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We read with much interest the recent observational study of Nougier et al., which aimed at studying thrombin generation (TG) and fibrinolysis profiles of COVID‐19 patients admitted to an intensive care unit (ICU) or to an internal medicine ward and receiving various schemes of prophylactic heparin.[1] They reported that thrombin potential remained within normal range despite heparin and that fibrinolysis was decreased in relation with increased plasminogen activator inhibitor 1 (PAI‐1) and thrombin‐activatable fibrinolysis inhibitor (TAFI) antigen plasma levels. Using the rotational thromboelastometry (ROTEM) delta device with EXTEM reagents and the addition of 0.625µg/mL tPA (referred to as ‘TEM‐tPA’), they reported decreased clot lysis in COVID‐19 patients, which was more pronounced in patients who presented a thrombotic event, compared to event‐free patients.

Introduction: Estimation of residual apixaban plasma concentrations may be requested in the management of emergencies. This study aims at assessing the performance of specific anti-Xa assays calibrated with apixaban on real-life samples with low apixaban plasma concentrations (®-Liquid Anti-Xa assay (STA ® LAX) and the low and normal procedures of the Biophen ®Direct Factor Xa Inhibitors (DiXaI) assay was tested on 134 blood samples, collected from patients on apixaban, wherefrom 74 patients received LMWH after apixaban cessation. The results were compared with the liquid chromatography coupled with tandem mass spectrometry (LC-MS/MS) measurements. Results: The Biophen ®DiXaI, Biophen ®DiXaI LOW, and STA ®LAX showed very good correlation with LC-MS/MS measurements in patients without LMWH administration (Spearman r.95,.99, and.98, respectively). Their limits of quantitation were defined at 48, 24, and 12 ng/mL, respectively. The Bland-Altman test measured mean bias (SD) at 5.6 (13.1), −2.5 (5.0), and −0.8 (6.1) ng/ml, respectively. The Spearman r of the Biophen ®DiXaI decreased to 0.64 in presence of low apixaban concentrations. The Spearman r of the Biophen ®DiXaI LOW and STA ®LAX decreased to 0.39 and 0.26, respectively, in presence of LMWH. Conclusions: The accuracy of the low methodologies (Biophen ®DiXaI LOW and STA ®LAX) is slightly improved for low apixaban plasma concentrations, compared with the normal procedure of Biophen ®DiXaI. The interference of LMWH on the low methodologies is measurable, however, less important than the previously reported interference of LMWH on rivaroxaban calibrated specific anti-Xa assays.

INTRODUCTION: Thrombin generation (TG) documents hypercoagulability. TG in platelet-poor plasma is exquisitely sensitive to heparins, which thus must be neutralized before testing. Heparinase and hexadimethrine bromide (polybrene) have been used for that purpose, but their effects per se on TG have been poorly studied so far.METHODS: (i) TG was studied in commercial normal pooled plasma (NPP; CryoCheck® , Cryopep) in absence or presence of neutralizing agents. (ii) NPP was spiked with increasing concentrations of unfractionated heparin (UFH; up to 1.0 IU/mL) or low-molecular-weight heparin (LMWH; enoxaparin up to 1.2 IU/mL) and TG studied after incubation of heparinase (Hepzyme® ; 15 minutes) or polybrene (0.025 mg/mL; 10 minutes).RESULTS: (i) With ThromboScreen reagent to initiate TG, addition of heparinase was associated with increased peak, whereas polybrene caused lengthening of lag time and time to peak, compared with nonsupplemented NPP. (ii) With polybrene, TG was completely restored over the whole range of UFH and LMWH studied. By contrast, heparinase failed to fully restore TG in presence of UFH concentrations ≥0.8 IU/mL or LMWH concentrations ≥1.0 IU/mL. Those effects were matched with detectable tiny residual amounts of non-neutralized heparin (as assessed with an anti-Xa assay) and were less pronounced with a higher picomolar concentration of tissue factor (DrugScreen reagent).CONCLUSION: Polybrene fully restored TG of heparinized plasma at the expense of an alteration of TG, pointing to the need to use adapted reference ranges. Heparinase failed to do so in presence of high concentrations of both heparins.

Purpose The concomitant use of direct oral anticoagulants (DOAC) and strong P-glycoprotein (P-gp) and cytochrome P450 3A4 (CYP3A4) inducers may lead to reduced DOAC levels and therapeutic failure. This study aimed to describe DOAC concentrations in patients receiving strong P-gp and CYP3A4 inducers, in relation to individual risk factors for high or low DOAC levels. Methods We retrospectively identified hospitalized patients simultaneously receiving a DOAC and carbamazepine, phenobarbital, phenytoin, or rifampicin between 2016 and 2021. Among them, patients who underwent DOAC measurement at steady state were included. DOAC peak or trough levels were compared with on-therapy ranges observed in pivotal trials. Individual risk factors for high or low DOAC levels were identified. Results We included 17 patients (median age 75 years), mainly receiving apixaban and carbamazepine. For 5 patients (29%), DOAC trough or peak level was below the expected range. Among the remaining 12 patients, 8 had at least one measurement in the lower quartile of the range. The median number of risk factors for drug accumulation was 0 (range 0-1) in patients with ≥1 measurement below the range and 2 (range 0-3) in other patients. DOAC measurement led to treatment adjustments in 9 patients (DOAC dose increase or switch). Conclusion Our data suggest a significant risk of reduced DOAC levels in patients taking strong P-gp and CYP3A4 inducers, especially those without risk factors for drug accumulation. DOAC measurement could help manage this relevant drug–drug interaction.

INTRODUCTION: Double centrifugation before freezing is recommended before thrombin generation assays (TGA). However, this procedure is not mandatory for routine hemostasis tests, precluding the use of these samples for TGA. The aim of this study is to assess the impact of single and double centrifugation on TGA performed on frozen samples from healthy volunteers (HVs) and patients receiving direct oral anticoagulants (DOACs).METHODS: Forty HVs and 57 patients receiving a DOAC (dabigatran, rivaroxaban, apixaban, or edoxaban) were included in this prospective double-center observational study. Blood was collected into 109 mmol/L citrated tubes and frozen at -70°C before TGA using ST Genesia with STG-DrugScreen reagent. Four pre-analytical conditions were studied: (A) single centrifugation (2000 g, 15 minutes) before freezing; (B) one centrifugation before freezing and another after thawing (2000 g, 15 minutes for both); (C) one centrifugation before freezing(2000 g, 15 minutes) and another after thawing (2000 g, 10 minutes); (D) double centrifugation (2000 g, 15 minutes) before freezing (reference). Centrifugation conditions (A), (B), and (C) were compared with the reference condition (D). Acceptable relative differences were defined at 6%, 8%, and 10% for normalized lag time, endogenous thrombin potential, and peak height, respectively.RESULTS: Centrifugation conditions had a small but acceptable impact on HVs samples, but single centrifugation always resulted in unacceptable reductions in normalized lag times for DOAC samples. A second centrifugation after thawing permitted the recovery of acceptable differences for the three TGA parameters for edoxaban but not for apixaban, rivaroxaban, nor dabigatran.CONCLUSION: Double centrifugation before freezing should remain the recommended pre-analytical condition before TGA.

In their recent article, Helin et al reported on a comparative study involving 43 frozen samples assessed on the ST-Genesia using the STG-DrugScreen among which 20 of them were also analyzed on the calibrated automated thrombogram (CAT) using the PPP-reagent. The authors concluded that “the ST-Genesia remains limited in its practical use, failing to measure over-anticoagulation using warfarin and heparin, as well as showing normal ETP with DOAC.” [...]

Resume Depuis plusieurs annees, des centres multidisciplinaires de traitement de la douleur chronique se sont developpes pour faire face a la complexite multidimensionnelle de cette pathologie. Malgre cette avancee majeure dans la prise en charge des patients, les soignants restent confrontes quotidiennement a des situations qui s’enlisent et perdurent au-dela de ce a quoi patients et soignants se seraient attendus ou auraient souhaite. Parmi les nombreuses causes possibles, certaines pourraient directement resulter de l’inadequation de la prise en charge elle-meme. Soignants et patients restent, en effet, souvent ancres dans un modele de prise en charge « aigu » dans lequel la douleur devrait naturellement s’attenuer – voire disparaitre – apres quelques semaines ou mois de traitement. Or, l’arsenal therapeutique disponible n’a souvent que peu d’emprise sur la douleur chronique elle-meme une fois installee. Par ailleurs, cette situation peut egalement etre source de souffrance pour le personnel soignant, confronte a ses propres limites ou s’epuisant dans une surenchere diagnostique ou therapeutique futile. La decentralisation du symptome douloureux, qui echappe trop souvent aux possibilites therapeutiques, pour se tourner vers des objectifs avant tout fonctionnels et adaptes au contexte particulier du patient, semble representer un cap a franchir, tant pour le soignant que pour le patient. Ceci implique la necessite d’un changement de paradigme et d’une prise en charge au long cours pour finalement se donner la possibilite d’atteindre des resultats concrets en termes de fonctionnement et de qualite de vie, et d’eviter l’epuisement et le desinvestissement tant du soignant que du patient.

Coronavirus disease 2019 (COVID-19) is associated with extreme inflammatory response, disordered hemostasis and high thrombotic risk. A high incidence of thromboembolic events has been reported despite thromboprophylaxis, raising the question of a more effective anticoagulation. First-line hemostasis tests such as activated partial thromboplastin time, prothrombin time, fibrinogen and D-dimers are proposed for assessing thrombotic risk and monitoring hemostasis, but are vulnerable to many drawbacks affecting their reliability and clinical relevance. Specialized hemostasis-related tests (soluble fibrin complexes, tests assessing fibrinolytic capacity, viscoelastic tests, thrombin generation) may have an interest to assess the thrombotic risk associated with COVID-19. Another challenge for the hemostasis laboratory is the monitoring of heparin treatment, especially unfractionated heparin in the setting of an extreme inflammatory response. This review aimed at evaluating the role of hemostasis tests in the management of COVID-19 and discussing their main limitations.

This data article accompanies the manuscript entitled: “Prothrombotic Disturbances of hemostasis of Patients with Severe COVID-19: a Prospective Longitudinal Observational Cohort Study” submitted to Thrombosis Research by the same authors. We report temporal changes of plasma levels of an extended set of laboratory parameters during the ICU stay of the 21 COVID-19 patients included in the monocentre cohort: CRP, platelet count, prothrombin time; Clauss fibrinogen and clotting factors II, V and VIII levels, D-dimers, antithrombin activity, protein C, free protein S, total and free tissue factor pathway inhibitor, PAI-1 levels, von Willebrand factor antigen and activity, ADAMTS-13 (plasma levels); and of two integrative tests of coagulation (thrombin generation with ST Genesia) and fibrinolysis (global fibrinolytic capacity - GFC). Regarding hemostasis, we used double-centrifuged frozen citrated plasma prospectively collected after daily performance of usual coagulation tests. Demographic and clinical characteristics of patients and thrombotic and hemorrhagic complications were also collected from patient's electronic medical reports.

Background The ability to identify horses at risk for catastrophic injuries continues to be a pressing issue for the racing industry, especially given recent events in North America. Objectives Since most catastrophic injuries occur in areas of existing pathology and this pathology is likely to elicit an inflammatory response, it was hypothesised that analysis of messenger RNA (mRNA) expression would detect significant changes in select genes in horses at risk for a catastrophic injury. Study design Prospective cohort study. Methods Five racing jurisdictions across the United States participated in this study. A total of 686 Tempus® RNA Blood Tube samples were collected for mRNA analysis from 107 catastrophically injured horses, as well as from non-injured horses sampled either pre-race (n=374) or post-race (n=205). A subset of horses (n=37) were sampled both pre-race and post-race for analysis of expression changes during the post-race period. Results Out of 21 genes analysed via RT-qPCR, the expression of 12 genes (ALOX5AP, CD14, IL-10, IL-1β, IL-6, IL-8, MMP1, PTGS2, TLR4, TNFα, TNFSF13B, and VEGFA) changed significantly within 45 minutes after a race and were excluded. Of the remaining nine genes (BMP-2, IGF-1, IL1RN, MMP2, MMP9, Osteoprotegrin, RANKL, SAA1, and TGFβ), three genes (IGF-1, IL1RN, and MMP2) were found to be significantly different between catastrophically injured and non-injured horses using multiple logistic regression modeling. Receiver operating characteristic analysis of models, which included mRNA expression, demonstrated sensitivities from 76-82% (95% CI: 67% - 93%) and specificities from 84-88% (95% CI: 71% - 94%) at the Youden Index. Main limitations Samples were collected as soon as possible post-injury (within 30 minutes). Conclusions Analysis of mRNA expression of specific genes in the future may be considered as an economical, accessible, and non-invasive means by which horses at risk for catastrophic injury can be identified.

Platelet count, indices (mean volume, young&mdash, immature platelet fraction) and aggregation are widely used laboratory parameters to investigate primary hemostasis. We performed a systematic, thorough evaluation of the influence of the time-interval since blood draw from 20 healthy individuals and of the anticoagulation of collected blood on such parameters. Blood was anticoagulated with citrate, K2-ethylenediaminetetraacetic acid (EDTA) and hirudin and analyzed 5, 30, 60, 120 and 180 min after blood draw. Multiple electrode aggregometry (MEA) was performed with either hirudin (half-diluted with NaCl) or citrate samples (half-diluted with NaCl or CaCl2 3 mM). Platelet count and indices (Sysmex XN-20) were rather stable over time with EDTA blood. MEA results were lower with citrate blood than with hirudin blood, supplementation with calcium was partially compensatory. MEA results were also lower when performed less than 30 or more than 120 min after blood draw. Platelet clumping, quantitatively estimated with microscope examination of blood smears, was more important in hirudin blood than citrate or EDTA blood and could explain some of the differences observed between preanalytical variables. The results stress once more the importance of preanalytical variables in hemostasis laboratory testing. Decision thresholds based on those tests are only applicable within specific preanalytical conditions.

P2Y12 inhibitor discontinuation is essential before most surgical interventions to limit bleeding complications. Based on pharmacokinetic data, fixed discontinuation durations have been recommended. However, as platelet function recovery is highly variable among patients, a more individualized approach based on platelet function testing (PFT) has been proposed. The aim of this retrospective single-centre study was to determine whether PFT using whole blood adenosine diphosphate&ndash, multiple electrode aggregometry (ADP&ndash, MEA) was associated with a safe reduction of preoperative waiting time. Preoperative ADP&ndash, MEA was performed for 29 patients on P2Y12 inhibitors. Among those, 17 patients underwent a coronary artery bypass graft. Twenty one were operated with an ADP&ndash, MEA &ge, 19 U (quantification of the area under the aggregation curve), and the waiting time was shorter by 1.6 days (median 1.8 days, IQR 0.5&ndash, 2.9), by comparison with the current recommendations (five days for clopidogrel and ticagrelor, seven days for prasugrel). Platelet function recovery was indeed highly variable among individuals. With the 19 U threshold, high residual platelet inhibition was associated with perioperative platelet transfusion. These results suggest that preoperative PFT with ADP&ndash, MEA could help reduce waiting time before urgent surgery for patients on P2Y12 inhibitors.

International audience; COVID-19 is associated with disturbances of hemostasis in the laboratory and an increased thrombotic risk. Routine laboratory tests - activated partial thromboplastin time (aPTT), prothrombin time, Clauss fibrinogen and D-dimers levels measurement - are used for the evaluation of the thrombotic risk and the monitoring of hemostasis, but are subject to several drawbacks that may affect the reliability and clinical relevance of the delivered results. Another challenge for the hemostasis laboratory is the monitoring of heparin treatment. For instance, the issue of the monitoring of unfractionated heparin remains debated, the more so when there is a tremendous inflammatory response. This brief review considers the role of laboratory tests of hemostasis in the management of COVID-19 and discusses their main limitations to be kept in mind.

INTRODUCTION: Andexanet alfa is a recombinant modified factor Xa protein that has been developed to reverse factor Xa inhibitors. Since May 2018, the FDA has approved its utilization in patients treated with apixaban and rivaroxaban in case of life-threatening or uncontrolled bleeding. On 28 of February 2019, the Committee for Medicinal Products for Human Use adopted a positive opinion, recommending the granting of a conditional marketing authorization for andexanet alfa in Europe. Area covered: The authors provide an overview of andexanet alfa development and its pharmacokinetic and pharmacodynamic properties. The results of the clinical phase III trial ANNEXA as well as current limitations related to andexanet alfa are also discussed. Expert opinion: Although phase I and II studies have proven that andexanet alfa can be effective in reversing the effect of factor Xa inhibitors, its efficacy in major bleeding patients has only been shown for apixaban and rivaroxaban, without any comparator group. Well-designed studies comparing the efficacy and safety of andexanet alfa to other reversal strategies are required to confirm preliminary data. The benefit of andexanet alfa in specific settings needs to be investigated and its use in clinical practice needs to be facilitated by the implementation of international guidelines.

e24147 Background: Early referral to palliative care (PC) for patients w/ adv cancer is supported by compelling evidence from large RCTs demonstrating a lower symptom burden, higher QOL, and increased OS. However, these studies reflect pts who have self-selected to accept a PC referral and attend a PC visit. WHO/NCCN/ASCO guidelines support early integration of PC. We sought to characterize the referral patterns in our heme/onc practice to identify and mitigate the barriers to early PC adoption in the community setting. We began a concerted effort to discuss early PC referral w/ pts recently diagnosed w/ adv cancer at the time systemic therapy was initiated. Methods: To ensure real-world applicability of this study, we identified a large satellite clinic of a major academic center w/ access to PC on the main campus (located 20 miles from the satellite clinic). We retrospectively reviewed new pts age 18y+ w/ adv cancer, characterized PC referrals and outcomes. Using qualitative methodology, we identified pt-reported barriers to accepting PC care. Results: 407 new pts were seen; 168 w/ benign heme, 145 w/ early cancers, 94 w/ adv cancers. Of the 94 pts w/ adv cancers, 25 pts had one-time 2ndopinion visits, and 16 pts were not candidates for, or did not desire cancer treatment and directly enrolled on hospice. Our analysis cohort consisted of 53 pts w/ adv cancer pursuing life prolonging therapy. At initiation of treatment, 57 % (n=30) were not offered a PC referral, 22% (n=12) received a PC referral and attended the appt, however 21% (n=11) received a PC referral but did not attend the appt. A qualitative analysis of the 11 patients referred to early PC who did not attend the appt revealed; 5 patients scheduled an appt but did not attend (3-unknown reason, 2-hospitalized during appt, 1-lack of transportation), 2 pts were unreachable, and 4 pts were contacted but declined to schedule stating: “I feel pressured” “I want to hear what other treatment options I have” “I want to be treated first and then see if I need it” "I am overwhelmed with too many new doctors and visits". Conclusions: Despite the benefit of early PC referral in pts w/ adv cancer, we identified a considerable gap in its adoption in our community practice despite access to proximate PC clinic. Further studies are under design to address institutional and pt-related factors to improve real-world adoption of this critical service.

How many permutations of the natural numbers are needed so that every conditionally convergent series of real numbers can be rearranged to no longer converge to the same sum? We define the rearrangement number, a new cardinal characteristic of the continuum, as the answer to this question. We compare the rearrangement number with several natural variants, for example one obtained by requiring the rearranged series to still converge but to a new, finite limit. We also compare the rearrangement number with several well-studied cardinal characteristics of the continuum. We present some new forcing constructions designed to add permutations that rearrange series from the ground model in particular ways, thereby obtaining consistency results going beyond those that follow from comparisons with familiar cardinal characteristics. Finally, we deal briefly with some variants concerning rearrangements by a special sort of permutation and with rearranging some divergent series to become (conditionally) convergent.

We prove some identities involving tangents, secants, and cosecants of infinite sums.

A simple partial version of the Fundamental Theorem of Calculus can be presented on the first day of the first-year calculus course, and then relied upon repeatedly in assigned problems throughout the course. With that experience behind them, students can use the partial version to understand the full-fledged Fundamental Theorem, with further assigned problems leading them through that process.

CcO (cytochrome c oxidase) is a multisubunit bigenomic protein complex which catalyses the last step of the mitochondrial electron transport chain. The nuclear-encoded subunits are thought to have roles either in regulation or in the structural stability of the enzyme. Subunit Vb is a peripheral nuclear-encoded subunit of mammalian CcO that is dramatically reduced under hypoxia. Although it has been shown to contain different ligand-binding sites and undergo modifications, its precise function is not known. In the present study we generated a cell line from RAW 264.7 murine macrophages that has a more than 80% reduced level of Vb. Functional analysis of these cells showed a loss of CcO activity, membrane potential and less ability to generate ATP. Resolution of complexes on blue native gel and two-dimensional electrophoretic analysis showed an accumulation of subcomplexes of CcO and also reduced association with supercomplexes of the electron transfer chain. Furthermore, the mitochondria from CcO Vb knock-down cells generated increased ROS (reactive oxygen species), and the cells were unable to grow on galactose-containing medium. Pulse-chase experiments suggest the role of the CcO Vb subunit in the assembly of the complex. We show for the first time the role of a peripheral, non-transmembrane subunit in the formation as well as function of the terminal CcO complex.

Scaled Boolean algebras are a category of mathematical objects that arose from attempts to understand why the conventional rules of probability should hold when probabilities are construed, not as frequencies or proportions or the like, but rather as degrees of belief in uncertain propositions. This paper separates the study of these objects from that not-entirely-mathematical problem that motivated them. That motivating problem is explicated in the first section, and the application of scaled Boolean algebras to it is explained in the last section. The intermediate sections deal only with the mathematics. It is hoped that this isolation of the mathematics from the motivating problem makes the mathematics clearer., 53 pages, 8 Postscript figures, Uses ajour.sty from Academic Press, To appear in Advances in Applied Mathematics

One warm friday night, A friend and I decided to pop into a Steak 'n Shake for a burger. We noticed an interesting claim on the place mat. According to the restaurant chain, a customer could order from the sandwich platter menu for a year without ordering the same meal twice. Wondering aloud whether the claim was true, my friend and I decided to have some fun and determine the answer. We arrived at a solution, quickly, but I also thought that this question would give middle school students an interesting opportunity to build connections between mathematical concepts and realistic situations.

This study used a wildfire loss simulation model to evaluate how different land use policies are likely to influence wildfire risk in the wildland urban interface (WUI) for Flathead County, Montana. The model accounts for the complex socio-ecological interactions among climate change, economic growth, land use change and policy, homeowner mitigations, and forest treatments in Flathead County's WUI over the five 10-year subperiods comprising the future evaluation period (i.e., 2010–2059). Wildfire risk, defined as expected residential losses from wildfire [E(RLW)], depends on the number of residential properties on parcels, the probability that parcels burn, the probability of wildfire losses to residential structures on properties given the parcels on which those properties are located burn, the average percentage of wildfire-related losses in aesthetic values of residential properties, and the total value (structures plus land) of residential properties. E(RLW) for the five subperiods is simulated for 2010 (referred to as the current), moderately restrictive, and highly restrictive land use policy scenarios, a moderate economic growth scenario and the A2 greenhouse gas emissions scenario. Results demonstrate that increasingly restrictive land use policy for Flathead County significantly reduces the amount and footprint of future residential development in the WUI. In addition, shifting from the current to a moderately restrictive land use policy for Flathead County significantly reduces wildfire risk for the WUI, but shifting from the current to a highly restrictive land use policy does not significantly reduce wildfire risk in the WUI. Both the methods and results of the study can help land and wildfire managers to better manage future wildfire risk and identify residential areas having potentially high wildfire risk.

Concentrated animal feeding operations around the globe generate large amounts of nitrous oxide (N(2)O) in the surrounding atmosphere. Liquid animal waste systems have received little attention with respect to N(2)O emissions. We hypothesized that the solution chemistry of animal waste aqueous suspensions would promote conditions that lead to N(2)O supersaturation at the liquid/air interface. The concentration of dissolved N(2)O in poultry litter (PL) aqueous suspensions at 25 degrees C was 0.36 microg N(2)O mL(-1), at least an order of magnitude greater than that measured in water in equilibrium with ambient air, suggesting N(2)O supersaturation. There was a nonlinear increase in the N(2)O Henry constants of PL from 2810 atm/mole fraction at 35 degrees C to 17 300 atm/mole fraction at 41 degrees C. The extremely high N(2)O Henry constants were partially ascribed to N(2)O complexation with aromatic moieties. Complexed N(2)O structures were unstable at temperatures > 35 degrees C, supplying the headspace with additional free N(2)O concentrations.

(2015). Using Infinitesimals to Differentiate Secant and Tangent. The American Mathematical Monthly: Vol. 122, No. 5, pp. 494-494.

We show that trigonometric identities arising from the most well known alternative to the arc-length parametrization of the circle share some of the same elaborate nature as the more famili...

The natural forms of the Leibniz rule for the $k$th derivative of a product and of Fa\`a di Bruno's formula for the $k$th derivative of a composition involve the differential operator $\partial^k/\partial x_1 ... \partial x_k$ rather than $d^k/dx^k$, with no assumptions about whether the variables $x_1,...,x_k$ are all distinct, or all identical, or partitioned into several distinguishable classes of indistinguishable variables. Coefficients appearing in forms of these identities in which some variables are indistinguishable are just multiplicities of indistinguishable terms (in particular, if all variables are distinct then all coefficients are 1). The computation of the multiplicities in this generalization of Fa\`a di Bruno's formula is a combinatorial enumeration problem that, although completely elementary, seems to have been neglected. We apply the results to cumulants of probability distributions., Comment: 13 pages

We show that the normalized likelihood function needed to get from a prior probability vector to the posterior that results from the minimum cross-entropy inference process has the highest entropy among all probability vectors satisfying an appropriate set of linear constraints. We regard the domains of the entropy and cross-entropy functions as groups. The relationship between the group structure and the entropy and cross-entropy functions is used to derive the result on entropies of likelihood functions.