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Thermodynamics Problem-Solving in Physical Chemistry: Study Guide and Map is an innovative and unique workbook that guides physical chemistry students through the decision-making process to assess a problem situation, create appropriate solutions, and gain confidence through practice in solving physical chemistry problems. The workbook includes six major sections with 20-30 solved problems in each section that span from easy, single-objective questions to difficult, multistep analysis problems. Each section of the workbook contains key points that highlight major features of the topic, to remind students of what they need to apply to solve problems in the topic area. Key Features: Includes a visual map that shows how all the \"equations\" used in thermodynamics are connected and how they are derived from the three major energy laws. Acts as a guide in deriving the correct solution to a problem. Illustrates the questions students should ask themselves about the critical features of the concepts to solve problems in physical chemistry Can be used as a stand-alone product for review of thermodynamics questions for major tests.

We study the Feynman integral for the three-banana graph defined as the scalar two-point self-energy at three-loop order. The Feynman integral is evaluated for all identical internal masses in two space-time dimensions. Two calculations are given for the Feynman integral: one based on an interpretation of the integral as an inhomogeneous solution of a classical Picard–Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the Feynman integral is a family of regulator periods associated to a family of $K3$ surfaces. We show that the integral is given by a sum of elliptic trilogarithms evaluated at sixth roots of unity. This elliptic trilogarithm value is related to the regulator of a class in the motivic cohomology of the $K3$ family. We prove a conjecture by David Broadhurst which states that at a special kinematical point the Feynman integral is given by a critical value of the Hasse–Weil $L$-function of the $K3$ surface. This result is shown to be a particular case of Deligne’s conjectures relating values of $L$-functions inside the critical strip to periods.

This review paper collects several results that form part of the theoretical foundation of isogeometric methods. We analyse variational techniques for the numerical resolution of PDEs based on splines or NURBS and we provide optimal approximation and error estimates in several cases of interest. The theory presented also includes estimates for T-splines, which are an extension of splines allowing for local refinement. In particular, we focus our attention on elliptic and saddle point problems, and we define spline edge and face elements. Our theoretical results are demonstrated by a rich set of numerical examples. Finally, we discuss implementation and efficiency together with preconditioning issues for the final linear system.

Multiple imputation is a straightforward method for handling missing data in a principled fashion. This paper presents an overview of multiple imputation, including important theoretical results and their practical implications for generating and using multiple imputations. A review of strategies for generating imputations follows, including recent developments in flexible joint modeling and sequential regression/chained equations/fully conditional specification approaches. Finally, we compare and contrast different methods for generating imputations on a range of criteria before identifying promising avenues for future research.

Statistical modeling is a powerful tool for developing and testing theories by way of causal explanation, prediction, and description. In many disciplines there is near-exclusive use of statistical modeling for causal explanation and the assumption that models with high explanatory power are inherently of high predictive power. Conflation between explanation and prediction is common, yet the distinction must be understood for progressing scientific knowledge. While this distinction has been recognized in the philosophy of science, the statistical literature lacks a thorough discussion of the many differences that arise in the process of modeling for an explanatory versus a predictive goal. The purpose of this article is to clarify the distinction between explanatory and predictive modeling, to discuss its sources, and to reveal the practical implications of the distinction to each step in the modeling process.

Under mild regularity assumptions, the transport problem is stable in the following sense: if a sequence of optimal transport plans π¹, π², . . . converges weakly to a transport plan π, then π is also optimal (between its marginals). Alfonsi, Corbetta and Jourdain (Ann. Inst. Henri Poincaré Probab. Stat. 56 (2020) 1706–1729) asked whether the same property is true for the martingale transport problem. This question seems particularly pressing since martingale transport is motivated by robust finance where data is naturally noisy. On a technical level, stability in the martingale case appears more intricate than for classical transport since martingale optimal transport plans are not characterized by a “monotonicity”-property of their supports. In this paper we give a positive answer and establish stability of the martingale transport problem. As a particular case, this recovers the stability of the left curtain coupling established by Juillet (In Séminaire de Probabilités XLVIII (2016) 13–32 Springer). An important auxiliary tool is an unconventional topology which takes the temporal structure of martingales into account. Our techniques also apply to the the weak transport problem introduced by Gozlan, Roberto, Samson and Tetali.

The performance of spectral clustering can be considerably improved via regularization, as demonstrated empirically in Amini et al. [Ann. Statist. 41 (2013) 2097-2122]. Here, we provide an attempt at quantifying this improvement through theoretical analysis. Under the stochastic block model (SBM), and its extensions, previous results on spectral clustering relied on the minimum degree of the graph being sufficiently large for its good performance. By examining the scenario where the regularization parameter τ is large, we show that the minimum degree assumption can potentially be removed. As a special case, for an SBM with two blocks, the results require the maximum degree to be large (grow faster than log n) as opposed to the minimum degree. More importantly, we show the usefulness of regularization in situations where not all nodes belong to well-defined clusters. Our results rely on a 'bias-variance' -like trade-off that arises from understanding the concentration of the sample Laplacian and the eigengap as a function of the regularization parameter. As a byproduct of our bounds, we propose a data-driven technique DKest (standing for estimated Davis–Kahan bounds) for choosing the regularization parameter. This technique is shown to work well through simulations and on a real data set.

In graph theory, a square path is a path in which each vertex is adjacent to the vertices two places ahead and behind. A square cycle is defined analogously. A path or cycle is Hamilton in a graph G if it includes every vertex of G. Pósa conjectured that every graph with minimum degree at least 2/3 its order has a Hamilton square cycle. This dissertation presents a proof of Pósa’s conjecture for graphs on at least 62,000 vertices, an improvement over the previous state of the art. I also explore some auxiliary questions that emerge naturally when exploring square paths. For example, it turns out that there exists a natural analogue of a component, called a triangle component, that seems to describe connectivity by square paths. This dissertation includes some initial results regarding triangle components and suggests avenues for further research. Also, one might seek to rearrange a square path using prefix reversals, and this leads to a novel combinatorial problem concerning prefix reversals of binary and quaternary strings. This dissertation presents a complete result for the binary case and some initial results for the quaternary case.

We investigate the energy-critical nonlinear Schr¨odinger equation, the generalized Korteweg– de Vries equation, and the energy-critical nonlinear wave equation: respectively,where u(t, x) is a function in spacetime Rt × R d x for d as above.For each of these models (assuming k ≥ 8 for (∗∗)), we prove that solutions exhibit pointwise-in-time dispersive decay, requiring only that initial data lie in a scaling-critical space. In particular, global, scattering solutions will decay at the same rate as the underlying linear model. In addition, for the model (∗), we demonstrate dispersive decay for the final-state problem, assuming only that scattering data lie in a scaling-critical space. Finally, for the model (∗∗) in the case 4 ≤ k < 8 (including the mass-critical k = 4) we prove dispersive decay with the additional assumption that initial data lie in a near-scaling-critical space.[Equation omitted].

Bi-Hermitian is an extension of Kähler geometry which arose first in the study of supersymmetricσ-models in the 1980s. It was later rediscovered by Gualtieri, within Hitchin’s generalized geometry program, and rechristened as generalized Kähler geometry. In this thesis, we study a wide variety of curvatures that appear in generalized Kähler geometry.Using the structure provided by the generalized Kähler geometry we identify Chern connections on the canonical bundles of the generalized complex structures and relate these to the Bismut connections of the underlying bi-Hermitian manifold. We then identify these connections as components of generalized Chern connections and as a result obtain symmetries of the generalized complex structures which may be described in terms of bi-Hermitian data. We identify a second type of generalized Chern connection arising from the holomorphic Poisson structures present on any generalized Kähler manifold. Using a description of these involving bi-Hermitian data we give a novel reformulation of generalized Kähler-Ricci flow in terms of purely generalized geometric data.The main tool that we use to accomplish this is the Roytenberg algebra, which is the algebra of functions on the graded symplectic manifold corresponding to the underlying Courant algebroid. We describe the structure inherited by the Roytenberg algebra in the presence of a generalized Kähler structure and give a characterization of the integrability condition in terms of the Roytenberg algebra.