recursion coefficients of the form $c(n+1)^p$ for $0 0$, as well as by recursion coefficients which correspond to polynomials orthonormal with respect to the exponential weight $W(x)=\exp(-|x|^\beta)$ for $\beta>1$. We use these results to show that, in a Hilbert space defined in a natural way by such a family of orthonormal polynomials, every two complex exponentials $e_{\omega}(t)={e}^{i \omega t}$ and $e_{\sigma}(t)={e}^{i \sigma t}$ of distinct frequencies $\omega,\sigma$ are mutually orthogonal. We finally formulate a surprising conjecture for the corresponding families of non-symmetric orthonormal polynomials; extensive numerical tests indicate that such a conjecture appears to be true.

We show that the logarithmic infrared divergences in electron self-energy and vertex function of massless QED in 2+1 dimensions can be removed at all orders of 1/N by an appropriate choice of a non-local gauge. Thus the infrared behaviour given by the leading order in 1/N is not modified by higher order corrections. Our analysis gives a computational scheme for the Amati-Testa model, resulting in a non-trivial conformal invariant field theory for all space-time dimensions 2 < d < 4.

We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus T with a complex line C, with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bundle extends to TxP1. The converse statement is also true, namely a holomorphic bundle on TxP1 which is flat on the torus at infinity, and satisfies a stability condition, comes from a doubly-periodic instanton. Finally, we study the hyperkahler geometry of the moduli space of doubly-periodic instantons, and prove that the Nahm transform previously defined by the second author is a hyperkahler isometry with the moduli space of certain meromorphic Higgs bundles on the dual torus.

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As demonstrated in our previous work on ${\boldsymbol T}_{4}$, the space of phylogenetic trees with four leaves, the global, as well as the local, topological structure of the space plays an important role in the non-classical limiting behaviour of the sample Fr\'echet means of a probability distribution on ${\boldsymbol T}_{4}$. Nevertheless, the techniques used in that paper were specific to ${\boldsymbol T}_{4}$ and cannot be adapted to analyse Fr\'echet means in the space ${\boldsymbol T}_{m}$ of phylogenetic trees with $m(\geqslant5)$ leaves. To investigate the latter, this paper first studies the log map of ${\boldsymbol T}_{m}$, a generalisation of the inverse of the exponential map on a Riemannian manifold. Then, in terms of a modified version of the log map, we characterise Fr\'echet means in ${\boldsymbol T}_{m}$ that lie in top-dimensional or co-dimension one strata. We derive the limiting distributions for the corresponding sample Fr\'echet means, generalising our previous results. In particular, the results show that, although they are related to the Gaussian distribution, the forms taken by the limiting distributions depend on the co-dimensions of the strata in which the Fr\'echet means lie.

It has been argued that charged Ads black holes have similar thermodynamic behavior as the Van der Waals fluid system, provided one treats the cosmological constant as a thermodynamic variable in an extended phase space. In this paper, we disclose the deep connection between charged AdS black holes and Van der Waals fluid system without extending the phase space. We keep the cosmological constant as a fixed parameter and instead, treat the square of the charge of black hole as a thermodynamic variable. Therefore, we write the equation of state as $Q^{2}=Q^{2}(T,\Psi)$ where $\Psi$ (conjugate of $Q^{2} $) is the inverse of the specific volume, $\Psi=1/v$. This allows us to complete the analogy of charged AdS black holes with Van der Waals fluid system and derive the phase transition as well as critical exponents of the system. We identify a thermodynamic instability in this new picture with real analogy to Van der Waals fluid with physically relevant Maxwell construction. We therefore study the critical behavior of isotherms in $% Q^2-\Psi$ diagram and deduce all the critical exponents of the system and determine that the system exhibits a small-large black hole phase transition at the critical point $(T_c,Q^2_c, \Psi_c)$. This alternative view is important as one can imagine such a change for a given single black hole i. e. acquiring charge which induces the phase transition. Finally, we disclose the microscopic properties of charged AdS black holes by using thermodynamic geometry. Interestingly, we find that scalar curvature has a gap between small and large black holes, and this gap becomes exceedingly large as one moves away from the critical point along the transition line. Therefore, we are able to attribute the sudden enlargement of the black hole to the strong repulsive nature of the internal constituents at the phase transition.

We have studied using the KLUN sample of 5171 spiral galaxies having Tully-Fisher distance moduli, the average radial space distribution of galaxies out to a distance of about 200 Mpc (for H_0=50 km/s/Mpc). One motivation came from the current debate on the fractal dimension D and maximum scale of fractality. To study this question, we used a new method based on photometric TF distances, independent of redshift, to construct the number density distribution. Our main results are: (1) While scattered below about 20 Mpc, at larger distances the radial distribution starts to follow, in terms of distance modulus mu_TF, the law log N = 0.46 mu + const., using diameter TF relation, and log N = 0.40 mu + const. for magnitudes. These are the predictions based on fractal dimensions 2.3 and 2.0, respectively. These radial density gradients are valid up to the limits of KLUN, or about 200 Mpc. (2) We have tried to understand the derived radial density behaviour as a result of some bias in KLUN or our analysis, however, without success. Numerical simulations have shown that the method itself works. If the density law is caused by incompleteness in the diameter limited KLUN sample, then the incompleteness should start at widely different angular diameters d_25 for different values of rotation parameter log V_M, which would be quite unexpected. (3) If correlation studies favoring long scale fractality (200 Mpc or more) and D ~ 2 are correct, the position of our Galaxy would be close to average in the Universe, with the galaxy density decreasing around us according to the expected law.

We use the Conformal Metric as described in Kar-Sinha work on Gravitational Bending of Light in a 5D Spacetime to recompute the equations of the 5D Force in Basini-Capozziello-Leon Formalism and we arrive at a result that possesses some advantages. The equations of the Extra Force as proposed by Leon are now more elegant in Conformal Formalism and many algebraic terms can be simplified or even suppressed. Also we recompute the Kar-Sinha Gravitational Bending of Light affected by the presence of the Extra Dimension and analyze the Superluminal Chung-Freese Features of this Formalism describing the advantages of the Chung-Freese BraneWorld when compared to other Superluminal spacetime metrics(eg:Warp Drive) and we describe why the Extra Dimension is invisible and how the Extra Dimension could be made visible at least in theory.We also examine the Maartens-Clarkson Black Holes in 5D(Black Strings) coupled to massive Kaluza-Klein graviton modes predicted by Extra Dimensions theories and we study experimental detection of Extra Dimensions on-board LIGO and LISA Space Telescopes.We also propose the use of International Space Station(ISS) to measure the additional terms(resulting from the presence of Extra Dimensions) in the Kar-Sinha Gravitational Bending of Light in Outer Space to verify if we really lives in a Higher Dimensional Spacetime.Also we demonstrate that Particle $Z$ can only exists if the 5D spacetime exists.

The study of path behaviour of stochastic processes is a classical topic in probability theory and related areas. In this frame, a natural question one can address is: whether or not sample paths belong to a critical H\"older space? The answer to this question is negative in the case of Brownian motion and many other stochastic processes: it is well-known that despite the fact that Brownian paths satisfy, on each compact interval $I$, a H\"older condition of any order strictly less than $1/2$, they fail to belong to the critical H\"older space $\mathcal{C}^{1/2}(I)$. In this article, we show that a different phenomenon happens in the case of linear multifractional stable motion (LMSM): for any given compact interval one can find a critical H\"older space to which sample paths belong. Among other things, this result improves an upper estimate, recently derived in Bierm\'e, Lacaux (2013), on behaviour of LMSM, by showing that the logarithmic factor in it is not needed.

The study of path behaviour of stochastic processes is a classical topic in probability theory and related areas. In this frame, a natural question one can address is: whether or not sample paths belong to a critical H\"older space? The answer to this question is negative in the case of Brownian motion and many other stochastic processes: it is well-known that despite the fact that Brownian paths satisfy, on each compact interval $I$, a H\"older condition of any order strictly less than $1/2$, they fail to belong to the critical H\"older space $\mathcal{C}^{1/2}(I)$. In this article, we show that a different phenomenon happens in the case of linear multifractional stable motion (LMSM): for any given compact interval one can find a critical H\"older space to which sample paths belong. Among other things, this result improves an upper estimate, recently derived in Bierm\'e, Lacaux (2013), on behaviour of LMSM, by showing that the logarithmic factor in it is not needed.

The study of path behaviour of stochastic processes is a classical topic in probability theory and related areas. In this frame, a natural question one can address is: whether or not sample paths belong to a critical H\"older space? The answer to this question is negative in the case of Brownian motion and many other stochastic processes: it is well-known that despite the fact that Brownian paths satisfy, on each compact interval $I$, a H\"older condition of any order strictly less than $1/2$, they fail to belong to the critical H\"older space $\mathcal{C}^{1/2}(I)$. In this article, we show that a different phenomenon happens in the case of linear multifractional stable motion (LMSM): for any given compact interval one can find a critical H\"older space to which sample paths belong. Among other things, this result improves an upper estimate, recently derived in Bierm\'e, Lacaux (2013), on behaviour of LMSM, by showing that the logarithmic factor in it is not needed.

We study the behaviour of spin-half particles in curved space-time. Since Dirac equation gives the dynamics of spin-half particles, we mainly study the Dirac equation in Schwarzschild, Kerr, Reissner-Nordstr\"om geometry. Due to the consideration of existence of black hole in space-time (the curved space-time), particles are influenced and equation will be modified. As a result the solution will be changed from that due to flat space.

he starting process with countable number of types \mu(t) generates a stopped branching process \xi(t). The starting process stops, by falling into the nonempty set S. It is assumed, that the starting process is subcritical, indecomposable and noncyclic. It is proved, that the extinction probability converges to the cyclic function with period 1.

he starting process with countable number of types \mu(t) generates a stopped branching process \xi(t). The starting process stops, by falling into the nonempty set S. It is assumed, that the starting process is subcritical, indecomposable and noncyclic. It is proved, that the extinction probability converges to the cyclic function with period 1.

We construct the fundamental solution of the Porous Medium Equation posed in the hyperbolic space $H^n$ and describe its asymptotic behaviour as $t\to\infty$. We also show that it describes the long time behaviour of integrable nonnegative solutions, and very accurately if the solutions are also radial and compactly supported. By radial we mean functions depending on the space variable only through the geodesic distance $r$ from a given point $O\in H^n$. We also construct an exact generalized traveling wave solution. We show that the location of the free boundary of compactly supported solutions grows logarithmically for large times, in contrast with the well-known power-like growth of the PME in the Euclidean space. Very slow propagation at long distances is a feature of porous medium flow in hyperbolic space.

We study the asymptotic behaviour of solutions of the fourth Pain\-lev\'e equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalisation of phase space described by Okamoto. We show that the limit set of each solution is compact and connected and, moreover, that any non-special solution has an infinite number of poles and infinite number of zeroes.

In the framework of the spatial coherence wavelets, different features of the first-order spatial coherence (Young's interference) are analysed by calculating the corresponding marginal power spectrum, a close related quantity to the classical Wigner Distribution Function (WDF) of the optical field. The consideration of radiant and virtual point sources evidences some effects, conventionally attributed to non-classical correlations of light, although such type of correlations are not explicitly included in the model. Specifically, a light state is produced that has similar morphology to the WDF of the well-known quantum Schr\"odinger cat state.

It is well known that the solutions to the non-viscous Burgers equation develop a gradient catastrophe at a critical time provided the initial data have a negative derivative in certain points. We consider this equation assuming that the particle paths in the medium are governed by a random process with a variance which depends in a polynomial way on the velocity. Given an initial distribution of the particles which is uniform in space and with the initial velocity linearly depending on the position we show both analytically and numerically that there exists a threshold effect: if the power in the above variance is less than 1, then the noise does not influence the solution behavior, in the following sense: the mean of the velocity when we keep the value of position fixed goes to infinity outside the origin. If however the power is larger or equal 1, then this mean decays to zero as the time tends to a critical value.

We study the asymptotic behaviour of the solutions of the fifth Painlev\'e equation as the independent variable approaches zero and infinity in the space of initial values. We show that the limit set of each solution is compact and connected and, moreover, that any solution with the essential singularity at zero has an infinite number of poles and zeroes, and any solution with the essential singularity at infinity has infinite number of poles and takes value $1$ infinitely many times.

The limitations of three-dimensional semi-classical gravity are explored in the context of a conformally invariant theory for a self-interacting scalar field. The analysis of the theory's scaling behaviour reveals that scalar-loop effects contribute to the conformal anomaly only at advanced orders allowing for a range of relevant energy scales which extend to those comparable to the non-perturbative scale of Planck mass.

This dissertation summarizes my investigations in operator theory during my PhD studies. The first chapter is an introduction to that field of operator theory which was developed by B. Sz.-Nagy and C. Foias, the theory of power-bounded Hilbert space operators. In the second and third chapter I characterize operators which arise from power-bounded operators asymptotically. Chapter 4 is devoted to provide a possible generalization of (the necessity part of) Sz.-Nagy's famous similarity theorem. In Chapter 5 I collected my results concerning the commutant mapping of asymptotically non-vanishing contractions. In the final chapter the reader can find results about cyclic properties of weighted shift operators on directed trees.

This dissertation summarizes my investigations in operator theory during my PhD studies. The first chapter is an introduction to that field of operator theory which was developed by B. Sz.-Nagy and C. Foias, the theory of power-bounded Hilbert space operators. In the second and third chapter I characterize operators which arise from power-bounded operators asymptotically. Chapter 4 is devoted to provide a possible generalization of (the necessity part of) Sz.-Nagy's famous similarity theorem. In Chapter 5 I collected my results concerning the commutant mapping of asymptotically non-vanishing contractions. In the final chapter the reader can find results about cyclic properties of weighted shift operators on directed trees.

This dissertation summarizes my investigations in operator theory during my PhD studies. The first chapter is an introduction to that field of operator theory which was developed by B. Sz.-Nagy and C. Foias, the theory of power-bounded Hilbert space operators. In the second and third chapter I characterize operators which arise from power-bounded operators asymptotically. Chapter 4 is devoted to provide a possible generalization of (the necessity part of) Sz.-Nagy's famous similarity theorem. In Chapter 5 I collected my results concerning the commutant mapping of asymptotically non-vanishing contractions. In the final chapter the reader can find results about cyclic properties of weighted shift operators on directed trees.

This dissertation summarizes my investigations in operator theory during my PhD studies. The first chapter is an introduction to that field of operator theory which was developed by B. Sz.-Nagy and C. Foias, the theory of power-bounded Hilbert space operators. In the second and third chapter I characterize operators which arise from power-bounded operators asymptotically. Chapter 4 is devoted to provide a possible generalization of (the necessity part of) Sz.-Nagy's famous similarity theorem. In Chapter 5 I collected my results concerning the commutant mapping of asymptotically non-vanishing contractions. In the final chapter the reader can find results about cyclic properties of weighted shift operators on directed trees.