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Let $R$ be a commutative ring and let $n \geq 1.$ We study $\Gamma(s)$, the generating function and Ann$(s)$, the ideal of characteristic polynomials of $s$, an $n$--dimensional sequence over $R$. We express $f(X_1,\ldots,X_n) \cdot \Gamma(s)(X_1^{-1},\ldots ,X_n^{-1})$ as a partitioned sum. That is, we give (i) a $2^n$--fold ``border'' partition (ii) an explicit expression for the product as a $2^n$--fold sum; the support of each summand is contained in precisely one member of the partition. A key summand is $\beta_0(f,s)$, the ``border polynomial'' of $f$ and $s$, which is divisible by $X_1\cdots X_n$. We say that $s$ is {\em eventually rectilinear} if the elimination ideals Ann$(s)\cap R[X_i]$ contain an $f_i(X_i)$ for $1 \leq i \leq n$. In this case, we show that $\mbox{Ann}(s)$ is the ideal quotient $(\sum_{i=1}^n(f_i)\ :\ \beta_0(f,s)/(X_1\cdots X_n)).$ When $R$ and $R[[X_1,X_2, \ldots ,X_n]]$ are factorial domains (e.g. $R$ a principal ideal domain or ${\Bbb F}[X_1,\ldots,X_n]$), we compute {\em the monic generator} $\gamma _i$ of $\mbox{Ann}(s) \cap R[X_i]$ from known $f_i \in \mbox{Ann}(s) \cap R[X_i]$ or from a finite number of $1$--dimensional linear recurring sequences over $R$. Over a field ${\Bbb F}$ this gives an $O(\prod_{i=1}^n \delta \gamma _i^3)$ algorithm to compute an ${\Bbb F}$--basis for $\mbox{Ann}(s)$., Comment: This is my original latex document submitted to Journal of Symbolic Computation without the typographical errors which were introduced: 'The Journal apologizes for the typographical errors in Norton (1995) introduced in the subediting process'; see this journal, (1995)20, 769-770

The elemental abundances of stars, particularly the refractory elements (e.g., Fe, Si, Mg), play an important role in connecting stars to their planets. Most Sun-like stars do not have refractory abundance measurements since obtaining a large sample of high-resolution spectra is difficult with oversubscribed observing resources. In this work we infer abundances for C, N, O, Na, Mn, Cr, Si, Fe, Ni, Mg, V, Ca, Ti, Al, and Y for solar analogs with Gaia RVS spectra (R=11,200) using the Cannon, a data-driven method. We train a linear model on a reference set of 34 stars observed by Gaia RVS with precise abundances measured from previous high resolution spectroscopic efforts (R > 30,000--110,000). We then apply this model to several thousand Gaia RVS solar analogs. This yields abundances with average upper limit precisions of 0.04--0.1 dex for 17,412 stars, 50 of which are identified planet (candidate) hosts. We subsequently test the relative refractory depletion of these stars with increasing element condensation temperature compared to the Sun. The Sun remains refractory depleted compared to other Sun-like stars regardless of our current knowledge of the planets they host. This is inconsistent with theories of various types of planets locking up or sequestering refractories. Furthermore, we find no significant abundance differences between identified close-in giant planet hosts, giant planet hosts, and terrestrial/small planet hosts and the rest of the sample within our precision limits. This work demonstrates the utility of data-driven learning for future exoplanet composition and demographics studies., Comment: 24 pages, 10 figures, 4 tables, 1 appendix. Accepted in ApJ. Tables 1 and 2 available upon request

The $5d^1$ ordered double perovskites present an exotic playground for studying novel multi-polar physics due to large spin-orbit coupling. We present Re L3 edge resonant inelastic X-ray scattering (RIXS) results that reveal the presence of the dynamic Jahn-Teller effect in the A$_2$MgReO$_6$ (A = Ca, Sr, Ba) family of $5d^1$ double perovskites. The spin-orbit excitations in these materials show a strongly asymmetric lineshape and exhibit substantial temperature dependence, indicating that they are dressed with lattice vibrations. Our experimental results are explained quantitatively through a RIXS calculation based on a spin-orbit-lattice entangled electronic ground state with the dynamic Jahn-Teller effect taken into consideration. We find that the spin-orbit-lattice entangled state is robust against magnetic and structural phase transitions as well as against significant static Jahn-Teller distortions. Our results illustrate the importance of including vibronic coupling for a complete description of the ground state physics of $5d^1$ double perovskites. Usage: Secondary publications and information retrieval purposes.

The prediction of new high entropy oxides (HEOs) remains a profound challenge due to their inherent chemical complexity. In this work, we combine experimental and computational methods to search for new HEOs in the tetravalent $A$O$_2$ family, using exclusively $d^0$ and $d^{10}$ cations, and to explain the observed phase stability of the $\alpha$-PbO$_2$ structure, as found for the medium entropy oxide (Ti, Zr, Hf, Sn)O$_2$. Using a pairwise approach to approximate the mixing enthalpy, we confirm that $\alpha$-PbO$_2$ is the expected lowest energy structure for this material above other candidates including rutile, baddeleyite, and fluorite structures. We also show that no other five-component compound composed of the tetravalent cations considered here is expected to form under solid state synthesis conditions, which we verify experimentally. Ultimately, we conclude that the flexible geometry of the $\alpha$-PbO$_2$ structure can be used to understand its stability among tetravalent HEOs.

Rueppel's conjecture on the linear complexity of the first $n$ terms of the sequence $(1,1,0,1,0^3,1,0^7,1,0^{15},\ldots)$ was first proved by Dai using the Euclidean algorithm. We have previously shown that we can attach a homogeneous (annihilator) ideal of $F[x,z]$ to the first $n$ terms of a sequence over a field $F$ and construct a pair of generating forms for it. This approach gives another proof of Rueppel's conjecture. We also prove additional properties of these forms and deduce the outputs of the LFSR synthesis algorithm applied to the first $n$ terms. Further, dehomogenising the leading generators yields the minimal polynomials of Dai.

Spinel oxides are an ideal setting to explore the interplay between configurational entropy, site selectivity, and magnetism in high entropy oxides. In this work we characterize the magnetic properties of the spinel (Cr,Mn,Fe,Co,Ni)$_3$O$_4$ and study the evolution of its magnetism as a function of non-magnetic gallium substitution. Across the range of compositions studied here, from 0% to 40% Ga, magnetic susceptibility and powder neutron diffraction measurements show that ferrimagnetic order is robust in the spinel HEO. However, we also find that the ferrimagnetic order is highly tunable, with the ordering temperature, saturated and sublattice moments, and magnetic hardness all varying significantly as a function of Ga concentration. Through x-ray absorption and magnetic circular dichroism, we are able to correlate this magnetic tunability with strong site selectivity between the various cations and the tetrahedral and octahedral sites in the spinel structure. In particular, we find that while Ni and Cr are largely unaffected by the substitution with Ga, the occupancies of Mn, Co, and Fe are each significantly redistributed. Ga substitution also requires an overall reduction in the transition metal valence, and this is entirely accommodated by Mn. Finally, we show that while site selectivity has an overall suppressing effect on the configurational entropy, over a certain range of compositions, Ga substitution yields a striking increase in the configurational entropy and may confer additional stabilization. Spinel oxides can be tuned seamlessly from the low-entropy to the high-entropy regime, making this an ideal platform for entropy engineering., Comment: 11 pages, 5 figures, 1 tables

For many students, introductory college science courses are often the only opportunity in their formal higher education to be exposed to science, shaping their view of the subject, their scientific literacy, and their attitudes towards their own ability in STEM. While science writing instruction has been demonstrated to impact attitudes and outlooks of STEM majors in their coursework, this instructional strategy has yet to be explored for non-majors. In this work, we investigate student attitudes towards STEM before and after taking a writing-intensive introductory astronomy course. We find that students cite writing about science as beneficial to their learning, deepening their understanding of science topics and their perspective on science as a field and finding writing to be a "bridge" between STEM content and their focus on humanities in their majors. Students also report increased perceptions of their own ability and confidence in engaging with STEM across multiple metrics, leaving the course more prepared to be informed, engaged, and science literate citizens., Comment: 22 pages; submitted to Astronomy Education Journal (AEJ)

The Games-Chan algorithm finds the minimal period of a periodic binary sequence of period $2^n$, in $n$ iterations. We generalise this to periodic $q$-ary sequences (where $q$ is a prime power) using generating functions and polynomials and apply this to find the multiplicity of $x-1$ in a $q$-ary polynomial $f$ in $\log_{\,q}\deg(f)$ iterations., Comment: Exposition and main theorem improved, typos corrected. Application to finding multiplicity of x-1 in any $q$-ary polynomial added

We simplify an earlier paper of the same title by not using syzygy polynomials and by not using a trichotomy of inverse forms. Let $\K$ be a field and $\M=\K[x^{-1},z^{-1}]$ denote Macaulay's $\K[x,z]$ module of inverse polynomials; here $z$ and $z^{-1}$ are homogenising variables. An inverse form $F\in\M$ has a homogeneous annihilator ideal, $\I_F$\,. In an earlier paper we inductively constructed an ordered pair ($f_1$\,,\,$f_2$) of forms in $\K[x,z]$ which generate $\I_F$. We used syzygy polynomials to show that the intermediate forms give a minimal grlex Groebner basis, which can be efficiently reduced. We give a significantly shorter proof that the intermediate forms are a minimal grlex Groebner basis for $\I_F$\,. We also simplify our proof that either $ F$ is already reduced or a monomial of $f_1$ can be reduced by $f_2$\,. The algorithm that computes $f_1\,,f_2$ yields a variant of the Berlekamp-Massey algorithm which does not use the last 'length change' approach of Massey. These new proofs avoid the three separate cases, 'triples' and the technical factorisation of intermediate 'essential' forms. We also show that $f_1,f_2$ is a maximal $\R$ regular sequence for $\I_F$\,, so that $\I_F$ is a complete intersection., Comment: arXiv admin note: substantial text overlap with arXiv:1710.07731

Let $K$ be a field. We simplify and extend work of Althaler \& D\"ur on finite sequences over $K$ by regarding $K[x^{-1},z^{-1}]$ as a $K[x,z]$ module, and studying forms in $K[x^{-1},z^{-1}]$ from first principles. Then we apply our results to finite sequences. First we define the annihilator ideal $I_F$ of a non-zero form $F\in K[x^{-1},z^{-1}]$, a homogeneous ideal. We inductively construct an ordered pair ($f_1$\,,\,$f_2$) of forms which generate $I_F$\,; our generators are special in that $z$ does not divide the leading grlex monomial of $f_1$ but $z$ divides $f_2$\,, and the sum of their total degrees is always $2-|F|$, where $|F|$ is the total degree of $F$. We show that $f_1,f_2$ is a maximal regular sequence for $I_F$, so that the height of $I_F$ is 2. The corresponding algorithm is $\sim |F|^2/2$. The row vector obtained by accumulating intermediate forms of the construction gives a minimal grlex Gr\"obner basis for $I_F$ for no extra computational cost other than storage and apply this to determining $\dim_K (K[x,z] /I_F)$\,. We show that either the form vector is reduced or a monomial of $f_1$ can be reduced by $f_2$\,. This enables us to efficiently construct the unique reduced Gr\"obner basis for $I_F$ from the vector extension of our algorithm. Then we specialise to the inverse form of a finite sequence, obtaining generator forms for its annihilator ideal and a corresponding algorithm which does not use the last 'length change' of Massey. We compute the intersection of two annihilator ideals using syzygies in $K[x,z]^5$. This improves a result of Althaler \& D\"ur. Finally, dehomogenisation induces a one-to-one correspondence ($f_1$\,,$f_2$) $\mapsto$ (minimal polynomial, auxiliary polynomial), the output of the author's variant of the Berlekamp-Massey algorithm. So we can also solve the LFSR synthesis problem via the corresponding algorithm for sequences., Comment: We have improved the proof of the main construction, made minor improvements to the presentation and corrected some typos. We have also moved Subsection 4.5 on the maximal regular sequence to ArXiv 1805.03995

OSIRIS-REx will return pristine samples of carbonaceous asteroid Bennu. This article describes how pristine was defined based on expectations of Bennu and on a realistic understanding of what is achievable with a constrained schedule and budget, and how that definition flowed to requirements and implementation. To return a pristine sample, the OSIRIS- REx spacecraft sampling hardware was maintained at level 100 A/2 and <180 ng/cm2 of amino acids and hydrazine on the sampler head through precision cleaning, control of materials, and vigilance. Contamination is further characterized via witness material exposed to the spacecraft assembly and testing environment as well as in space. This characterization provided knowledge of the expected background and will be used in conjunction with archived spacecraft components for comparison with the samples when they are delivered to Earth for analysis. Most of all, the cleanliness of the OSIRIS-REx spacecraft was achieved through communication among scientists, engineers, managers, and technicians., Comment: 75 pages, 28 figures, 2 supplements, accepted for publication in Space Science Reviews

Our overall goal is to unify and extend some results in the literature related to the approximation of generating functions of finite and infinite sequences over a field by rational functions. In our approach, numerators play a significant role. We revisit a theorem of Niederreiter on (i) linear complexities and (ii) '$n^{th}$ minimal polynomials' of an infinite sequence, proved using partial quotients. We prove (i) and its converse from first principles and generalise (ii) to rational functions where the denominator need not have minimal degree. We prove (ii) in two parts: firstly for geometric sequences and then for sequences with a jump in linear complexity. The basic idea is to decompose the denominator as a sum of polynomial multiples of two polynomials of minimal degree; there is a similar decomposition for the numerators. The decomposition is unique when the denominator has degree at most the length of the sequence. The proof also applies to rational functions related to finite sequences, generalising a result of Massey. We give a number of applications to rational functions associated to sequences., Comment: Several more typos corrected. To appear in J. Applied Algebra in Engineering, Communication and Computing. The final publication version is available at Springer via http://dx.doi.org/10.1007/s00200-015-0256-5

We derive B\'ezout identities for the minimal polynomials of a finite sequence and use them to prove a theorem of Wang and Massey on binary sequences with a perfect linear complexity profile. We give a new proof of Rueppel's conjecture and simplify Dai's original proof. We obtain short proofs of results of Niederreiter relating the linear complexity of a sequence s and K(s), which was defined using continued fractions. We give an upper bound for the sum of the linear complexities of any sequence. This bound is tight for sequences with a perfect linear complexity profile and we apply it to characterise these sequences in two new ways., Comment: 19 pages, 3 tables

We consider finite sequences $s\in D^n$ where $D$ is a commutative, unital, integral domain. We prove three sets of identities (possibly with repetitions), each involving $2n$ polynomials associated to $s$. The right-hand side of these identities is a recursively-defined (non-zero) 'product-of-discrepancies'. There are implied iterative algorithms (of quadratic complexity) for the left-hand side coefficients; when the ground domain is factorial, the identities are in effect B\'ezout identities. We give a number of applications: an algorithm to compute B\'ezout coefficients over a field; the outputs of the Berlekamp-Massey algorithm; sequences with perfect linear complexity profile; annihilating polynomials which do not vanish at zero and have minimal degree: we simplify and extend an algorithm of Salagean to sequences over $D$. In the Appendix, we give a new proof of a theorem of Imamura and Yoshida on the linear complexity of reverse sequences, initially proved using Hankel matrices over a field and now valid for sequences over a factorial domain., Comment: Major revision, based on recursive index function and minimal polynomial theorem of 1001.1597. Improved notation. Contains new identities and applications. References expanded

We present a recursive minimal polynomial theorem for finite sequences over a commutative integral domain $D$. This theorem is relative to any element of $D$. The ingredients are: the arithmetic of Laurent polynomials over $D$, a recursive 'index function' and simple mathematical induction. Taking reciprocals gives a 'Berlekamp-Massey theorem' i.e. a recursive construction of the polynomials arising in the Berlekamp-Massey algorithm, relative to any element of $D$. The recursive theorem readily yields the iterative minimal polynomial algorithm due to the author and a transparent derivation of the iterative Berlekamp-Massey algorithm. We give an upper bound for the sum of the linear complexities of $s$ which is tight if $s$ has a perfect linear complexity profile. This implies that over a field, both iterative algorithms require at most $2\lfloor \frac{n^2}{4}\rfloor$ multiplications., Comment: Major revision of earlier versions: Introduction expanded, main theorem is now a recursive construction using a recursively defined index function and is relative to any element of $D$. Includes some simpler proofs, a recursive Berlekamp-Massey 'theorem' and additional references

In this paper we draw on data from a large mathematics competition, for the years 1987 to 2000 and use two different but closely related measures to investigate possible gender differences in performance. Our analyses revealed that small gender differences in favour of males persisted but had decreased over time. Consistent with reports from previous studies, gender differences in performance were more marked at senior high school than at junior high school grades. (Contains 1 footnote and 4 tables.) [For complete proceedings, see ED500858.]

Let $s$ be a finite sequence over a field of length $n$. It is well-known that if $s$ satisfies a linear recurrence of order $d$ with non-zero constant term, then the reverse of $s$ also satisfies a recurrence of order $d$ (with coefficients in reverse order). A recent article of A. Salagean proposed an algorithm to find such a shortest 'two-way' recurrence -- which may be longer than a linear recurrence for $s$ of shortest length $\LC_n$. We give a new and simpler algorithm to compute a shortest two-way linear recurrence. First we show that the pairs of polynomials we use to construct a minimal polynomial iteratively are always relatively prime; we also give the extended multipliers. Then we combine degree lower bounds with a straightforward rewrite of a published algorithm due to the author to obtain our simpler algorithm. The increase in shortest length is $\max\{n+1-2\LC_n,0\}$., Comment: This paper has been withdrawn by the author as the proof of Part (b) of Theorem 4.10(ii) is incorrect

We show that a straightforward rewrite of a known minimal polynomial algorithm yields a simpler version of a recent algorithm of A. Salagean., Comment: Section 2 added, remarks and references expanded. To appear in IEEE Transactions on Information Theory.

OSIRIS-REx will return pristine samples of carbonaceous asteroid Bennu. This manuscript describes how pristine was defined based on expectations of Bennu and on a realistic understanding of what is achievable with a constrained schedule and budget, and how that definition flowed to requirements and implementation. To return a pristine sample, the OSIRIS-REx spacecraft sampling hardware was maintained at Level 100 A/2 and less than 180 nanograms per square centimeter of amino acids and hydrazine on the sampler head through precision cleaning, control of materials, and vigilance. Contamination is further characterized via witness material exposed to the spacecraft assembly and testing environment as well as in space. This characterization provided knowledge of the expected background and will be used in conjunction with archived spacecraft components for comparison with the samples when they are delivered to Earth for analysis. Most of all, the cleanliness of the OSIRIS-REx spacecraft was achieved through communication between scientists, engineers, managers, and technicians.

The search for reduced carbon has been a major focus of past and present missions to Mars. Thermal evolved gas analysis was used by the Viking and Phoenix landers and is currently in use by the Sample Analysis at Mars (SAM) instrument suite on the Mars Science Laboratory (MSL) to characterize volatiles evolved from solid samples, including those associated with reduced organic species. SAM has the additional capability to perform a combustion experiment, in which a sample of Mars regolith is heated in the presence of oxygen and the composition of the evolved gases is measured using quadrupole mass spectrometry (QMS) and tunable laser spectrometry (TLS) [1]. Organics detection on the Martian surface has been complicated by oxidation and destruction during heating by soil oxidants [2], including oxychlorine compounds, and terrestrial organics in the SAM background contributed by one of the SAM wet chemistry reagents MTBSTFA (N-Methyl-N-tertbutyldimethylsilyl- trifluoroacetamide) [3,4]. Thermal Evolved Gas Analysis (TEGA) results from Phoenix show a mid temperature CO2 release between 400 C - 680 C speculated to be carbonate, CO2 adsorbed to grains, or combustion of organics by soil oxidants [5]. Low temperature CO2 evolutions (approx. 200 C - 400 C) were also present at all three sites in Gale Crater where SAM Evolved Gas Analysis (EGA) was performed, and potential sources include combustion of terrestrial organics from SAM, as well as combustion and/or decarboxylation either indigenous martian or exogenous organic carbon [4,6]. By performing an experiment to intentionally combust all reduced materials in the sample, we hope to compare the bulk abundance of CO2 and other oxidized species evolved by combustion to that evolved during an EGA experiment to estimate how much CO2 could be contributed by reduced carbon sources. In addition, C, O, and H isotopic compositions of CO2 and H2O measured by TLS can contribute information regarding the potential sources of these volatiles.

The combustion experiment on the Sample Analysis at Mars (SAM) suite on Curiosity will heat a sample of Mars regolith in the presence of oxygen and measure composition of the evolved gases using quadrupole mass spectrometry (QMS) and tunable laser spectrometry (TLS). QMS will enable detection of combustion products such as CO, CO2, NO, and other oxidized species, while TLS will enable precise measurements of the abundance and carbon isotopic composition (delta(sup 13)C) of the evolved CO2 and hydrogen isotopic composition (deltaD) of H2O. SAM will perform a two-step combustion to isolate combustible materials below approx.550 C and above approx.550 C. The combustion experiment on SAM, if properly designed and executed, has the potential to answer multiple questions regarding the origins of volatiles seen thus far in SAM evolved gas analysis (EGA) on Mars. Constraints imposed by SAM and MSL time and power resources, as well as SAM consumables (oxygen gas), will limit the number of SAM combustion experiments, so it is imperative to design an experiment targeting the most pressing science questions. Low temperature combustion experiments will primarily target the quantification of carbon (and nitrogen) contributed by SAM wet chemistry reagants MTBSTFA (N-Methyl-N-tert-butyldimethylsilyltrifluoroacetamide) and DMF (Dimethylformamide), which have been identified in the background of blank and sample runs and may adsorb to the sample while the cup is in the Sample Manipulation System (SMS). In addition, differences between the sample and "blank" may yield information regarding abundance and delta(sup 13)C of bulk (both organic and inorganic) martian carbon. High temperature combustion experiments primarily aim to detect refractory organic matter, if present in Cumberland fines, as well as address the question of quantification and deltaD value of water evolution associated with hydroxyl hydrogen in clay minerals.

Results are presented for an investigation designed to characterize the microstructure of controlled nucleation thermomechanical deposition (CNTD) produced SiC material with respect to grain structure, stoichiometry, phase analysis, etc., and to evaluate the room-temperature and high-temperature fracture and oxidation behavior. By using the CNTD process, ultrafine-grained SiC is deposited on tungsten wires as substrates, with superior surface smoothness and without the macrocolumnar growth commonly observed in conventional CVD materials. The results suggest that the high-purity, fully dense, and stable grain size SiC material produced by CNTD shows potential for high-temperature structural applications, provided that pertinent problems are resolved.

Cathode ray oscilloscope displays curves or contours of complex shapes corresponding to sets of x,y coordinates. It requires few storage facilities and produces a rapid display of complex curves with a fewer number of commands than previous systems.

On-line graphical display system for remote consoles of time-sharing computer facility

Graphical display technique for complex curve display, discussing system components, experimental results and relevant software