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In this article we give an overview of the Plancherel theory for Riemannian symmetric spaces Z = G/K. In particular we illustrate recently developed methods in Plancherel theory for real spherical spaces by explicating them for Riemannian symmetric spaces, and we explain how Harish-Chandra's Plancherel theorem for Z can be proven from these methods.

In this paper we give an overview on $L^p$-factorizations of Lie group representations and introduce the notion of smooth $L^p$-factorization., Comment: This article is dedicated to the fond memories of Gerrit van Dijk

Chirped pulse amplification (CPA) and subsequent nonlinear optical (NLO) systems constitute the backbone of myriad advancements in semiconductor and additive manufacturing, communication networks, biology and medicine, defense and national security, and a host of other sectors over the past decades. Accurately and efficiently modeling CPA and NLO-based laser systems is challenging because of the multitude of coupled linear and nonlinear processes and high variability in simulation frameworks. The lack of fully-integrated models severely hampers further advances in tailoring existing or materializing new CPA+NLO systems. Such tools are the key to enabling emerging optimization and inverse design approaches reliant on data-driven machine learning methods. Here, we present a modular start-to-end software model encompassing an array of amplifier designs and nonlinear optics techniques. The simulator renders time- and frequency-resolved electromagnetic fields alongside essential physical characteristics of energy, fluence, and spectral distribution. To demonstrate its robustness and real-world applicability -- specifically, reverse engineering, system optimization, and inverse design -- we present a case study on the LCLS-II photo-injector laser, representative of a high-power and spectro-temporally non-trivial CPA+NLO system.

Our concern is with Riemannian symmetric spaces $Z=G/K$ of the non-compact type and more precisely with the Poisson transform $\mathcal{P}_\lambda$ which maps generalized functions on the boundary $\partial Z$ to $\lambda$-eigenfunctions on $Z$. Special emphasis is given to a maximal unipotent group $N0})^r

Complex socio-environmental interdependencies drive biological invasions, causing damages across large spatial scales. For widespread invasions, targeting of management activities based on optimization approaches may fail due to computational or data constraints. Here we evaluate an alternative approach that embraces complexity by representing the invasion as a network and using network structure to inform management locations. We compare optimal versus network-guided invasive species management at a landscape-scale, considering siting of boat decontamination stations targeting 1.6 million boater movements among 9,182 lakes in Minnesota, USA. Studying performance for 58 counties, we find that when full information is known on invasion status and boater movements, the best-performing network-guided metric achieves a median and lower quartile performance of 100% of optimal. We also find that performance remains relatively high using different network metrics or with less information (median above 80% and lower quartile above 60% of optimal for most metrics), but is more variable, particularly at the lower quartile. Additionally, performance is generally stable across counties with varying lake counts, suggesting viability for large-scale invasion management., Comment: 40 pages, 8 figures, 7 tables

We give an example of a semisimple symmetric space $G/H$ and an irreducible representation of $G$ which has multiplicity 1 in $L^2(G/H)$ and multiplicity 2 in $C^\infty(G/H)$.

We formulate and prove a Paley-Wiener theorem for Harish-Chandra modules for a real reductive group. As a corollary we obtain a new and elementary proof of the Helgason conjecture., Comment: Submitted version; with two appendices on the Helgason conjecture and an application

We explain by elementary means why the existence of a discrete series representation of a real reductive group $G$ implies the existence of a compact Cartan subgroup of $G$. The presented approach has the potential to generalize to real spherical spaces.

Several methods have been proposed in the literature to solve reliability-based optimization problems, where failure probabilities are design constraints. However, few methods address the problem of life-cycle cost or risk optimization, where failure probabilities are part of the objective function. Moreover, few papers in the literature address time-variant reliability problems in life-cycle cost or risk optimization formulations; in particular, because most often computationally expensive Monte Carlo simulation is required. This paper proposes a numerical framework for solving general risk optimization problems involving time-variant reliability analysis. To alleviate the computational burden of Monte Carlo simulation, two adaptive coupled surrogate models are used: the first one to approximate the objective function, and the second one to approximate the quasi-static limit state function. An iterative procedure is implemented for choosing additional support points to increase the accuracy of the surrogate models. Three application problems are used to illustrate the proposed approach. Two examples involve random load and random resistance degradation processes. The third problem is related to load-path dependent failures. This subject had not yet been addressed in the context of risk-based optimization. It is shown herein that accurate solutions are obtained, with extremely limited numbers of objective function and limit state functions calls.

We define Sobolev norms of arbitrary real order for a Banach representation $(\pi, E)$ of a Lie group, with regard to a single differential operator $D=d\pi(R^2+\Delta)$. Here, $\Delta$ is a Laplace element in the universal enveloping algebra, and $R>0$ depends explicitly on the growth rate of the representation. In particular, we obtain a spectral gap for $D$ on the space of smooth vectors of $E$. The main tool is a novel factorization of the delta distribution on a Lie group., Comment: 10 pages, to appear in Journal of Functional Analysis

Given a unimodular real spherical space $Z=G/H$ we construct for each boundary degeneration $Z_I=G/H_I$ of $Z$ a Bernstein morphism $B_I: L^2(Z_I)_{\rm disc }\to L^2(Z)$. We show that $B:=\bigoplus_I B_I$ provides an isospectral $G$-equivariant morphism onto $L^2(Z)$. Further, the maps $B_I$ are finite linear combinations of orthogonal projections which translates in the known cases where $Z$ is a group or a symmetric space into the familiar Maass-Selberg relations. As a corollary we obtain that $L^2(Z)_{\rm disc }\neq \emptyset$ provided that ${\mathfrak h}^\perp$ contains elliptic elements in its interior., Comment: 101 pages. Final version. Accepted to J. Amer. Math. Soc

We characterize a transition from normal to ballistic diffusion in a bouncing ball dynamics. The system is composed of a particle, or an ensemble of non-interacting particles, experiencing elastic collisions with a heavy and periodically moving wall under the influence of a constant gravitational field. The dynamics lead to a mixed phase space where chaotic orbits have a free path to move along the velocity axis, presenting a normal diffusion behavior. Depending on the control parameter, one can observe the presence of featured resonances, known as accelerator modes, that lead to a ballistic growth of velocity. Through statistical and numerical analysis of the velocity of the particle, we are able to characterize a transition between the two regimes, where transport properties were used to characterize the scenario of the ballistic regime. Also, in an analysis of the probability of an orbit to reach an accelerator mode as a function of the velocity, we observe a competition between the normal and ballistic transport in the mid range velocity., Comment: To appear in Physical Review E, 2018

Let $Z=G/H$ be the homogeneous space of a real reductive group and a unimodular real spherical subgroup, and consider the regular representation of $G$ on $L^2(Z)$. It is shown that all representations of the discrete series, that is, the irreducible subrepresentations of $L^2(Z)$, have infinitesimal characters which are real and belong to a lattice. Moreover, let $K$ be a maximal compact subgroup of $G$. Then each irreducible representation of $K$ occurs in a finite set of such discrete series representations only. Similar results are obtained for the twisted discrete series, that is, the discrete components of the space of square integrable sections of a line bundle, given by a unitary character on an abelian extension of $H$., Comment: To appear in GAFA

Nowadays the Science progress depends on the numerical calculus, due to the possibility of obtention of solutions using simulations which would be impracticable, or even impossible, to be analitically obtained. In this aspect, it becomes important to dominate the fundamental tools of numerical calculus applied to some programming language, Fortran for example. In this work we presented few basic concepts of the Fortran 77. We show how this concepts can be applied in the simulation and analysis of physical systems, trying to offer a first contact, a guide, with this programming language. We presented the construction of routines that model the behavior of spring-mass system, trigonometric functions and a more complex system, the standart map. We presented a discussion about meaningful learning involving physical concepts and programming in Fortran., Comment: 12 pages, in Portuguese, 12 figures

Nowadays, divertors are used in the main tokamaks to control the magnetic field and to improve the plasma confinement. In this article, we present analytical symplectic maps describing Poincar\'e maps of the magnetic field lines in confined plasmas with a single null poloidal divertor. Initially, we present a divertor map and the tokamap for a diverted configuration. We also introduce the Ullmann map for a diverted plasma, whose control parameters are determined from tokamak experiments. Finally, an explicit, area-preserving and integrable magnetic field line map for a single-null divertor tokamak is obtained using a trajectory integration method to represent toroidal equilibrium magnetic surfaces. In this method, we also give examples of onset of chaotic field lines at the plasma edge due to resonant perturbations.

We provide $L^p$-versus $L^\infty$-bounds for eigenfunctions on a real spherical space $Z$ of wavefront type. It is shown that these bounds imply a non-trivial error term estimate for lattice counting on $Z$. The paper also serves as an introduction to geometric counting on spaces of the mentioned type. Section 7 on higher rank is new and extends the result from v1 to higher rank. Final version. To appear in Acta Math. Sinica., Comment: 46 pages

If ${\mathfrak g}$ is a real reductive Lie algebra and ${\mathfrak h} < {\mathfrak g}$ is a subalgebra, then $({\mathfrak g}, {\mathfrak h})$ is called real spherical provided that ${\mathfrak g} = {\mathfrak h} + {\mathfrak p}$ for some choice of a minimal parabolic subalgebra ${\mathfrak p} \subset {\mathfrak g}$. In this paper we classify all real spherical pairs $({\mathfrak g}, {\mathfrak h})$ where ${\mathfrak g}$ is semi-simple but not simple and ${\mathfrak h}$ is a reductive real algebraic subalgebra. The paper is based on the classification of the case where ${\mathfrak g}$ is simple (see arXiv:1609.00963) and generalizes the results of Brion and Mikityuk in the (complex) spherical case., Comment: Extended revised version. Section 6 and Appendix B are new. To appear in Transformation Groups. 40p

Let $Z$ be a unimodular real spherical space. We develop a theory of constant terms for tempered functions on $Z$ which parallels the work of Harish-Chandra. The constant terms $f_I$ of an eigenfunction $f$ are parametrized by subsets $I$ of the set $S$ of spherical roots which determine the fine geometry of $Z$ at infinity. Constant terms are transitive i.e. $(f_J)_I=f_I$ for $I\subset J$, and our main result is a quantitative bound of the difference $f-f_I$, which is uniform in the parameter of the eigenfunction., Comment: Final version. 72 pages. Accepted to IMRN

We give an introduction to basic harmonic analysis and representation theory for homogeneous spaces $Z=G/H$ attached to a real reductive Lie group $G$. A special emphasis is made to the case where $Z$ is real spherical., Comment: Shortened title, typos fixed, more details on dual smooth Frobenius reciprocity (now Lemma 6.6). 38 pages, lecture notes for the Sanya meeting on spherical varieties. To appear in Acta Math. Sinica

This paper gives a classification of all pairs $(\mathfrak g, \mathfrak h)$ with $\mathfrak g$ a simple real Lie algebra and $\mathfrak h < \mathfrak g$ a reductive subalgebra for which there exists a minimal parabolic subalgebra $\mathfrak p < \mathfrak g$ such that $\mathfrak g = \mathfrak h + \mathfrak p$ as vector sum., Comment: 44 pages, 8 tables. Minor corrections in Table 8. Published version

We present a comprehensive study of laser pulse amplification with respect to operation regimes, gain dynamics, and highly stable operation points of Ho:YLF regenerative amplifiers (RAs). The findings are expected to be more generic than for this specific case. Operation regimes are distinguished with respect to pulse energy and the appearance of pulse instability, and are studied as a function of the repetition rate, seed energy, and pump intensity. The corresponding gain dynamics are presented, identifying highly stable operation points related to high gain build -up during pumping and high gain depletion during pulse amplification. These operation points are studied numerically and experimentally as a function of several parameters, thereby achieving, for our Ho:YLF RA, highly stable output pulses with measured fluctuations of only 0.19% (standard deviation).

In this paper, we study rationality properties of reductive group actions which are defined over an arbitrary field of characteristic zero. Thereby, we unify Luna's theory of spherical systems and Borel-Tits' theory of reductive groups. In particular, we define for any reductive group action a generalized Tits index whose main constituents are a root system and a generalization of the anisotropic kernel. The index controls to a large extent the behavior at infinity (i.e., embeddings). For k-spherical varieties (i.e., where a minimal parabolic has an open orbit) we obtain explicit (wonderful) completions of the set of rational points. For local fields this means honest compactifications generalizing the maximal Satake compactification of a symmetric space. Our main tool is a k-version of the local structure theorem., Comment: v1: 62 pages; v2: 66 pages, some material added (especially 13.5-13.8), minor corrections; v3: 69 pages, section 12 on boundary degenerations rewritten, mistake in Prop. 13.1 fixed, Ex. 13.9 and Prop. 14.7 added

We demonstrate a Ho:YLF regenerative amplifier (RA) overcoming bifurcation instability and consequently achieving high extraction energies of 6.9 mJ at a repetition rate of 1 kHz with pulse-to-pulse fluctuations of 1.1%. Measurements of the output pulse energy, corroborated by numerical simulations, identify an operation point that allows high-energy pulse extraction at a minimum noise level. Complete suppression of the onset of bifurcation was achieved by gain saturation after each pumping cycle in the Ho:YLF crystal via lowering the repetition rate and cooling the crystal. Even for moderate cooling, a significant temperature dependence of the Ho:YLF RA performance was observed.

Let G/H be a unimodular real spherical space which is either absolutely spherical or wave-front. It is shown that every tempered representation of G/H embeds into a relative discrete series of a boundary degeneration of G/H. If in addition G/H is of wave-front type it follows that the tempered representation is parabolically induced from a discrete series representation of a lower dimensional real spherical space. Final version. To appear in Acta Math., Comment: 64 pages

By the collective name of {\it lattice counting} we refer to a setup introduced in Duke-Rudnick-Sarnak that aim to establish a relationship between arithmetic and randomness in the context of affine symmetric spaces. In this paper we extend the geometric setup from symmetric to real spherical spaces and continue to develop the approach with harmonic analysis which was initiated in Duke-Rudnick-Sarnak., Comment: Extended and revised version with 4 additional pages. To appear in Documenta math

We apply the local structure theorem and the polar decomposition to a real spherical space Z=G/H and control the volume growth on Z. We define the Harish-Chandra Schwartz space on Z. We give a geometric criterion to ensure $L^p$-integrability of matrix coefficients on Z., Comment: Additional material of 4 pages added. To appear in J. Funct. Analysis

Let $H$ be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold $X$ and a real algebraic bundle $\mathcal{E}$ on $X$. Let $\mathfrak{h}$ be the Lie algebra of $H$. Let $\mathcal{S}(X,\mathcal{E})$ be the space of Schwartz sections of $\mathcal{E}$. We prove that $\mathfrak{h}\mathcal{S}(X,\mathcal{E})$ is a closed subspace of $\mathcal{S}(X,\mathcal{E})$ of finite codimension. We give an application of this result in the case when $H$ is a real spherical subgroup of a real reductive group $G$. We deduce an equivalence of two old conjectures due to Casselman: the automatic continuity and the comparison conjecture for zero homology. Namely, let $\pi$ be a Casselman-Wallach representation of $G$ and $V$ be the corresponding Harish-Chandra module. Then the natural morphism of coinvariants $V_{\mathfrak{h}}\to \pi_{\mathfrak{h}}$ is an isomorphism if and only if any linear $\mathfrak{h}$-invariant functional on $V$ is continuous in the topology induced from $\pi$. The latter statement is known to hold in two important special cases: if $H$ includes a symmetric subgroup, and if $H$ includes the nilradical of a minimal parabolic subgroup of $G$., Comment: v4: version appearing in Math. Z + erratum added in the end

Let Z be an algebraic homogeneous space Z=G/H attached to real reductive Lie group G. We assume that Z is real spherical, i.e., minimal parabolic subgroups have open orbits on Z. For such spaces we investigate their large scale geometry and provide a polar decomposition. This is obtained from the existence of simple compactifications of Z which is established in this paper., Comment: Extended revised version. Section 5 with a systematic study of the compression (or valuation) cone is new