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   Alonso, Ricardo_J; Čolić, Milana; Gamba, Irene_M (January 2024, Journal of Statistical Physics)

   Abstract From a unified vision of vector valued solutions in weighted Banach spaces, this paper establishes the existence and uniqueness for space homogeneous Boltzmann bi-linear systems with conservative collisional forms arising in complex gas dynamical structures. This broader vision is directly applied to dilute multi-component gas mixtures composed of both monatomic and polyatomic gases. Such models can be viewed as extensions of scalar Boltzmann binary elastic flows, as much as monatomic gas mixtures with disparate masses and single polyatomic gases, providing a unified approach for vector valued solutions in weighted Banach spaces. Novel aspects of this work include developing the extension of a general ODE theory in vector valued weighted Banach spaces, precise lower bounds for the collision frequency in terms of the weighted Banach norm, energy identities, angular or compact manifold averaging lemmas which provide coerciveness resulting into global in time stability, a new combinatorics estimate forp-binomial forms producing sharper estimates for thek-moments of bi-linear collisional forms. These techniques enable the Cauchy problem improvement that resolves the model with initial data corresponding to strictly positive and bounded initial vector valued mass and total energy, in addition to only a$$2^+$$ $2^{+}$moment determined by the hard potential rates discrepancy, a result comparable in generality to the classical Cauchy theory of the scalar homogeneous Boltzmann equation. 

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2. Neural Network Method for Integral Fractional Laplace Equations

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   Hao, Zhaopeng; Park, Moongyu; Cai, Zhiqiang (January 2023, East Asian Journal on Applied Mathematics)

   A neural network method for fractional order diffusion equations with integral fractional Laplacian is studied. We employ the Ritz formulation for the corresponding fractional equation and then derive an approximate solution of an optimization problem in the function class of neural network sets. Connecting the neural network sets with weighted Sobolev spaces, we prove the convergence and establish error estimates of the neural network method in the energy norm. To verify the theoretical results, we carry out numerical experiments and report their outcome. 

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   Feng, Xiaobing; Sutton, Mitchell (July 2021, Analysis and Applications)

   null (Ed.)

   This paper presents a self-contained new theory of weak fractional differential calculus in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives; it also helps to unify multiple existing fractional derivative definitions and characterize what functions are fractionally differentiable. Various calculus rules including a fundamental theorem calculus, product and chain rules, and integration by parts formulas are established for weak fractional derivatives. Additionally, relationships with classical fractional derivatives and detailed characterizations of weakly fractional differentiable functions are also established. Furthermore, the notion of weak fractional derivatives is also systematically extended to general distributions instead of only to some special distributions. This new theory lays down a solid theoretical foundation for systematically and rigorously developing new theories of fractional Sobolev spaces, fractional calculus of variations, and fractional PDEs as well as their numerical solutions in subsequent works. 

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4. Formal Privacy for Functional Data with Gaussian Perturbations

   Ardalan Mirshani, Matthew Reimherr (January 2019, Proceedings of the 36th International Conference on Machine Learning)

   Motivated by the rapid rise in statistical tools in Functional Data Analysis, we consider the Gaussian mechanism for achieving differential privacy (DP) with parameter estimates taking values in a, potentially infinite-dimensional, separable Banach space. Using classic results from probability theory, we show how densities over function spaces can be utilized to achieve the desired DP bounds. This extends prior results of Hall et al (2013) to a much broader class of statistical estimates and summaries, including “path level" summaries, nonlinear functionals, and full function releases. By focusing on Banach spaces, we provide a deeper picture of the challenges for privacy with complex data, especially the role regularization plays in balancing utility and privacy. Using an application to penalized smoothing, we highlight this balance in the context of mean function estimation. Simulations and an application to {diffusion tensor imaging} are briefly presented, with extensive additions included in a supplement. 

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5. The Role of Neural Network Activation Functions

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   Parhi, Rahul; Nowak, Robert D (January 2020, IEEE Signal Processing Letters)

   A wide variety of activation functions have been proposed for neural networks. The Rectified Linear Unit (ReLU) is especially popular today. There are many practical reasons that motivate the use of the ReLU. This paper provides new theoretical characterizations that support the use of the ReLU, its variants such as the leaky ReLU, as well as other activation functions in the case of univariate, single-hidden layer feedforward neural networks. Our results also explain the importance of commonly used strategies in the design and training of neural networks such as “weight decay” and “path-norm” regularization, and provide a new justification for the use of “skip connections” in network architectures. These new insights are obtained through the lens of spline theory. In particular, we show how neural network training problems are related to infinite-dimensional optimizations posed over Banach spaces of functions whose solutions are well-known to be fractional and polynomial splines, where the particular Banach space (which controls the order of the spline) depends on the choice of activation function. 

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