Igniting the Spark: Creating Effective Next Generation Boards

For family foundations and a growing number of donor-advised funds, preparing the next generation for involvement brings special concerns -- and exciting opportunities. Succession is reported to be the single most important issue facing family foundations, according to nearly half (48%) of respondents to the Association for Small Foundations 2011 Foundation Operations and Management Report. At the same time, there is a broad range of experience in families with regard to how next generation family members are involved in family philanthropy. According to the National Center for Family Philanthropy's 2011 study, Current Practices in Family Foundations, a near equal number of respondents "strongly agreed" (34%) or "strongly disagreed" (33%) with the statement "next generation members are playing a significant role in the foundation.

Fluctuation Moments for Regular Functions of Wigner Matrices

We compute the deterministic approximation for mixed fluctuation moments of products of deterministic matrices and general Sobolev functions of Wigner matrices. Restricting to polynomials, our formulas reproduce recent results of [Male, Mingo, Pech\'e, Speicher 2022], showing that the underlying combinatorics of non-crossing partitions and annular non-crossing permutations continue to stay valid beyond the setting of second-order free probability theory. The formulas obtained further characterize the variance in the functional central limit theorem obtained recently in the companion paper [Reker 2023].Comment: 52 pages (including appendix), 20 figure

On the operator norm of a Hermitian random matrix with correlated entries

We consider a correlated $N\times N$ Hermitian random matrix with a polynomially decaying metric correlation structure. By calculating the trace of the moments of the matrix and using the summable decay of the cumulants, we show that its operator norm is stochastically dominated by one

Multi-Point Functional Central Limit Theorem for Wigner Matrices

Consider the random variable $\mathrm{Tr}( f_1(W)A_1\dots f_k(W)A_k)$ where $W$ is an $N\times N$ Hermitian Wigner matrix, $k\in\mathbb{N}$, and choose (possibly $N$-dependent) regular functions $f_1,\dots, f_k$ as well as bounded deterministic matrices $A_1,\dots,A_k$. We give a functional central limit theorem showing that the fluctuations around the expectation are Gaussian. Moreover, we determine the limiting covariance structure and give explicit error bounds in terms of the scaling of $f_1,\dots,f_k$ and the number of traceless matrices among $A_1,\dots,A_k$, thus extending the results of [Cipolloni, Erd\H{o}s, Schr\"oder 2023] to products of arbitrary length $k\geq2$. As an application, we consider the fluctuation of $\mathrm{Tr}(\mathrm{e}^{\mathrm{i} tW}A_1\mathrm{e}^{-\mathrm{i} tW}A_2)$ around its thermal value $\mathrm{Tr}(A_1)\mathrm{Tr}(A_2)$ when $t$ is large and give an explicit formula for the variance.Comment: 48 pages (including appendix

Short-time behavior of solutions to L\'evy-driven SDEs

We consider solutions of L\'evy-driven stochastic differential equations of the form $\mathrm{d} X_t=\sigma(X_{t-})\mathrm{d} L_t$, $X_0=x$ where the function $\sigma$ is twice continuously differentiable and maximal of linear growth and the driving L\'evy process $L=(L_t)_{t\geq0}$ is either vector or matrix-valued. While the almost sure short-time behavior of L\'evy processes is well-known and can be characterized in terms of the characteristic triplet, there is no complete characterization of the behavior of the process $X$. Using methods from stochastic calculus, we derive limiting results for stochastic integrals of the from $\smash{t^{-p}\int_{0+}^t\sigma(X_{t-})\mathrm{d} L_t}$ to show that the behavior of the quantity $t^{-p}(X_t-X_0)$ for $t\downarrow0$ almost surely mirrors the behavior of $t^{-p}L_t$. Generalizing $t^p$ to a suitable function $f:[0,\infty)\rightarrow\mathbb{R}$ then yields a tool to derive explicit LIL-type results for the solution from the behavior of the driving L\'evy process

Lemon: an MPI parallel I/O library for data encapsulation using LIME

We introduce Lemon, an MPI parallel I/O library that is intended to allow for efficient parallel I/O of both binary and metadata on massively parallel architectures. Motivated by the demands of the Lattice Quantum Chromodynamics community, the data is stored in the SciDAC Lattice QCD Interchange Message Encapsulation format. This format allows for storing large blocks of binary data and corresponding metadata in the same file. Even if designed for LQCD needs, this format might be useful for any application with this type of data profile. The design, implementation and application of Lemon are described. We conclude with presenting the excellent scaling properties of Lemon on state of the art high performance computers