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We study "partition Eisenstein series", extensions of the Eisenstein series $G_{2k}(\tau),$ defined by $$\lambda=(1^{m_1}, 2^{m_2},\dots, k^{m_k}) \vdash k \ \ \ \ \ \longmapsto \ \ \ \ \ G_{\lambda}(\tau):= G_2(\tau)^{m_1} G_4(\tau)^{m_2}\cdots G_{2k}(\tau)^{m_k}. $$ For functions $\phi: \mathcal{P}\rightarrow \mathbb{C}$ on partitions, the weight $2k$ "partition Eisenstein trace" is the quasimodular form $$ {\mathrm{Tr}}_k(\phi;\tau):=\sum_{\lambda \vdash k} \phi(\lambda)G_{\lambda}(\tau). $$ These traces give explicit formulas for some well-known generating functions, such as the $k$th elementary symmetric functions of the inverse points of 2-dimensional complex lattices $\mathbb{Z}\oplus \mathbb{Z}\tau,$ as well as the $2k$th power moments of the Andrews-Garvan crank function. To underscore the ubiquity of such traces, we show that their generalizations give the Taylor coefficients of generic Jacobi forms with torsional divisor.

Ramanujan's celebrated partition congruences modulo $\ell\in \{5, 7, 11\}$ assert that $$ p(\ell n+\delta_{\ell})\equiv 0\pmod{\ell}, $$ where $0<\delta_{\ell}<\ell$ satisfies $24\delta_{\ell}\equiv 1\pmod{\ell}.$ By proving Subbarao's Conjecture, Radu showed that there are no such congruences when it comes to parity. There are infinitely many odd (resp. even) partition numbers in every arithmetic progression. For primes $\ell \geq 5,$ we give a new proof of the conclusion that there are infinitely many $m$ for which $p(\ell m+\delta_{\ell})$ is odd. This proof uses a generalization, due to the second author and Ramsey, of a result of Mazur in his classic paper on the Eisenstein ideal. We also refine a classical criterion of Sturm for modular form congruences, which allows us to show that the smallest such $m$ satisfies $m<(\ell^2-1)/24,$ representing a significant improvement to the previous bound., Comment: Corrects minor typos found by the referees

Differences in staining and imaging procedures can cause significant color variations in histopathology images, leading to poor generalization when deploying deep-learning models trained from a different data source. Various color augmentation methods have been proposed to generate synthetic images during training to make models more robust, eliminating the need for stain normalization during test time. Many color augmentation methods leverage domain labels to generate synthetic images. This approach causes three significant challenges to scaling such a model. Firstly, incorporating data from a new domain into deep-learning models trained on existing domain labels is not straightforward. Secondly, dependency on domain labels prevents the use of pathology images without domain labels to improve model performance. Finally, implementation of these methods becomes complicated when multiple domain labels (e.g., patient identification, medical center, etc) are associated with a single image. We introduce ContriMix, a novel domain label free stain color augmentation method based on DRIT++, a style-transfer method. Contrimix leverages sample stain color variation within a training minibatch and random mixing to extract content and attribute information from pathology images. This information can be used by a trained ContriMix model to create synthetic images to improve the performance of existing classifiers. ContriMix outperforms competing methods on the Camelyon17-WILDS dataset. Its performance is consistent across different slides in the test set while being robust to the color variation from rare substances in pathology images. We make our code and trained ContriMix models available for research use. The code for ContriMix can be found at https://gitlab.com/huutan86/contrimix

Motivated by the many roles that hook lengths play in mathematics, we study the distribution of the number of $t$-hooks in the partitions of $n$. We prove that the limiting distribution is normal with mean $\mu_t(n)\sim \frac{\sqrt{6n}}{\pi}-\frac{t}{2}$ and variance $\sigma_t^2(n)\sim \frac{(\pi^2-6)\sqrt{6n}}{2\pi^3}.$ Furthermore, we prove that the distribution of the number of hook lengths that are multiples of a fixed $t\geq 4$ in partitions of $n$ converge to a shifted Gamma distribution with parameter $k=(t-1)/2$ and scale $\theta=\sqrt{2/(t-1)}.$, Comment: Minor edits and a new first paragraph on previous related work (added at the suggestion of a referee)

Hausel and Rodriguez-Villegas recently observed that work of G\"ottsche, combined with a classical result of Erd\H{o}s and Lehner on integer partitions, implies that the limiting Betti distribution for the Hilbert schemes $(\mathbb{C}^2)^{[n]}$ on $n$ points, as $n\rightarrow +\infty,$ is a \textit{Gumbel distribution}. In view of this example, they ask for further such Betti distributions. We answer this question for the quasihomogeneous Hilbert schemes $((\mathbb{C}^2)^{[n]})^{T_{\alpha,\beta}}$ that are cut out by torus actions. We prove that their limiting distributions are also of Gumbel type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on these Hilbert schemes with our generalization of the result of Erd\H{o}s and Lehner, which gives the distribution of the number of parts in partitions that are multiples of a fixed integer $A\geq 2.$ Furthermore, if $p_k(A;n)$ denotes the number of partitions of $n$ with exactly $k$ parts that are multiples of $A$, then we obtain the asymptotic $$ p_k(A,n)\sim \frac{24^{\frac k2-\frac14}(n-Ak)^{\frac k2-\frac34}}{\sqrt2\left(1-\frac1A\right)^{\frac k2-\frac14}k!A^{k+\frac12}(2\pi)^k}e^{2\pi\sqrt{\frac1{6}\left(1-\frac1A\right)(n-Ak)}}, $$ a result which is of independent interest., Comment: Minor revision addressing referee comments

The classical $\mathrm{AGM}$ produces wonderful interdependent infinite sequences of arithmetic and geometric means with common limit. For finite fields $\mathbb{F}_q,$ with $q\equiv 3\pmod 4,$ we introduce a finite field analogue $\mathrm{AGM}_{\mathbb{F}_q}$ that spawns directed finite graphs instead of infinite sequences. The compilation of these graphs reminds one of a $\mathit{jellyfish~swarm},$ as the 3D renderings of the connected components resemble $\mathit{jellyfish}$ (i.e. tentacles connected to a bell head). These swarms turn out to be more than the stuff of child's play; they are taxonomical devices in number theory. Each jellyfish is an isogeny graph of elliptic curves with isomorphic groups of $\mathbb{F}_q$-points, which can be used to prove that each swarm has at least $(1/2-\varepsilon)\sqrt{q}$ jellyfish. Additionally, this interpretation gives a description of the $\mathit{class~numbers}$ of Gauss, Hurwitz, and Kronecker which is akin to counting types of spots on jellyfish., Comment: Corrects minor typos

We explicitly construct the Dirichlet series $$L_{\mathrm{Tam}}(s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(m)}{m^s},$$ where $P_{\mathrm{Tam}}(m)$ is the proportion of elliptic curves $E/\mathbb{Q}$ in short Weierstrass form with Tamagawa product $m.$ Although there are no $E/\mathbb{Q}$ with everywhere good reduction, we prove that the proportion with trivial Tamagawa product is $P_{\mathrm{Tam}}(1)=0.5053\dots.$ As a corollary, we find that $L_{\mathrm{Tam}}(-1)=1.8193\dots$ is the average Tamagawa product for elliptic curves over $\mathbb{Q}.$ We give an application of these results to canonical and Weil heights., Comment: This version corrects a typographical error in a Remark on page 3

A modular grid is a pair of sequences $(f_m)_m$ and $(g_n)_n$ of weakly holomorphic modular forms such that for almost all $m$ and $n$, the coefficient of $q^n$ in $f_m$ is the negative of the coefficient of $q^m$ in $g_n$. Zagier proved this coefficient duality in weights $1/2$ and $3/2$ in the Kohnen plus space, and such grids have appeared for Poincar\'{e} series, for modular forms of integral weight, and in many other situations. We give a general proof of coefficient duality for canonical row-reduced bases of spaces of weakly holomorphic modular forms of integral or half-integral weight for every group $\Gamma \subseteq {\text{SL}}_2(\mathbb{R})$ commensurable with ${\text{SL}}_2(\mathbb{Z})$. We construct bivariate generate functions that encode these modular forms, and study linear operations on the resulting modular grids., Comment: Second revision

In this note we obtain effective lower bounds for the canonical heights of non-torsion points on $E(\mathbb{Q})$ by making use of suitable elliptic curve ideal class pairings $$\Psi_{E,-D}: E(\mathbb{Q})\times E_{-D}(\mathbb{Q})\mapsto \mathrm{CL}(-D).$$ In terms of the class number $H(-D)$ and $T_E(-D)$, a logarithmic function in $D$, we prove $$ \widehat{h}(P)> \frac{|E_{\mathrm{tor}}(\mathbb{Q})|^2}{\left( H(-D)+ |E_{\mathrm{tor}}(\mathbb{Q})|\right)^2}\cdot T_E(-D). $$, Comment: Comments. 1) This paper was published as Proceedings of the American Mathematical Society 149 (2021), 5093-5100. 2) After the paper was published, we discovered a typographical error in the example after Theorem 1.1. The number 0.043 should be 0.035

For positive rank $r$ elliptic curves $E(\mathbb{Q})$, we employ ideal class pairings $$ E(\mathbb{Q})\times E_{-D}(\mathbb{Q}) \rightarrow \mathrm{CL}(-D), $$ for quadratic twists $E_{-D}(\mathbb{Q})$ with a suitable ``small $y$-height'' rational point, to obtain effective class number lower bounds. For the curves $E^{(a)}: \ y^2=x^3-a,$ with rank $r(a),$ this gives $$ h(-D) \geq \frac{1}{10}\cdot \frac{|E_{\mathrm{tor}}(\mathbb{Q})|}{\sqrt{R_{\mathbb{Q}}(E)}}\cdot \frac{\Gamma\left (\frac{r(a)}{2}+1\right)}{(4\pi)^{\frac{r(a)}{2}}} \cdot \frac{\log(D)^{\frac{r(a)}{2}}}{\log \log D}, $$ representing an improvement to the classical lower bound of Goldfeld, Gross and Zagier when $r(a)\geq 3$. We prove that the number of twists $E_{-D}^{(a)}(\mathbb{Q})$ with such a point (resp. with such a point and rank $\geq 2$ under the Parity Conjecture) is $\gg_{a,\varepsilon} X^{\frac{1}{2}-\varepsilon}.$ We give infinitely many cases where $r(a)\geq 6$. These results can be viewed as an analogue of the classical estimate of Gouv\^ea and Mazur for the number of rank $\geq 2$ quadratic twists, where in addition we obtain ``log-power'' improvements to the Goldfeld-Gross-Zagier class number lower bound., Comment: We correct minor typographical errors, including the formula in the abstract and equation (1.3)

The modularity theorem implies that for every elliptic curve $E /\mathbb{Q}$ there exist rational maps from the modular curve $X_0(N)$ to $E$, where $N$ is the conductor of $E$. These maps may be expressed in terms of pairs of modular functions $X(z)$ and $Y(z)$ where $X(z)$ and $Y(z)$ satisfy the Weierstrass equation for $E$ as well as a certain differential equation. Using these two relations, a recursive algorithm can be used to calculate the $q$ - expansions of these parametrizations at any cusp. %These functions are algebraic over $\mathbb{Q}(j(z))$ and satisfy modular polynomials where each of the coefficient functions are rational functions in $j(z)$. Using these functions, we determine the divisor of the parametrization and the preimage of rational points on $E$. We give a sufficient condition for when these preimages correspond to CM points on $X_0(N)$. We also examine a connection between the algebras generated by these functions for related elliptic curves, and describe sufficient conditions to determine congruences in the $q$-expansions of these objects.

Ideal class pairings map the rational points of rank $r\geq 1$ elliptic curves $E/\Q$ to the ideal class groups $\CL(-D)$ of certain imaginary quadratic fields. These pairings imply that $$h(-D) \geq \frac{1}{2}(c(E)-\varepsilon)(\log D)^{\frac{r}{2}} $$ for sufficiently large discriminants $-D$ in certain families, where $c(E)$ is a natural constant. These bounds are effective, and they offer improvements to known lower bounds for many discriminants., Comment: Corrected a typo appearing in bounds

We investigate Riemann's xi function $\xi(s):=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)$ (here $\zeta(s)$ is the Riemann zeta function). The Riemann Hypothesis (RH) asserts that if $\xi(s)=0$, then $\mathrm{Re}(s)=\frac{1}{2}$. P\'olya proved that RH is equivalent to the hyperbolicity of the Jensen polynomials $J^{d,n}(X)$ constructed from certain Taylor coefficients of $\xi(s)$. For each $d\geq 1$, recent work proves that $J^{d,n}(X)$ is hyperbolic for sufficiently large $n$. Here we make this result effective. Moreover, we show how the low-lying zeros of the derivatives $\xi^{(n)}(s)$ influence the hyperbolicity of $J^{d,n}(X)$., Comment: 13 pages. This revision represents a major revision of the previous version. The exposition has been improved and many clarifications have been added. Moreover, Theorem 1.1 has been improved

In 1927 P\'olya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function $\zeta(s)$ at its point of symmetry. This hyperbolicity has been proved for degrees $d\leq 3$. We obtain an asymptotic formula for the central derivatives $\zeta^{(2n)}(1/2)$ that is accurate to all orders, which allows us to prove the hyperbolicity of a density $1$ subset of the Jensen polynomials of each degree. Moreover, we establish hyperbolicity for all $d\leq 8$. These results follow from a general theorem which models such polynomials by Hermite polynomials. In the case of the Riemann zeta function, this proves the GUE random matrix model prediction in derivative aspect. The general theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function., Comment: 11 pages

Euler showed that there are infinitely many triangular numbers that are three times other triangular numbers. In general, it is an easy consequence of the Pell equation that for a given square-free m > 1, the relation P=mP' is satisfied by infinitely many pairs of triangular numbers P, P'. After recalling what is known about triangular numbers, we shall study this problem for higher polygonal numbers. Whereas there are always infinitely many triangular numbers which are fixed multiples of other triangular numbers, we give an example that this is false for higher polygonal numbers. However, as we will show, if there is one such solution, there are infinitely many. We will give conditions which conjecturally assure the existence of a solution. But due to the erratic behavior of the fundamental unit in quadratic number fields, finding such a solution is exceedingly difficult. Finally, we also show in this paper that, given m > n > 1 with obvious exceptions, the system of simultaneous relations P = mP', P = nP'' has only finitely many possibilities not just for triangular numbers, but for triplets P, P', P'' of polygonal numbers, and give examples of such solutions., Comment: 17 pages, 1 figure, 2 tables. New version added a table of solutions to the second problem

The transition from globular to elongated states of biopolymers in shear flow occurs at a distinct critical shear rate, $\dot{\gamma }^{*}$. The magnitude of $\dot{\gamma }^{*}$ depends on the internal potential and the polymer length. For example, the critical shear is much larger for von Willebrand Factor (vWF) compared to DNA. Furthermore, it is shown through computational analysis of vWF (model-vWF) that $\dot{\gamma }^{*}\sim N^{1/3}$, where N is the number of dimeric units in the vWF. In this study, we show that in the presence of charged nanoparticles (CNP) and polymer, the critical shear rate scales differently depending on the polymer length and the charge-strength of the CNP. It is shown that CNP alter the conformational dynamics of polymers under shear flow when the polymer beads have an opposite charge. The introduction of CNP shifts the critical shear rate and alters the scaling. Furthermore, it is shown that the critical zeta potential of CNP, $\zeta^{*}$, scales linearly with shear rate, and scales cubically with ratio $\beta$ between final CNP-polymer composite size and original polymer size, that is $\zeta^{*}\sim \dot{\gamma }\beta ^{-3}$.

Modular and mock modular forms possess many striking $p$-adic properties, as studied by Bringmann, Guerzhoy, Kane, Kent, Ono, and others. Candelori developed a geometric theory of harmonic Maass forms arising from the de Rham cohomology of modular curves. In the setting of over-convergent $p$-adic modular forms, Candelori and Castella showed this leads to $p$-adic analogs of harmonic Maass forms. In this paper we take an analytic approach to construct $p$-adic analogs of harmonic Maass forms of weight $0$ %and $1/2$ with square free level. Although our approaches differ, where the two theories intersect the forms constructed are the same. However our analytic construction defines these functions on the full super singular locus as well as on the ordinary locus. As with classical harmonic Maass forms, these $p$-adic analogs are connected to weight $2$ cusp forms and their modular derivatives are weight $2$ weakly holomorphic modular forms. Traces of their CM values also interpolate the coefficients of half integer weight modular and mock modular forms. We demonstrate this through the construction of $p$-adic analogs of two families of theta lifts for these forms.

In this paper we prove the existence of an infinite dimensional graded super-module for the finite sporadic Thompson group $Th$ whose McKay-Thompson series are weakly holomorphic modular forms of weight $\frac 12$ satisfying properties conjectured by Harvey and Rayhaun., Comment: 36 pages, 8 tables v2 addresses reviewer's comments and corrects a mistake in the proof of Lemma 2.10

Bloch-Okounkov studied certain functions on partitions $f$ called shifted symmetric polynomials. They showed that certain $q$-series arising from these functions (the so-called \emph{$q$-brackets} $\left_q$) are quasimodular forms. We revisit a family of such functions, denoted $Q_k$, and study the $p$-adic properties of their $q$-brackets. To do this, we define regularized versions $Q_k^{(p)}$ for primes $p.$ We also use Jacobi forms to show that the $\left_q$ are quasimodular and find explicit expressions for them in terms of the $\left_q$., Comment: 16 pages

The Umbral Moonshine Conjectures assert that there are infinite-dimensional graded modules, for prescribed finite groups, whose McKay-Thompson series are certain distinguished mock modular forms. Gannon has proved this for the special case involving the largest sporadic simple Mathieu group. Here we establish the existence of the umbral moonshine modules in the remaining 22 cases., Comment: 56 pages, to appear in Research in the Mathematical Sciences

Monstrous moonshine relates distinguished modular functions to the representation theory of the monster. The celebrated observations that 196884=1+196883 and 21493760=1+196883+21296876, etc., illustrate the case of the modular function j-744, whose coefficients turn out to be sums of the dimensions of the 194 irreducible representations of the monster. Such formulas are dictated by the structure of the graded monstrous moonshine modules. Recent works in moonshine suggest deep relations between number theory and physics. Number theoretic Kloosterman sums have reappeared in quantum gravity, and mock modular forms have emerged as candidates for the computation of black hole degeneracies. This paper is a survey of past and present research on moonshine. We also compute the quantum dimensions of the monster orbifold, and obtain exact formulas for the multiplicities of the irreducible components of the moonshine modules. These formulas imply that such multiplicities are asymptotically proportional to dimensions., Comment: 67 pages; a number of revisions and corrections in v.2, including a new result (Cor. 8.3) on the quantum dimensions of the monster orbifold, obtained following a suggestion of an anonymous referee

Mock modular forms, which give the theoretical framework for Ramanujan's enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves $E/\mathbb{Q}$. We show that mock modular forms which arise from Weierstrass $\zeta$-functions encode the central $L$-values and $L$-derivatives which occur in the Birch and Swinnerton-Dyer Conjecture. By defining a theta lift using a kernel recently studied by H\"ovel, we obtain canonical weight 1/2 harmonic Maass forms whose Fourier coefficients encode the vanishing of these values for the quadratic twists of $E$. We employ results of Bruinier and the third author, which builds on seminal work of Gross, Kohnen, Shimura, Waldspurger, and Zagier. We also obtain $p$-adic formulas for the corresponding weight 2 newform using the action of the Hecke algebra on the Weierstrass mock modular form., Comment: To appear in Research in Number Theory

The two Rogers-Ramanujan $q$-series \[ \sum_{n=0}^{\infty}\frac{q^{n(n+\sigma)}}{(1-q)\cdots (1-q^n)}, \] where $\sigma=0,1$, play many roles in mathematics and physics. By the Rogers-Ramanujan identities, they are essentially modular functions. Their quotient, the Rogers-Ramanujan continued fraction, has the special property that its singular values are algebraic integral units. We find a framework which extends the Rogers-Ramanujan identities to doubly-infinite families of $q$-series identities. If $a\in\{1,2\}$ and $m,n\geq 1$, then we have \[ \sum_{\substack{\lambda \lambda_1\leq m}} q^{a|\lambda|} P_{2\lambda}(1,q,q^2,\dots;q^n) =\textrm{"infinite product modular function"}, \] where the $P_{\lambda}(x_1,x_2,\dots;q)$ are Hall-Littlewood polynomials. These $q$-series are specialized characters of affine Kac--Moody algebras. Generalizing the Rogers-Ramanujan continued fraction, we prove in the case of $\textrm{A}_{2n}^{(2)}$ that the relevant $q$-series quotients are integral units., Comment: 44 pages. This paper supersedes the paper arXiv:1309.5216 "The A_{2n}^{(2)} Rogers-Ramanujan identities"; Improved introduction, typos corrected and references added; Final version to appear in Duke Mathematical Journal; Typographical errors are corrected

There are two families of Donaldson invariants for the complex projective plane, corresponding to the SU(2)-gauge theory and the SO(3)-gauge theory with non-trivial Stiefel-Whitney class. In 1997 Moore and Witten conjectured that the regularized u-plane integral on the complex projective plane gives the generating functions for these invariants. In earlier work the second two authors proved the conjecture for the SO(3)-gauge theory. Here we complete the proof of the conjecture by confirming the claim for the SU(2)-gauge theory. As a consequence, we find that the SU(2) Donaldson invariants for CP^2 are explicit linear combinations of the Hurwitz class numbers which arise in the theory of imaginary quadratic fields and orders., Comment: 18 pages. arXiv admin note: text overlap with arXiv:0808.1442, arXiv:1008.0175

Reports on a visual content analysis of 1,104 Gulf War-related pictures. Finds that a narrowly limited range of images (with a special emphasis on cataloging military weaponry and technology) dominated the pictorial coverage. Suggests that the scarcity of pictures depicting ongoing events in the Gulf contradicts the impression of first-hand media coverage promoted during the conflict. (SR)

In membrane-aerated biofilm reactors (MABRs), hollow fibers are used to supply oxygen to the biofilms and bulk fluid. A pressure and concentration gradient between the inner volume of the fibers and the reactor reservoir drives oxygen mass transport across the fibers toward the bulk solution, providing the fiber-adhered biofilm with oxygen. Conversely, bacterial metabolic gases from the bulk liquid, as well as from the biofilm, move opposite to the flow of oxygen, entering the hollow fiber and out of the reactor. Metabolic gases are excellent indicators of biofilm vitality, and can aid in microbial identification. Certain gases can be indicative of system perturbations and control anomalies, or potentially unwanted biological processes occurring within the reactor. In confined environments, such as those found during spaceflight, it is important to understand what compounds are being stripped from the reactor and potentially released into the crew cabin to determine the appropriateness or the requirement for additional mitigation factors. Reactor effluent gas analysis focused on samples provided from Kennedy Space Center's sub-scale MABRs, as well as Johnson Space Center's full-scale MABRs, using infrared spectroscopy and gas chromatography techniques. Process gases, such as carbon dioxide, oxygen, nitrogen, nitrogen dioxide, and nitrous oxide, were quantified to monitor reactor operations. Solid Phase Microextraction (SPME) GC-MS analysis was used to identify trace volatile compounds. Compounds of interest were subsequently quantified. Reactor supply air was examined to establish target compound baseline concentrations. Concentration levels were compared to average ISS concentration values and/or Spacecraft Maximum Allowable Concentration (SMAC) levels where appropriate. Based on a review of to-date results, current trace contaminant control systems (TCCS) currently on board the ISS should be able to handle the added load from bioreactor systems without the need for secondary mitigation.

Current International Space Station water recovery regimes produce a sizable portion of waste water brine. This brine is highly toxic and water recovery is poor: a highly wasteful proposition. With new biological techniques that do not require waste water chemical pretreatment, the resulting brine would be chromium-free and nitrate rich which can allow possible fertilizer recovery for future plant systems. Using a system of ion exchange resins we can remove hardness, sulfate, phosphate and nitrate from these brines to leave only sodium and potassium chloride. At this point modern chlor-alkali cells can be utilized to produce a low salt stream as well as an acid and base stream. The first stream can be used to gain higher water recovery through recycle to the water separation stage while the last two streams can be used to regenerate the ion exchange beds used here, as well as other ion exchange beds in the ISS. Conveniently these waste products from ion exchange regeneration would be suitable as plant fertilizer. In this report we go over the performance of state of the art resins designed for high selectivity of target ions under brine conditions. Using ersatz ISS waste water we can evaluate the performance of specific resins and calculate mass balances to determine resin effectiveness and process viability. If this system is feasible then we will be one step closer to closed loop environmental control and life support systems (ECLSS) for current or future applications.

Speech topics include: Leadership in Space; Space Exploration: Real and Acceptable Reasons; Why Explore Space?; Space Exploration: Filling up the Canvas; Continuing the Voyage: The Spirit of Endeavour; Incorporating Space into Our Economic Sphere of Influence; The Role of Space Exploration in the Global Economy; Partnership in Space Activities; International Space Cooperation; National Strategy and the Civil Space Program; What the Hubble Space Telescope Teaches Us about Ourselves; The Rocket Team; NASA's Direction; Science and NASA; Science Priorities and Program Management; NASA and the Commercial Space Industry; NASA and the Business of Space; American Competitiveness: NASA's Role & Everyone's Responsibility; Space Exploration: A Frontier for American Collaboration; The Next Generation of Engineers; System Engineering and the "Two Cultures" of Engineering; Generalship of Engineering; NASA and Engineering Integrity; The Constellation Architecture; Then and Now: Fifty Years in Space; The Reality of Tomorrow; and Human Space Exploration: The Next 50 Years.

The purpose of this NASA Earth Science Technology Office funded effort was to flight validate an on-board cloud detection algorithm and to determine the performance that can be achieved with a Mongoose V flight computer. This validation was performed on the EO-1 satellite, which is operational, by uploading new flight code to perform the cloud detection. The algorithm was developed by MIT/Lincoln Lab and is based on the use of the Hyperion hyperspectral instrument using selected spectral bands from 0.4 to 2.5 microns. The Technology Readiness Level (TRL) of this technology at the beginning of the task was level 5 and was TRL 6 upon completion. In the final validation, an 8 second (0.75 Gbytes) Hyperion image was processed on-board and assessed for percentage cloud cover within 30 minutes. It was expected to take many hours and perhaps a day considering that the Mongoose V is only a 6-8 MIP machine in performance. To accomplish this test, the image taken had to have level 0 and level 1 processing performed on-board before the cloud algorithm was applied. For almost all of the ground test cases and all of the flight cases, the cloud assessment was within 5% of the correct value and in most cases within 1-2%.

Nearly four decades into the space age, our biggest problem is still the high cost of transportation to space. Close behind are the relatively infrequent flight opportunities and high risk of the transportation systems we have. These problems are of course not independent; rather, they are interrelated measures of the difficulty of accelerating a payload through an ideal velocity increment of over 9 km/s and a height of 200 km or more in just a few minutes. These difficulties are not unrecognized, and the demand for newer, cheaper, more reliable, more frequently flying launch systems has been unending. Hundreds, maybe thousands of studies have been done on various approaches offering improved access to space, and each approach has its ardent supporters, convinced that no other approach can work as well. In the government, hardly a year passes without the attempted initiation of a new program to lower the cost and improve the reliability of space transportation. This paper does not attempt to decide the merits and demerits of various transportation concepts but examines the cost of access to space from a slightly different perspective, and attempts to use that perspective to reach some conclusions about characteristics that a next-generation system must have to yield lower cost, regardless of its design.

In November of 1991, the First ISCCP (International Satellite Cloud Climatology Project) Regional Experiment (FIRE) Phase II cirrus study took place at Coffeyville, Kansas. The field experiment incorporated instrumentation from surface, aircraft, and satellite to attempt to define the optical, radiative, and microphysical characteristics of these high altitude, predominantly ice clouds. The NCAR King Air research aircraft was outfitted with a variety of radiative and microphysical instrumentation for the FIRE II project. Included for this project was the SSH-2, a 16-channel passive radiometer. The SSH-2 was originally designed as a space-qualified infrared (IR) temperature and water vapor sounder for deployment onboard the Defense Meteorological Satellite Program (DMSP) series of environmental satellites. For this experiment, only those channels associated with the water vapor profiling function have been examined although downwelling radiance measurements were taken at all channels during the project. With supporting information from the aircraft telemetry observations it may be possible to relate these SSH-2 measurements to cloud radiative and microphysical properties. The following sections will describe the spectral characteristics of the instrument, the calibration scheme used to convert the raw measured counts into calibrated radiances, and the case studies that will be covered in this paper. This will be followed by a discussion of the results of this preliminary investigation and a description of future work to be done.

This paper addresses the dilemma which NASA faces in starting a major new initiative within the constraints of the current national budget. It addressed the fact that unlike previous NASA programs, the major mission constraints come from management factors as opposed to technologies. An action plan is presented, along with some results from early management simplification processes.

Activities and National Aeronautics and Space Administration (NASA) programs, both ongoing and planned, are described by NASA administrative personnel from the offices of Space Science and Applications, Space Systems Development, Space Flight, Exploration, and from the Johnson Space Center. NASA's multi-year strategic plan, called Vision 21, is also discussed. It proposes to use the unique perspective of space to better understand Earth. Among the NASA programs mentioned are the Magellan to Venus and Galileo to Jupiter spacecraft, the Cosmic Background Explorer, Pegsat (the first Pegasus payload), Hubble, the Joint U.S./German ROSAT X-ray Mission, Ulysses to Jupiter and over the sun, the Astro-Spacelab Mission, and the Gamma Ray Observatory. Copies of viewgraphs that illustrate some of these missions, and others, are provided. Also discussed were life science research plans, economic factors as they relate to space missions, and the outlook for international cooperation.

A technique is described which combines up to three channels of imagery into a single 8-bit composite color image. Each image controls a specific color or colors (red, green, or blue) in the resultant color composite image, allowing for improved interpretation of scene characteristics. The technique is a useful tool for the forecaster and satellite image analyst. In addition to providing the capability of displaying multichannel images on an 8-bit desktop color workstation, the technique allows sophisticated image manipulation to be performed.

The optical and microphysical properties of subvisual cirrus clouds are derived from ground-based polarization lidar, shortwave radiation flux, and solar corona measurements of two approximately 0.75 km deep cirrus located near the tropopause. The first cloud produced no visual manifestations under excellent viewing conditions, and the second appeared to be a persistent aircraft contrail that was generally visible except in the zenith direction. Average lidar linear depolarization ratios and volume backscatter coefficients for the two clouds were 0.19 and 0.35, and 0.6 x 10 to the -3 and 1.4 x 10 to the -3 /km sr, respectively. It is estimated that the zenith-subvisual cirrus contained ice crystals of 25-micron effective diameter at a mean concentration of 25/1 and ice mass content of 0.2 mg/cu m. The threshold cloud optical thickness for visual-versus-invisible cirrus, derived from both broadband shortwave flux and 0.694 micrometer lidar data, is found to be tau sub c approx equal 0.03. Such tau values are comparable to those of 5 to 10 km deep stratospheric aerosol clouds of volcanic origin and polar stratospheric clouds, which are episodic in nature. Hence, we conclude that if these clouds are a fairly common feature of the upper troposphere, as recent SAGE satellite measurements would suggest, then the impact of natural and contrail subvisual cirrus on the planet's radiation balance may be relatively significant.

The optical and microphysical properties of subvisual cirrus clouds are derived from ground-based polarization lidar, shortwave radiation flux, and solar corona measurements of two approximately 0.75 km deep cirrus located near the tropopause. The first cloud produced no visual manifestations under excellent viewing conditions, and the second appeared to be a persistent aircraft contrail that was generally visible except in the zenith direction. Average lidar linear depolarization ratios and volume backscatter coefficients for the two clouds were 0.19 and 0.35, and 0.6x10 to the -3 and 1.4x10 to the -3 /km sr, respectively. It is estimated that the zenith-subvisual cirrus contained ice crystals of 25 micron effective diameter at a mean concentration of 25/l and ice mass content of 0.2 mg/cu m. The threshold cloud optical thickness for visual-versus-invisible cirrus, derived from both broadband shortwave flux and 0.694 micrometer lidar data, is found to be tau sub c approx equal 0.03. Such tau values are comparable to those of 5 to 10 km deep stratospheric aerosol clouds of volcanic origin and polar stratospheric clouds, which are episodic in nature. Hence, we conclude that if these clouds are a fairly common feature of the upper troposphere, as recent SAGE satellite measurements would suggest, then the impact of natural and contrail subvisual cirrus on the planet's radiation balance may be relatively significant.