Abstract:
The inflationary potential and its derivatives determine the spectrum of scalar and tensor metric perturbations that arise from quantum fluctuations during inflation. The CBR anisotropy offers a promising means of determining the spectra of metric perturbations and thereby a means of constraining the inflationary potential. The relation between the metric perturbations and CBR anisotropy depends upon cosmological parameters -- most notably the possibility of a cosmological constant. Motivated by some observational evidence for a cosmological constant (large-scale structure, cluster-baryon fraction, measurements of the Hubble constant and age of the Universe) we derive the reconstruction equations and consistency relation to second order in the presence of a cosmological constant. We also clarify previous notation and discuss alternative schemes for reconstruction.

Abstract:
We present Monte Carlo reconstruction, a new method for ``inverting'' observational data to constrain the form of the scalar field potential responsible for inflation. This stochastic technique is based on the flow equation formalism and has distinct advantages over reconstruction methods based on a Taylor expansion of the potential. The primary ansatz required for Monte Carlo reconstruction is simply that inflation is driven by a single scalar field. We also require a very mild slow roll constraint, which can be made arbitrarily weak since Monte Carlo reconstruction is implemented at arbitrary order in the slow roll expansion. While our method cannot evade fundamental limits on the accuracy of reconstruction, it can be simply and consistently applied to poor data sets, and it takes advantage of the attractor properties of single-field inflation models to constrain the potential outside the small region directly probed by observations. We show examples of Monte Carlo reconstruction for data sets similar to that expected from the Planck satellite, and for a hypothetical measurement with a factor of five better parameter discrimination than Planck.

Abstract:
To first order in the deviation from scale invariance the inflationary potential and its first two derivatives can be expressed in terms of the spectral indices of the scalar and tensor perturbations, $n$ and $n_T$, and their contributions to the variance of the quadrupole CBR temperature anisotropy, $S$ and $T$. In addition, there is a ``consistency relation'' between these quantities: $n_T= -{1\over 7}{T\over S}$. We discuss the overall strategy of perturbative reconstruction and derive the second-order expressions for the inflationary potential and its first two derivatives and the first-order expression for its third derivative, all in terms of $n$, $n_T$, $S$, $T$, and $dn/d\ln k$. We also obtain the second-order consistency relation, $n_T =-{1\over 7}{T\over S}[1+0.11{T\over S} + 0.15 (n-1)]$. As an example we consider the exponential potential, the only known case where exact analytic solutions for the perturbation spectra exist. We reconstruct the potential via Taylor expansion (with coefficients calculated at both first and second order), and introduce the Pad\'{e} approximant as a greatly improved alternative.

Abstract:
In this paper, we present quantitative constraints on the scalar field potential for a general class of inflationary models. (1) We first consider the reconstruction of the inflationary potential for given primordial density fluctuation spectra. Our work differs from previous work on reconstruction in that we find a semi-analytic solution for the potential for the case of density fluctuations with power-law spectra. In addition, for the case of more general spectra, we show how constraints on the density fluctuation spectra imply corresponding constraints on the potential. We present a series of figures which show how the shape of the potential depends on the shape of the perturbation spectrum and on the relative contribution of tensor modes. (2) We show that the average ratio $\rave$ of the amplitude of tensor perturbations (gravity wave perturbations) to scalar density perturbations is bounded from above: $\rave \le$ 1.6. We also show that the ratio $\rave$ is proportional to the change $\Delta \phi$ in the field: $\rave \approx 0.42 \Delta \phi/\mp$. Thus, if tensor perturbations are important for the formation of structure, then the width $\Delta \phi$ must be comparable to the Planck mass. (3) We constrain the change $\Delta V$ of the potential and the change $\Delta \phi$ of the inflation field during the portion of inflation when cosmological structure is produced. We find both upper and lower bounds for $\Delta \phi$ and for $\Delta V$. In addition, these constraints are then used to derive a bound on the scale $\Lambda$, which is the scale of the height of the potential

Abstract:
We make a more general determination of the inflationary observables in the standard 4-D and 5-D single-field inflationary scenarios, by the exact reconstruction of the dynamics of the inflation potential during the observable inflation with minimal number of assumptions: the computation does not assume the slow-roll approximation and is valid in all regimes if the field is monotonically rolling down its potential. Making use of the {\em Hamilton-Jacobi} formalism developed for the 5-D single-field inflation model,we compute the scale dependence of the amplitudes of the scalarand tensor perturbations by integrating the exact mode equation. We analyze the implications of the theoretical uncertainty in the determination of the reheating temperature after inflation on the observable predictions of inflation and evaluate its impact on the degeneracy of the standard inflation consistency relation.

Abstract:
To a Lie groupoid over a compact base, the associated group of bisection is an (infinite-dimensional) Lie group. Moreover, under certain circumstances one can reconstruct the Lie groupoid from its Lie group of bisections. In the present article we consider functorial aspects of these construction principles. The first observation is that this procedure is functorial (for morphisms fixing the base). Moreover, it gives rise to an adjunction between the category of Lie groupoids over a fixed base and the category of Lie groups acting on the base. In the last section we then show how to promote this adjunction to almost an equivalence of categories.

Abstract:
In this review I consider several different issues related to inflation. I will begin with the wave function of the Universe. This issue is pretty old, but recently there were some new insights based on the theory of the self-reproducing inflationary universe. Then we will discuss stationarity of inflationary universe and the possibility to make predictions in the context of quantum cosmology using stochastic approach to inflation. Returning to more pragmatic aspects of inflationary theory, we will discuss inflationary models with $\Omega < 1$. Finally, we will describe several aspects of the theory of reheating of the Universe based on the effect of parametric resonance.

Abstract:
We derive an analogue of the Berry phase associated with inflationary cosmological perturbations of quantum mechanical origin by obtaining the corresponding wavefunction. We have further shown that cosmological Berry phase can be completely envisioned through the observable parameters, viz. spectral indices. Finally, physical significance of this phase is discussed from the point of view of theoretical and observational aspects with some possible consequences of this quantity in inflationary cosmology.

Abstract:
Oncologic mandibular reconstruction has changed significantly over the years and continues to evolve with the introduction of newer technologies and techniques. Patient demographic, reconstructive, and complication data were obtained from a prospectively maintained clinical database of patients who underwent head and neck reconstruction at our institution. The free fibular flap is now considered the gold standard for mandibular reconstruction. However, in patients with multiple comorbidities, lengthy procedures may be less optimal and pedicled flaps, with specific modifications, can yield reasonable outcomes. Technical aspects and comorbidity profiles are examined in the oncological mandibular reconstruction cohort. 1. Introduction Oncologic mandibular reconstruction has changed significantly over the years and continues to evolve with the introduction of newer technologies and techniques. The goals of reconstruction, following oncologic resection, are both functional and aesthetic. Functional considerations include successful wound closure of the oropharynx, preservation of a patent upper airway, phonation, mastication, and potential for dental rehabilitation, in addition to restoration of aesthetic impairment. The principles that guide oncologic mandibular reconstruction focus on optimizing outcomes and identifying ideal flap reconstruction, with consideration of patient co-morbidities and reconstructive requirements. Reconstruction with pedicled pectoralis major myocutaneous and deltopectoral flaps used to be the standard of care and continues to be in selected cases [1]. Free-flap reconstruction of oncologic defects has become the modern standard of care, largely due to superior functional and aesthetic outcomes [2]. However, in patients with multiple co-morbidities, who cannot tolerate lengthy surgery or fluid shifts, pedicled flaps may be best suited to meet the reconstructive requirements. 2. Materials and Methods Patient demographic, reconstructive, and complication data were obtained from a prospectively maintained clinical database of patients who underwent head and neck reconstruction at the University of Illinois at Chicago Medical Center. Institutional Review Board approval was obtained. All patients who underwent oncological mandibular reconstruction were included in this study, representing a single surgeon’s experience (A. K. Antony) from October of 2010 to May of 2011. Medical records were retrospectively reviewed to further characterize comorbid conditions and technical modifications employed to optimize results. The following