Orbital Sets Research Articles

Abstract 5549: Multicellular spheroid structural and metabolic characterization through radial line profiling

Abstract Three-dimensional multicellular tumor spheroids have emerged as a step between in vitro and in vivo models that can offer biological insight in fields such as tumor biology. Spheroids closely resemble in vivo solid tumors regarding their heterogeneous architecture and gradients of nutrients and oxygenation. These characteristics give the spheroid model great potential for studying basic tumor biology. However, traditional characterization methods investigate changes in the discrete regions within the spheroid, limiting the amount of spatial information regarding cellular metabolism. In this study, multicellular spheroids were created through a combination of hanging drops and gentle agitation using murine RAW 264.7 macrophages and colorectal adenocarcinoma CT26 cells. Hanging drops were maintained for three days then transferred to 6-well plates and placed on an orbital shaker set to 70 RPM for seven days. Prior to metabolic imaging, spheroids were moved to a microincubator with controllable temperature and humidified gas delivery (5% CO2). A custom inverted multiphoton imaging system (Bruker) equipped with an Ultrafast Ti:Sapphire via a 60x/1.2NA water immersion objective (Olympus) and four close-proximity, high-efficiency GaAsP detectors. NADH fluorescence was captured with a 460 (± 20) nm band pass filter at 755 nm excitation and FAD fluorescence with a 525 (± 25) nm band pass filter at 855 nm excitation. NADH and FAD fluorescence were normalized by PMT gain and laser power. An integrated fluorescence lifetime imaging microscopy (FLIM) module was used to measure mitochondrial function regarding the different components contributing to NADH autofluorescence. Immunofluorescence was used to characterize macrophage populations (M1 vs M2) and cellular proliferation (Ki67) and apoptosis (CC3). CD206, a M2 marker, showed a lower normalized fluorescent intensity at the spheroid core (0.429) compared to the proliferative edge (0.570). Similar trends are also observed for CD80, a M1 marker and Ki67. CC3 showed a higher normalized fluorescent intensity at the spheroid core (0.823) compared to the proliferative edge (0.004). The optical redox ratio displays a slightly lower value at the core (0.429) compared to the proliferative edge (0.570). Overall, these results indicate that the proliferative edge shows higher numbers of immune cells and more oxidative metabolism compared to the core. Future work includes exposing this spheroid model to low levels of oxygen to investigate how environmental changes in oxygen can affect spheroid structure and metabolism. Citation Format: Shelby N. Bess, Matthew J. Igoe, Gaven K. Smart, Timothy J. Muldoon. Multicellular spheroid structural and metabolic characterization through radial line profiling [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2024; Part 1 (Regular Abstracts); 2024 Apr 5-10; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2024;84(6_Suppl):Abstract nr 5549.

Age of Geminids derived from the statistics of meteoroid orbits

ABSTRACT Statistical analysis of samples of the orbits of celestial bodies is complicated by the fact that the Keplerian orbit is a multidimensional object, the coordinate representation of which non-linearly depends on the choice of orbital elements. In this work, using the construction of the Fréchet mean, concepts of mean orbit and dispersion of the orbit family are introduced, consistent with the distance function on the orbit set. The introduced statistical characteristics serve as analogues of sample mean and variance of a one-dimensional random variable. Exact formulas for calculating the elements of mean orbits and dispersion quantities with respect to two metrics on the orbit space are derived. For a large sample of meteoroid orbits from the Geminid stream, numerical simulations of orbit evolution over 20 000 yr in the past were conducted. By analysing the dependency of statistical characteristics on time, estimates for the age of the stream and the gas outflow velocity are obtained under the assumption of the birth of the Geminids due to the rapid destruction of the cometary nucleus. The estimate of the age of the stream lies in the interval from 1200 to 2400 yr, and the speed of gas outflow at perihelion should have been more than 1.2 km s−1.

Elevation Anomalies of the Volcanic Floor Unit and Their Relationships to the Multiple Lakes of Jezero Crater, Mars

AbstractWe reassessed several orbital topographic data sets for the Perseverance rover landing site at Jezero Crater, Mars to better understand its floor units. Tens‐of‐meters deep topographic anomalies occur in the volcanic floor of Jezero crater and are not a result of impact cratering. Eight km‐scale steep escarpment‐bounded depressions may be locations of paleotopographic highs that were embayed by the volcanic floor lava flows, forming inverted topography from either contemporaneous upward inflation of embaying lavas or later deep scour due to differential erosion over 107−9 years. Five multi km‐scale shallow‐sloped depressions linked by channel‐like forms may record locations of buried paleolakes and channels that predate the volcanic floor units or a drained magma system. These results indicate Jezero experienced multiple closed‐basin or dry phases, allowing erosion of the crater floor and creation of topography, which provides new geologic context for the samples gathered by Perseverance.

The study of atomic structure parameters for [formula omitted] = 4 - [formula omitted] = 3 transitions in Mg-like ions with [formula omitted

Multiconfiguration Dirac–Hartree–Fock (MCDHF) and relativistic configuration interaction (RCI) methods are utilized for theoretical calculations of energy levels, wavelengths, line strengths, absorption oscillator strengths, and transition probabilities in 3s2-3s4p, 3s3p-3s4s, 3s3p-3s4d, 3s3p-3p4p, 3s3d-3s4f, 3p2-3p4s, 3p2-3p4d, and 3p3d-3p4f transitions in Mg-like ions with 15≤Z≤30. The calculations of energies account for Breit interaction and quantum electrodynamics (QED) contributions. The active space approximation is employed for the calculations, and energy converges when the active orbital set is increased to n = 7. The calculated lowest 78 levels including valence and core–valence correlations in Mg-like ions with Z=15–30 show good agreement with experiment results and other calculations. The relative difference between calculated wavelengths and experiment values exhibits a decreasing trend as the atomic numbers increase, and the difference is better than 1% for a majority of transitions. To evaluate the accuracy of the wave functions and transition parameters, the quantity dT is analyzed within 0.1 for most strong transitions. The transition probabilities are compared with other theoretical values to analyze with line strength S. Such calculations may provide valuable data for the experimental study of plasma diagnostics and modeling as there are others.

Revisiting the Kepler problem with linear drag using the blowup method and normal form theory

In this paper, we revisit the Kepler problem with linear drag. With dissipation, the energy and the angular momentum are both decreasing, but in Margheri et al (2017 Celest. Mech. Dyn. Astron. 127 35–48) it was shown that the eccentricity vector has a well-defined limit in the case of linear drag. This limiting eccentricity vector defines a conserved quantity, and in the present paper, we prove that the corresponding invariant sets are smooth manifolds. These results rely on normal form theory and a blowup transformation, which reveals that the invariant manifolds are (nonhyperbolic) stable sets of (limiting) periodic orbits. Moreover, we identify a separate invariant manifold which corresponds to a zero limiting eccentricity vector. This manifold is obtained as a generalized center manifold over the zero eigenspace of a zero-Hopf point. Finally, we present a detailed blowup analysis, which provides a geometric picture of the dynamics. We believe that our approach and results will have general interest in problems with blowup dynamics, including the Kepler problem with generalized nonlinear drag.

Understanding extended homometry based on complementary crystallographic orbit sets.

Extended homometry is a phenomenon in which distinct structures have the same X-ray diffraction (XRD) intensities, which may lead to incorrect results of structural analysis based on XRD methods. It is proposed and proved herein that half of a crystallographic orbit has the same powder X-ray diffraction intensity as its complementary set; three more theorems are deduced. These results are conducive to understanding the formation of extended homometric structures. Also analyzed are some reported or potential homometric or weakly homometric structures in the Inorganic Crystal Structure Database to confirm the theorems. This work presents a quick approach to analyze and construct extended homometric structures based on crystallographic orbits.

Coupled-cluster theory for trapped bosonic mixtures.

We develop a coupled-cluster theory for bosonic mixtures of binary species in external traps, providing a promising theoretical approach to demonstrate highly accurately the many-body physics of mixtures of Bose-Einstein condensates. The coupled-cluster wavefunction for the binary species is obtained when an exponential cluster operator eT, where T = T(1) + T(2) + T(12) and T(1) accounts for excitations in species-1, T(2) for excitations in species-2, and T(12) for combined excitations in both species, acts on the ground state configuration prepared by accumulating all bosons in a single orbital for each species. We have explicitly derived the working equations for bosonic mixtures by truncating the cluster operator up to the single and double excitations and using arbitrary sets of orthonormal orbitals for each of the species. Furthermore, the comparatively simplified version of the working equations are formulated using the Fock-like operators. Finally, using an exactly solvable many-body model for bosonic mixtures that exists in the literature allows us to implement and test the performance and accuracy of the coupled-cluster theory for situations with balanced as well as imbalanced boson numbers and for weak to moderately strong intra- and interspecies interaction strengths. The comparison between our computed results using coupled-cluster theory with the respective analytical exact results displays remarkable agreement exhibiting excellent success of the coupled-cluster theory for bosonic mixtures. All in all, the correlation exhaustive coupled-cluster theory shows encouraging results and could be a promising approach in paving the way for high-accuracy modeling of various bosonic mixture systems.

Dynamics of infinitesimal body in the concentric restricted five-body problem

This research presents a model of a specific restricted five-body problem. In this model, there are four main bodies, referred to as primary bodies. The first two primary bodies have the same mass while the third and fourth primary bodies also share an identical mass, which is lesser than the former. These primary bodies are lined up in a straight line along a designated axis. They revolve in circular paths with different distances from a common central point that represents their shared center of mass. The first set of orbits has a shorter radius compared to the second set. Simplifying the motion equations of an infinitesimally small mass reveals a single governing parameter, λ, constrained within λ ∊ (λ0, 1), λ0 = 0.417221. Within the orbital plane of these primaries, seven equilibrium points are identified: four along the x-axis, two on the y-axis, and one at the system's origin. Notably, no equilibrium points were found outside this orbital plane. The study concludes that collinear equilibrium points are linearly unstable, while the non-collinear points maintain stability for values λc < λ < 1, λc = 0.971105.

Basis Set Calculations of Heavy Atoms

Most modern calculations of many-electron atoms use basis sets of atomic orbitals. An accurate account for electronic correlations in heavy atoms is a very difficult computational problem, and an optimization of the basis sets can reduce computational costs and increase final accuracy. Here, we propose a simple differential ansatz to form virtual orbitals from the Dirac–Fock orbitals of the core and valence electrons. We use basis sets with such orbitals to calculate different properties in Cs including hyperfine structure constants and QED corrections to the valence energies and to the E1 transition amplitudes.

Removing Chaos while preserving equilibria by means of 2-contraction

Conditions for designing a feedback control law to remove the dense set of Unstable Periodic Orbits and chaotic attractors while preserving equilibria are formulated in terms of 2-contractivity of the closed loop system, by means of the second additive compound of the system's Jacobian. Matrix inequalities to allow computing the control gain matrix are discussed. An example of application of the method to the Thomas’ system is presented.

Quantum embedding for molecules using auxiliary particles - the ghost Gutzwiller Ansatz.

Strong/static electronic correlation mediates the emergence of remarkable phases of matter, and underlies the exceptional reactivity properties in transition metal-based catalysts. Modeling strongly correlated molecules and solids calls for multi-reference Ansätze, which explicitly capture the competition of energy scales characteristic of such systems. With the efficient computational screening of correlated solids in mind, the ghost Gutzwiller (gGut) Ansatz has been recently developed. This is a variational Ansatz which can be formulated as a self-consistent embedding approach, describing the system within a non-interacting, quasiparticle model, yet providing accurate spectra in both low and high energy regimes. Crucially, small fragments of the system are identified as responsible for the strong correlation, and are therefore enhanced by adding a set of auxiliary orbitals, the ghosts. These capture many-body correlations through one-body fluctuations and subsequent out-projection when computing physical observables. gGut has been shown to accurately describe multi-orbital lattice models at modest computational cost. In this work, we extend the gGut framework to strongly correlated molecules, for which it holds special promise. Indeed, despite the asymmetric embedding treatment, the quasiparticle Hamiltonian effectively describes all major sources of correlation in the molecule: strong correlation through the ghosts in the fragment, and dynamical correlation through the quasiparticle description of its environment. To adapt the gGut Ansatz for molecules, we address the fact that, unlike in the lattice model previously considered, electronic interactions in molecules are not local. Hence, we explore a hierarchy of approximations of increasing accuracy capturing interactions between fragments and environment, and within the environment, and discuss how these affect the embedding description of correlations in the whole molecule. We will compare the accuracy of the gGut model with established methods to capture strong correlation within active space formulations, and assess the realistic use of this novel approximation to the theoretical description of correlated molecular clusters.

Examples of projective billiards with open sets of periodic orbits

Examples of projective billiards with open sets of periodic orbits

Finite orbits for large groups of automorphisms of projective surfaces

We study finite orbits of non-elementary groups of automorphisms of compact projective surfaces. We prove that if the surface and the group are defined over a number field $\mathbf {k}$ and the group contains parabolic elements, then the set of finite orbits is not Zariski dense, except in certain very rigid situations, known as Kummer examples. Related results are also established when $\mathbf {k} = \mathbf {C}$. An application is given to the description of ‘canonical vector heights’ associated to such automorphism groups.

Motivic stable cohomotopy and unimodular rows

We relate the group structure of van der Kallen on orbit sets of unimodular rows ([26]) with values in a smooth algebra A over a field k with the motivic cohomotopy groups of X=SpecA with coefficients in An∖0 in the sense of [5]. In the last section, we compare the motivic cohomotopy theory studied in this paper and defined by An+1∖0 or, equivalently, by an A1-weakly equivalent quadric Q2n+1 to that considered in [5], defined by a quadric Q2n, by means of explicit morphisms Q2n+1→Q2n, Q2n×Gm→Q2n+1 of quadrics.

On the coadjoint Virasoro action

The set of coadjoint orbits of the Virasoro algebra at level 1 is in bijection with the set of conjugacy classes in a certain open subset SL˜(2,R)+ of the universal cover of SL(2,R). We strengthen this bijection to a Morita equivalence of quasi-symplectic groupoids, integrating the Poisson structure on vir1⁎(S1) and the Cartan-Dirac structure on SL˜(2,R)+, respectively.

Automated mixing of maximally localized Wannier functions into target manifolds

Maximally localized Wannier functions (MLWFs) are widely used in electronic-structure calculations. We have recently developed automated approaches to generate MLWFs that represent natural tight-binding sets of atomic-like orbitals; these describe accurately both the occupied states and the complementary unoccupied ones. For many applications, it is required to use MLWFs that describe instead certain target groups of bands: the valence or the conduction bands, or correlated manifolds. Here, we start from these tight-binding sets of MLWFs, and mix them using a combination of parallel transport and maximal localization to construct manifold-remixed Wannier functions (MRWFs): these are orthogonal sets of MLWFs that fully and only span desired target submanifolds. The algorithm is simple and robust, and is showcased here in reference applications (silicon, MoS2, and SrVO3) and in a mid-throughput study of 77 insulators.

Direct Determination of Optimal Real-Space Orbitals for Correlated Electronic Structure of Molecules.

We demonstrate how to determine numerically nearly exact orthonormal orbitals that are optimal for the evaluation of the energy of arbitrary (correlated) states of atoms and molecules by minimization of the energy Lagrangian. Orbitals are expressed in real space using a multiresolution spectral element basis that is refined adaptively to achieve the user-specified target precision while avoiding the ill-conditioning issues that plague AO basis set expansions traditionally used for correlated models of molecular electronic structure. For light atoms, the orbital solver, in conjunction with a variational electronic structure model [selected Configuration Interaction (CI)] provides energies of comparable precision to a state-of-the-art atomic CI solver. The computed electronic energies of atoms and molecules are significantly more accurate than the counterparts obtained with the orbital sets of the same rank expanded in Gaussian AO bases, and can be determined even when linear dependence issues preclude the use of the AO bases. It is feasible to optimize more than 100 fully correlated numerical orbitals on a single computer node, and significant room exists for additional improvement. These findings suggest that real-space orbital representations might be the preferred alternative to AO representations for high-end models of correlated electronic states of molecules and materials.

Field- and Temperature-Dependent Paramagnetic Relaxation Enhancements in Co(II) Trispyrazolylmethanes.

A comprehensive field- and temperature-dependent examination of nuclear magnetic resonance paramagnetic relaxation enhancements (PREs) for the constitutive protons of [Co(Tpm)2][BF4]2 is presented. Data for an apically substituted derivative clearly establish that bis-Tpm complexes of Co(II) undergo Jahn-Teller dynamics about the molecular threefold axis. PREs from the parent Tpm complex were used to numerically extract the electron relaxation times (T1e). The Tpm complex showed field-dependent behavior, with an approximately 40% higher activation barrier than the related trispyrazolylborate (Tp) complex, based on fits to the T1e vs T, B0 data. Analysis of the field-dependent line widths revealed a surprisingly large contribution from susceptibility (Curie) relaxation (20-35% at the highest field), and a molecular radius (9.5 Å) that is consistent with a tightly associated counterion slowing rotation in solution. Density functional theory showed a shared vibration that is consistent with the Jahn-Teller and appears proportionately higher in energy in [Co(Tpm)2]2+. Complete active-space self-consistent field calculations support ascribing electron relaxation to enhanced mixing of the two Eg orbital sets that accompanies the tetragonal distortion and the differences in electron correlation times to the higher Jahn-Teller activation barrier in [Co(Tpm)2]2+.

Dimensions of Kleinian orbital sets

Given a non-empty bounded subset of hyperbolic space and a Kleinian group acting on that space, the orbital set is the orbit of the given set under the action of the group. We may view orbital sets as bounded (often fractal) subsets of Euclidean space. We prove that the upper box dimension of an orbital set is given by the maximum of three quantities: the upper box dimension of the given set, the Poincaré exponent of the Kleinian group, and the upper box dimension of the limit set of the Kleinian group. Since we do not make any assumptions about the Kleinian group, none of the terms in the maximum can be removed in general. We show by constructing an explicit example that our assumption that the given set is bounded (in the hyperbolic metric) cannot be removed in general.

Complete and orthonormal sets of exponential-type orbitals with non-integer quantum numbers

Atomic and molecular orbitals show exponential decrease at long range. Complete orthonormal basis sets for atoms should satisfy this criterion. A number of such bases have been used in physics (e.g. Coulomb Sturmians). The challenge of this work is first adapting Slater type Orbitals for this role, as they are not radially orthogonal. Even more important is their generalization to non-integer quantum numbers that have applications for configuration interaction. This generalization requires the whole apparatus of non-integer calculus that is presented using the Riemann–Liouville approach.