A Second Adjoint Theorem for SL(2,R)

We formulate a second adjoint theorem in the context of tempered representations of real reductive groups, and prove it in the case of SL(2,R).Comment: 38 page

The mathematics of functional differentiation under conservation constraint

The mathematics of K-conserving functional differentiation, with K being the integral of some invertible function of the functional variable, is clarified. The most general form for constrained functional derivatives is derived from the requirement that two functionals that are equal over a restricted domain have equal derivatives over that domain. It is shown that the K-conserving derivative formula is the one that yields no effect of K-conservation on the differentiation of K-independent functionals, which gives the basis for its generalization for multiple constraints. Connections with the derivative with respect to the shape of the functional variable and with the shape-conserving derivative, together with their use in the density-functional theory of many-electron systems, are discussed. Yielding an intuitive interpretation of K-conserving functional derivatives, it is also shown that K-conserving derivatives emerge as directional derivatives along K-conserving paths, which is achieved via a generalization of the Gateaux derivative for that kind of paths. These results constitute the background for the practical application of K-conserving differentiation.Comment: final version, published in J Math Chem; with an Appendix with the proof of (17) added, and some errata to [1] inserte

A proof of convergence of multi-class logistic regression network

This paper revisits the special type of a neural network known under two names. In the statistics and machine learning community it is known as a multi-class logistic regression neural network. In the neural network community, it is simply the soft-max layer. The importance is underscored by its role in deep learning: as the last layer, whose autput is actually the classification of the input patterns, such as images. Our exposition focuses on mathematically rigorous derivation of the key equation expressing the gradient. The fringe benefit of our approach is a fully vectorized expression, which is a basis of an efficient implementation. The second result of this paper is the positivity of the second derivative of the cross-entropy loss function as function of the weights. This result proves that optimization methods based on convexity may be used to train this network. As a corollary, we demonstrate that no $L^2$-regularizer is needed to guarantee convergence of gradient descent

Electronic coupling mechanisms and characteristics for optically nonlinear photoactive nanomaterials

In a range of nanophotonic energy harvesting materials, resonance energy transfer (RET) is the mechanism for the intermolecular and intramolecular transfer of electronic excitation following the absorption of ultraviolet/visible radiation. In the nonlinear intensity regime, suitably designed materials can exhibit two quite different types of mechanism for channeling the excitation energy to an acceptor that is optically transparent at the input frequency. Both mechanisms are associated with two-photon optical excitation - of either a single donor, or a pair of donor chromophores, located close to the acceptor. In the former case the mechanism is two-photon resonance energy transfer, initiated by two-photon absorption at a donor, and followed by RET directly to the acceptor. The probability for fulfilling the initial conditions for this mechanism (for the donors to exhibit two-photon absorption) is enhanced at high levels of optical input. In the latter twin-donor mechanism, following initial one-photon excitations of two electronically distinct donors, energy pooling results in a collective channeling of their energy to an acceptor chromophore. This mechanism also becomes effective under high intensity conditions due to the enhanced probability of exciting donor chromophores within close proximity of each other and the acceptor. In this paper we describe the detailed balance of factors that determines the favored mechanism for these forms of optical nonlinearity, especially electronic factors. Attention is focused on dendrimeric nanostar materials with a propensity for optical nonlinearity

Analyticity in spaces of convergent power series and applications

We study the analytic structure of the space of germs of an analytic function at the origin of \ww C^{\times m} , namely the space \germ{\mathbf{z}} where \mathbf{z}=\left(z\_{1},\cdots,z\_{m}\right) , equipped with a convenient locally convex topology. We are particularly interested in studying the properties of analytic sets of \germ{\mathbf{z}} as defined by the vanishing locus of analytic maps. While we notice that \germ{\mathbf{z}} is not Baire we also prove it enjoys the analytic Baire property: the countable union of proper analytic sets of \germ{\mathbf{z}} has empty interior. This property underlies a quite natural notion of a generic property of \germ{\mathbf{z}} , for which we prove some dynamics-related theorems. We also initiate a program to tackle the task of characterizing glocal objects in some situations