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In this article, we investigate properties of digital H-spaces in the graph theoretic model of digital topology. As in prior work, the results obtained often depend fundamentally on the choice between NP$_1$ and NP$_2$ product adjacencies. We explore algebraic properties of digital H-spaces preserved under digital homotopy equivalence, and we give a general construction that produces examples of digital H-spaces which are not homotopy-equivalent to digital topological groups in both categories. Further, we show that this construction essentially classifies all NP$_2$-digital H-spaces. In a short appendix, we resolve a question that was left unresolved in [17], and complete the full classification of digital topological groups.

For a semisimple, simply-connected linear algebraic group, $G$, and parabolic subgroup, $P\subseteq G$, we use the fact that the Hilbert polynomial of the equivariant embedding of $G/P$ is equal to the Hilbert function to compute an explicit formula for the Hilbert series of $G/P$ in terms of the dimensions of finitely many irreducible representations of $G$. As an example, we compute the Hilbert series of the adjoint variety of $SL(n+1,\mathbb{C})$. We conclude by computing the linear term of the numerator of the Hilbert series of any fundamental representation in type $A$.

We use the ordinary Euler operator to compute the Ehrhart series for an arbitrary lattice polytope. The resulting formula involves the coefficients of the Ehrhart polynomial, combined via Eulerian numbers. We use this to compute $h^*_{d-1}$ in terms of the coefficients of the Ehrhart polynomial, resulting in a new linear inequality satisfied by the coefficents of the Ehrhart polynomial.

Consider a hierarchical log-linear model, given by a simplicial complex, $\Gamma$, and integer matrix $A_\Gamma$. We give a new characterization of the rank of $A_\Gamma$ given by a logarithmic transformation on the exponential Hilbert series of $\Gamma$. We show that, if each random variable in $X$ has the same number of possible outcomes, then this formula reduces to a simple description in terms of the face vector of $\Gamma$. If $\Gamma$ further satisfies the Dehn-Sommerville relations, then we give an exceptionally simple formula for computing the rank of $A_\Gamma$, and thus the dimension and the number of degrees of freedom of the model., Comment: Corrected typo in Corollary 2. Added comment on cyclic models

Mid-Air Helicopter Delivery (MAHD) is a new Entry, Descent and Landing (EDL) architecture to enable in situ mobility for Mars science at lower cost than previous missions. It uses a jetpack to slow down a Mars Science Helicopter (MSH) after separation from the backshell, and reach aerodynamic conditions suitable for helicopter take-off in mid air. For given aeroshell dimensions, only MAHD's lander-free approach leaves enough room in the aeroshell to accommodate the largest rotor option for MSH. This drastically improves flight performance, notably allowing +150\% increased science payload mass. Compared to heritage EDL approaches, the simpler MAHD architecture is also likely to reduce cost, and enables access to more hazardous and higher-elevation terrains on Mars. This paper introduces a design for the MAHD system architecture and operations. We present a mechanical configuration that fits both MSH and the jetpack within the 2.65-m Mars heritage aeroshell, and a jetpack control architecture which fully leverages the available helicopter avionics. We discuss preliminary numerical models of the flow dynamics resulting from the interaction between the jets, the rotors and the side winds. We define a force-torque sensing architecture capable of handling the wind and trimming the rotors to prepare for safe take-off. Finally, we analyze the dynamic environment and closed-loop control simulation results to demonstrate the preliminary feasibility of MAHD., Comment: Accepted in 2022 IEEE Aerospace Conference

Mid-Air Deployment (MAD) of a rotorcraft during Entry, Descent and Landing (EDL) on Mars eliminates the need to carry a propulsion or airbag landing system. This reduces the total mass inside the aeroshell by more than 100 kg and simplifies the aeroshell architecture. MAD's lighter and simpler design is likely to bring the risk and cost associated with the mission down. Moreover, the lighter entry mass enables landing in the Martian highlands, at elevations inaccessible to current EDL technologies. This paper proposes a novel MAD concept for a Mars helicopter. We suggest a minimum science payload package to perform relevant science in the highlands. A variant of the Ingenuity helicopter is proposed to provide increased deceleration during MAD, and enough lift to fly the science payload in the highlands. We show in simulation that the lighter aeroshell results in a lower terminal velocity (30 m/s) at the end of the parachute phase of the EDL, and at higher altitudes than other approaches. After discussing the aerodynamics, controls, guidance, and mechanical challenges associated with deploying at such speed, we propose a backshell architecture that addresses them to release the helicopter in the safest conditions. Finally, we implemented the helicopter model and aerodynamic descent perturbations in the JPL Dynamics and Real-Time Simulation (DARTS)framework. Preliminary performance evaluation indicates landing and helicopter operation scan be achieved up to 5 km MOLA (Mars Orbiter Laser Altimeter reference).

The use of computer-aided and web-based educational technologies such as Virtual Learning Environments (VLE) has increased significantly in the recent past. One example of such a VLE is Virtual Interactive Engineering on the Web (VIEW). VIEW is a 3D virtual, interactive, student centered, framework of web-based modules based on the Extensible 3D standard. These modules are dedicated to the improvement of student success and learning. In this paper, an overview of the recent developments in VIEW along with associated assessment results is presented. An experimental study was also performed to compare the learning experience and performance of students in a physical dissection activity vs. that in a virtual dissection activity using a VIEW module. The results of this study show that students can meet given learning objectives and that there is limited difference in their learning and performance irrespective of a physical or virtual setting.

We study the exponential Hilbert series (both coarsely- and finely-graded) of the Stanley-Reisner ring of an abstract simplicial complex, $\Delta$, and we introduce the $e$-vector of $\Delta$, which relates to the coefficients of the exponential Hilbert series. We explore the relationship of the $e$-vector with the classical $f$-vector and $h$-vector of $\Delta$ while simultaneously investigating the geometric information that the $e$-vector encodes about $\Delta$. We then prove a simple combinatorial identity for the $e$-vector in the case where $\Delta$ is an Eulerian manifold., Comment: Along with substantial updates, the new version merges the previous version with arXiv:2211.03765

In this article, we study properties of the exponential Hilbert series of a $G$-equivariant projective variety, where $G$ is a semisimple, simply-connected complex linear algebraic group. We prove a relationship between the exponential Hilbert series and the degree and dimension of the variety. We then prove a combinatorial identity for the coefficients of the polynomial representing the exponential Hilbert series. This formula is used in examples to prove further combinatorial identities involving Stirling numbers of the first and second kinds., Comment: 10 pages

Let $N$ be a connected and simply connected nilpotent Lie group, and let $K$ be a subgroup of the automorphism group of $N$. We say that the pair $(K,N)$ is a nilpotent Gelfand pair if $L^1_K(N)$ is an abelian algebra under convolution. In this document we establish a geometric model for the Gelfand spectra of nilpotent Gelfand pairs $(K,N)$ where the $K$-orbits in the center of $N$ have a one-parameter cross section and satisfy a certain non-degeneracy condition. More specifically, we show that the one-to-one correspondence between the set $\Delta(K,N)$ of bounded $K$-spherical functions on $N$ and the set $\mathcal{A}(K,N)$ of $K$-orbits in the dual $\mathfrak{n}^*$ of the Lie algebra for $N$ established by Benson and Ratcliff is a homeomorphism for this class of nilpotent Gelfand pairs. This result had previously been shown for $N$ a free group and $N$ a Heisenberg group, and was conjectured to hold for all nilpotent Gelfand pairs., Comment: 24 pages; This is a pre-print of an article published in Transformation Groups. The final authenticated version is available online at: https://doi.org/10.1007/s00031-019-09541-8

A pair of variables that tend to rise and fall either together or in opposition are said to be monotonically associated. For certain phenomena, this tendency is causally restricted to a subpopulation, as, for example, an allergic reaction to an irritant. Previously, Yu et al. (2011) devised a method of rearranging observations to test paired data to see if such an association might be present in a subpopulation. However, the computational intensity of the method limited its application to relatively small samples of data, and the test itself only judges if association is present in some subpopulation; it does not clearly identify the subsample that came from this subpopulation, especially when the whole sample tests positive. The present paper adds a "top-K" feature (Sampath and Verducci (2013)) based on a multistage ranking model, that identifies a concise subsample that is likely to contain a high proportion of observations from the subpopulation in which the association is supported. Computational improvements incorporated into this top-K tau-path (TKTP) algorithm now allow the method to be extended to thousands of pairs of variables measured on sample sizes in the thousands. A description of the new algorithm along with measures of computational complexity and practical efficiency help to gauge its potential use in different settings. Simulation studies catalog its accuracy in various settings, and an example from finance illustrates its step-by-step use.

We present a closed form for a multi-variate generating function for the dimensions of the irreducible representations of a semisimple, simply connected linear algebraic group over $\mathbb{C}$ whose highest weights lie in a finitely generated lattice cone in the dominant chamber. This result generalizes the formula for the Hilbert series of an equivariant embedding of a homogeneous projective variety. As a special case, we show how the multi-variate series can be used to compute the Hilbert series of the determinantal varieties.

Mid-Air Helicopter Delivery (MAHD) is a new Entry, Descent and Landing (EDL) architecture to enable in situ mobility for Mars science at lower cost than previous rover missions. It uses a jetpack to slow down a Mars Science Helicopter (MSH) after separation from the backshell, and reach aerodynamic conditions suitable for helicopter take-off in mid air. MAHD's lander-free approach leaves enough room in the aeroshell to accommodate larger rotors. This drastically improves flight performance compared to heritage EDL approaches, notably +60\% science payload mass. MAHD also brings cost savings, a simpler architecture, improved surface access and can reach higher elevations on Mars. This paper introduces a design for the MAHD system architecture and operations. We present a mechanical configuration which fits both MSH and the jetpack within the 2.65-m Mars heritage aeroshell, and a jetpack control architecture which fully leverages the available helicopter avionics. We discuss preliminary numerical models of the flow dynamics resulting from the interaction between the jets, the rotors and the side winds. We define a force-torque sensing architecture capable of handling the wind and trimming the rotors to prepare for safe take-off. Finally, we analyze the dynamic environment and closed-loop control simulation results to demonstrate the preliminary feasibility of MAHD.