Small Numbers Research Articles

Assessment of Prognostic Factors, Clinical Features Including the Microbiome, and Treatment Outcomes in Patients with Cancer of Unknown Primary Site

Introductions: cancer of unknown primary site (CUP) is a heterogeneous group of cancers in which metastases are found, and the primary tumor is not detected with available diagnostic methods. CUP is a disease that has not been fully researched, and its biology is unclear. The clinical characteristics of CUP are variable, but the prognosis of patients is usually unfavorable, and the possibilities of radical treatment are limited. The microbiome is the genes and gene products of microorganisms residing in a human body. In recent years, thanks to the use of next-generation sequencing, it is possible to assess the impact of the microbiome on human body functions. Head and neck cancers, due to the rich microbiome of this area, are influenced by it, and dysbiosis may be a risk factor for the development of cancer. Objective of this work: the aim of this study was to evaluate prognostic factors, clinical features including the microbiome, and treatment outcomes in patients with cancer of unknown primary site. Results: in the study group, increased numbers of bacteria of the phyla Bacteroides, Fusobacteria, Bacillota, Actinomycetota, Actinobacteria, and Candidatus were detected, while Firmicutes and Proteobacteria were detected in smaller numbers. Independent predictors of CUP occurrence were the following: leukocyte count of at most 6.49 × 103/mm, bacteria from the Proteobacteria phylum in the microbiome below 11.6%, Firmicutes below 22.1%, and Actinobacteria at least 11.0%. Increased numbers of Porphyromonas and Fusobacterium bacteria were associated with the risk of radiotherapy complications and shortened survival rate. Conclusions: clinical diagnosis and treatment of patients with CUP is complicated and difficult due to the lack of consensus on this issue. Treatment and prognosis of patients with CUP is unsatisfactory. The clinical value of the influence of the microbiome on the development, course, and treatment of cancer is becoming increasingly important. The microbiome may become a marker of response to anticancer treatment and the risk of its complications. Immunity modulation with the microbiome provides opportunities for further research on improving the effectiveness of oncological treatment. Fusobacterium and Porphyromonas seem to be the bacteria most important for the development of cancer, also worsening the prognosis of patients by increasing the risk of complications of radiotherapy and shortening the survival rate of patients. Streptococcus and Lactobacillus seem to be bacteria that reduce the risk of cancer, reduce the risk of complications, and improve the prognosis of patients. Total protein deficiency and elevated inflammatory markers are also important predictors of cancer risk.

THIN AND ULTRATHIN STRUCTURE OF ENVELOPES OF STEPHANOSTOMUM BACCATUM (TREMATODA: ACANTHOCOLPIDAE) METACERCARIAE IN YELLOWFIN SOLE, LIMANDA ASPERA.

The structure of the envelopes (capsule and cyst) surrounding metacercariae of Stephanostomum baccatum (Nicoll, 1907) in the second intermediate host, the yellowfin sole Limanda aspera (Pallas 1814), is examined with the methods of light and transmission electron microscopy. The cyst, presumably formed by secretions of the metacercarial tegument, consists of 2 layers: the outer, very thin layer of an electron-dense, finely granular substance and the inner layer composed of loose material of a moderate electron density that includes dense bodies varying in size, shape, and localization. The capsule, formed by the host's cells, is also organized into 2 distinct layers. The inner layer of the capsule is loose, consisting of evenly spaced debris of degenerated cells and lipid droplets with inclusions of intact macrophages between them. The outer layer of the capsule consists of parallel rows of cells arranged around the parasite, with fibroblasts and macrophages being dominant types and granulocytes and lymphocytes found in smaller numbers. Aggregations of collagen fibers are located in narrow spaces between the cells. The number of lipid droplets in the outer layer is significantly smaller than in the inner layer. The capsules formed around the examined trematodes have several structural features that distinguish them from those of S. baccatum and Stephanostomum sp. metacercariae recovered from other fishes of the family Pleuronectidae. The major morphological features of such capsules are the lack of epithelioid or giant multinucleated cells and the presence of numerous lipid droplets. Investigating the structural details of the envelopes surrounding metacercariae in trematodes, as well as other helminths, contributes to our scientific understanding of parasite biology, which can, in turn, have broader implications for understanding host-parasite interactions and evolutionary biology.

Optimal L(d,1)-Labeling of Certain Direct Graph Bundles Cycles over Cycles and Cartesian Graph Bundles Cycles over Cycles

An L(d,1)-labeling of a graph G=(V,E) is a function f from the vertex set V(G) to the set of nonnegative integers such that the labels on adjacent vertices differ by at least d and the labels on vertices at distance two differ by at least one, where d≥1. The span of f is the difference between the largest and the smallest numbers in f(V). The λ1d-number of G, denoted by λ1d(G), is the minimum span over all L(d,1)-labelings of G. We prove that λ1d(X)≤2d+2, with equality if 1≤d≤4, for direct graph bundle X=Cm×σℓCn and Cartesian graph bundle X=Cm□σℓCn, if certain conditions are imposed on the lengths of the cycles and on the cyclic ℓ-shift σℓ.

A proof of the Paz conjecture for 6 × 6 matrices

Let Mn(F) be the algebra of n×n matrices over a field F and let S be its generating set (as an F-algebra). The length of S is the smallest number k such that Mn(F) equals the F-linear span of all products of the length at most k of matrices from S. The length of Mn(F), denoted by l(Mn(F)), is defined to be the maximal length of any of its generating sets. In 1984, Paz conjectured that l(Mn(F))=2n−2, for any field F. This conjecture has been verified only for n⩽5. In this paper, we prove Paz's conjecture for n=6, meaning that l(M6(F))=10. We also prove that 12⩽l(M7(F))⩽13.

Particle response to oscillatory flows at finite Reynolds numbers

The response of spherical particles to oscillatory fluid flow forcing at finite Reynolds numbers exhibits significant deviations from classical analytical predictions due to nonlinear convective contributions. This study employs finite element simulations to explore the long-term (stationary) behavior of such particles across a wide range of conditions, including various external and particle Reynolds numbers, Strouhal numbers, and fluid-to-particle density ratios. Key contributions of this work include determining the range of validity of Tchen's equation of motion for infinitesimal and finite Reynolds numbers and correlating particle response for a wide range of density ratios and flow conditions at high frequency oscillations. This work introduces a modified form of the history drag term in a newly proposed Lagrangian equation of motion. The new equation incorporates a parameter-dependent fractional-order derivative tailored to accommodate nonlinearities due to convective effects. These novel correlations not only extend the operational range of existing model equations but also provide accurate estimates of particle response under a range of external flow conditions, as validated by comparison with numerical solutions of the Navier–Stokes flow around the particles.

PANORAMA DA FISIOTERAPIA OFERTADA A PACIENTES COM SEQUELAS DE QUEIMADURA

Introduction: Burns are thermal, electrical, chemical or radioactive traumas that damage body tissues and can lead to death. Objective: To analyze the data and information collected in the system, referring to the quantity of physiotherapeutic care offered, by the SUS, to patients with burn sequelae, in Brazil, in the last 10 years. Materials and methods: Data were obtained from the DATASUS database. The collected data was organized in an Excel spreadsheet, displayed in graphs and analyzed based on scientific literature. Result: In Brazil, in the last 10 years, 97,685 physiotherapeutic services were offered to medium and large burn patients, through the SUS. The largest number of care for medium and large burn patients was concentrated in the northeast region of Brazil and the smallest number in the Central-West region. Conclusion: The study provided a broad overview of the physiotherapeutic care provided to patients with burns in Brazil over the last ten years.

Apomictic seed formation and viability in Ilex paraguariensis A.St.-Hil. (Aquifoliaceae)

The objective of this study was to investigate fruit and seed formation in Ilex paraguariensis A.St.-Hil. without sexual reproduction in addition to evaluating their viability. This study was based on the hypothesis that the formation of apomictic fruit and seeds occurs, but the seeds are non-viable, contributing to the low germination percentage of the species. Five branches of 20 pistillate mother plants were bagged before anthesis (T1) and five immediately after fruit formation (TC). Once they reach maturity, the fruit were collected and counted. Their seeds were cut lengthwise and classified as intact (with embryo and endosperm) or non-intact (empty and/or deteriorated). The viability of intact seeds was evaluated with the tetrazolium test when they were classified as viable or non-viable. The data were submitted to the Kruskal-Wallis Test. The formation of viable fruit and seeds occurred in smaller numbers in T1, all statistically different from TC (p<0.05). Although in smaller quantities and with a statistical difference, the formation of viable fruit and seeds occurs without sexual reproduction, suggesting that the process responsible for their formation is the facultative apomixis. The higher percentage of viable fruit and seeds in TC is probably a result of the occurrence of both apomixis and sexual reproduction. The percentage of non-viable seeds does not differ between the two treatments, suggesting that apomixis may not be the cause of the low percentage of viable seeds and germination, requiring further studies to understand its causes.

Is polyphagy of a specific cryptic Bemisia tabaci species driving the high whitefly populations on cassava in eastern Africa?

AbstractSince the 1990s, the cryptic whitefly (Bemisia tabaci) has been linked to severe viral disease pandemics affecting cassava, a crucial staple crop in eastern Africa. This surge in whitefly populations has also been observed in other crops and uncultivated plants. While previous surveys have connected the increase on cassava to two specific populations, SSA1 and SSA2, the dynamics behind the population growth on other plants remain unclear. Additionally, other B. tabaci species, including EA1, IO, MED, SSA9, and SSA10, have been found on cassava in smaller numbers. This study aimed to identify the host plants that support the growth and development of different B. tabaci in Uganda by collecting fourth-instar nymphs from cassava and 20 other common host plants. Host transfer experiments were conducted to test the ability of seven species (EA1, MEAM1, MED-Africa Silver Leafing (ASL), SSA1-subgroup1, SSA1-Hoslundia, SSA6, and SSA12) to develop on cassava. The identities of the nymphs were determined using partial mitochondrial cytochrome oxidase 1 sequences. Twelve B. tabaci species were identified, including two novel species, based on the 3.5% nucleotide sequence divergence. Cassava was colonised by SSA1-SG1, SSA1-SG2, and SSA2. The most prevalent species were SSA1-SG1, MED-ASL, and SSA13, which were also the most polyphagous, colonising multiple plant species. Several whitefly species colonised specific weeds, such as Aspilia africana and Commelina benghalensis. The polyphagous nature of these species supports continuous habitats and virus reservoirs. Effective management of whitefly populations in eastern Africa requires an integrated approach that considers their polyphagy and the environmental factors sustaining host plants.

Quantification of eosinophils in the lower gastrointestinal tract of adults: a review of surgical specimens with normal histology from the Free State province, South Africa.

Eosinophils are normal residents of the gastrointestinal tract (GIT). They are noted in small numbers with significant variation between anatomic locations. An idiopathic increase of eosinophils is known as eosinophilic gastrointestinal disease (EGID). EGIDs are a heterologous group of disorders that produce a range of enteric and colonic syndromes. Their incidence has been increasing worldwide. Our study aimed to quantify eosinophils in each segment of the GIT in surgical specimens with normal histology to facilitate the histological diagnosis of EGID. Similarly, we aimed to describe the effect of race and gender on gastrointestinal eosinophil numbers. A retrospective, quantitative comparative study was performed. We assessed 360 surgical specimens with normal histology from the lower gastrointestinal tract of African and Caucasian adults from the Free State Province, South Africa. The number of eosinophils per mm² was counted. Overall, comparable eosinophil values were noted for both males and females, and African and Caucasian South Africans. However, Caucasians recorded a higher concentration of eosinophils in the appendix and the left colon. Eosinophils were most numerous in the lamina propria, with only small numbers present in the epithelium. Our results show that the South African population has similar eosinophil distribution trends to international studies. However, South Africans had far fewer eosinophils than Japanese and North American adults in each segment. Specific eosinophil reference ranges were formulated to quantify reference ranges of eosinophils in the lower GIT, allowing for the accurate diagnosis of EGIDs in our population in future.

The Impact of Biliary Injury on the Recurrence of Biliary Cancer and Benign Disease after Liver Transplantation: Risk Factors and Mechanisms.

Liver transplantation is known to generate significant inflammation in the entire organ based on the metabolic profile and the tissue's ability to recover from the ischemia-reperfusion injury (IRI). This cascade contributes to post-transplant complications, affecting both the synthetic liver function (immediate) and the scar development in the biliary tree. The new occurrence of biliary strictures, and the recurrence of malignant and benign liver diseases, such as cholangiocarcinoma (CCA) and primary sclerosing cholangitis (PSC), are direct consequences linked to this inflammation. The accumulation of toxic metabolites, such as succinate, causes undirected electron flows, triggering the releases of reactive oxygen species (ROS) from a severely dysfunctional mitochondrial complex 1. This initiates the inflammatory IRI cascade, with subsequent ischemic biliary stricturing, and the upregulation of pro-tumorigenic signaling. Such inflammation is both local and systemic, promoting an immunocompromised status that can lead to the recurrence of underlying liver disease, both malignant and benign in nature. The traditional treatment for CCA was resection, when possible, followed by cytotoxic chemotherapy. Liver transplant oncology is increasingly recognized as a potentially curative approach for patients with intrahepatic (iCCA) and perihilar (pCCA) cholangiocarcinoma. The link between IRI and disease recurrence is increasingly recognized in transplant oncology for hepatocellular carcinoma. However, smaller numbers have prevented similar analyses for CCA. The mechanistic link may be even more critical in this disease, as IRI causes the most profound damage to the intrahepatic bile ducts. This article reviews the underlying mechanisms associated with biliary inflammation and biliary pathology after liver transplantation. One main focus is on the link between transplant-related IRI-associated inflammation and the recurrence of cholangiocarcinoma and benign liver diseases of the biliary tree. Risk factors and protective strategies are highlighted.

High‐status versus low‐status stakeholders

AbstractThe literature on stakeholder theory has largely ignored the difficult and central issue of how judges and firms should resolve disputes among stakeholders. When the issue is addressed, focus has largely been on the potential for management to use stakeholder theory as cover for rent‐seeking or on disputes between classes of stakeholders. Sharply underappreciated is the potential for disparate interests within a stakeholder class. That potential is particularly acute due to a (largely education‐driven) stark and growing class divide in the United States. There is a substantial difference between the interests of a highly educated professional and managerial elite and a pink‐collar and blue‐collar working class who mostly do not hold 4‐year degrees. Despite their smaller numbers, the professional and managerial elite will frequently win out in intra‐stakeholder disputes with working‐class stakeholders due to their greater status, power, and influence. Because this class divide is cultural, social, and political, as well as economic, these disputes will go beyond financial pie‐splitting to culture war issues. This threatens to be destabilizing for both the republic and individual firms and undermines both the practical and ethical arguments for the stakeholder theory.

Optimizing Trichiasis Case Finding to Attain the Elimination of Trachoma as a Public Health Problem.

As national trachoma programmes increase efforts to reduce the burden of trachomatous trichiasis (TT), TT case finding and referral are critical public health programme components. Our research aimed to explore the most effective and efficient approaches to finding, referring, and managing TT cases. This was a prospective descriptive study, utilizing both routine programme data and primary data collection. This study compared four different approaches to finding TT cases across three different local government areas (LGAs) in Kano State, Nigeria. Each of the study LGAs was divided into four sub-units to accommodate the four different approaches. The number of outreach attendees was 4795 across the four case finding approaches, and this varied hugely, with the smallest number and proportion (403, 0.26%) in settings only employing house-to-house case finding and the largest number and proportion (1901, 0.99%) when town criers were used. That said, the proportion of TT cases among people presenting at outreach was highest (32.5%) when house-to-house case finding was used and lowest (10.3%) when town criers were used. More female TT patients were found (53-70%) and had surgery (79-85%) compared to male cases, across all approaches. The average project expenditure for finding one TT case was similar for approaches that included house-to-house case finding (USD 5.4-6.3), while it was 3.5 times higher (USD 21.5 per TT case found) when town criers were used. This study found that the house-to-house TT case finding approaches were the most efficient method with the highest yield of TT cases. Including other eye condition and/or vision testing yielded similar results but required more personnel and cost.

Ecosystem restoration coupled with poverty alleviation in eThekwini Municipality, KwaZulu‐Natal Province, South Africa

Ecosystem restoration is a necessity for addressing socio‐ecological challenges by improving ecosystem resilience and alleviating poverty through local community involvement. Here, we review a community‐based, large‐scale ecosystem restoration program, Working for Ecosystems, implemented by eThekwini Municipality in KwaZulu‐Natal, South Africa, between 2007 and 2022. For this study, we interrogated monthly and annual reports generated by the program between 2011 and 2021 to describe how the program worked to suppress invasive alien plants (IAPs) and provided employment opportunities to marginalized groups (women, youth, military veterans, and people living with disabilities). Annual budget was positively correlated with the area cleared of IAPs and effort (person‐days in the field). The program successfully incorporated marginalized groups, but military veterans and people living with disabilities remained in smaller numbers and were not well quantified. Participants were encouraged to constitute companies—Small, Medium, and Micro Enterprises—that were then subcontracted to the program or could progress to other endeavors, allowing other community members to enter the program. A steady increase in the implementation budget emphasized the need for a long‐term plan to ensure accountability and sustainability of restoration and socio‐ecological systems.

A Mini-Review of Diagnostic Methods for the Antigen and Antibody Detection of Rocky Mountain and Brazilian Spotted Fever.

Rocky Mountain or Brazilian spotted fever, caused by Rickettsia rickettsii, is a fulminant, seasonal, and neglected disease that occurs in focal points of North America and South America. Its rapid detection is essential for the better prognosis and survival rate of infected individuals. However, disease diagnosis still faces challenges as the accuracy of many of the available laboratory tests fluctuates. This review aimed to analyze methods for antibody or antigen detection, their gaps, and their evolution over time. A search was conducted to find all studies in the Pubmed database that described the antibody or antigen detection of R. rickettsii infections. Initially, a total of 403 articles were screened. Of these articles, only 17 fulfilled the pre-established inclusion criteria and were selected. Among the different methods applied, the IFA technique was the one most frequently found in the studies. However, it presented varied results such as a low specificity when using the indirect method. Other techniques, such as ELISA and immunohistochemistry, were also found, although in smaller numbers and with their own limitations. Although some studies showed promising results, there is a pressing need to find new techniques to develop a rapid and effective diagnosis of R. rickettssi infection.

Comparative Analysis and Phylogeny of the Complete Chloroplast Genomes of Nine Cynanchum (Apocynaceae) Species.

Cynanchum belongs to the Apocynaceae family and is a morphologically diverse genus that includes around 200 shrub or perennial herb species. Despite the utilization of CPGs, few molecular phylogenetic studies have endeavored to elucidate infrafamilial relationships within Cynanchum through extensive taxon sampling. In this research, we constructed a phylogeny and estimated divergence time based on the chloroplast genomes (CPGs) of nine Cynanchum species. We sequenced and annotated nine chloroplast (CP) genomes in this study. The comparative analysis of these genomes from these Cynanchum species revealed a typical quadripartite structure, with a total sequence length ranging from 158,283 to 161,241 base pairs (bp). The CP genome (CPG) was highly conserved and moderately differentiated. Through annotation, we identified a total of 129-132 genes. Analysis of the boundaries of inverted repeat (IR) regions showed consistent positioning: the rps19 gene was located in the IRb region, varying from 46 to 50 bp. IRb/SSC junctions were located between the trnN and ndhF genes. We did not detect major expansions or contractions in the IR region or rearrangements or insertions in the CPGs of the nine Cynanchum species. The results of SSR analysis revealed a variation in the number of SSRs, ranging from 112 to 150. In five types of SSRs, the largest number was mononucleotide repeats, and the smallest number was hexanucleotide repeats. The number of long repeats in the cp genomes of nine Cynanchum species was from 35 to 80. In nine species of Cynanchum, the GC3s values ranged from 26.80% to 27.00%, indicating a strong bias towards A/U-ending codons. Comparative analyses revealed four hotspot regions in the CPG, ndhA-ndhH, trnI-GAU-rrn16, psbI-trnS-GCU, and rps7-ndhB, which could potentially serve as molecular markers. In addition, phylogenetic tree construction based on the CPG indicated that the nine Cynanchum species formed a monophyletic group. Molecular dating suggested that Cynanchum diverged from its sister genus approximately 18.87 million years ago (Mya) and species diversification within the Cynanchum species primarily occurred during the recent Miocene epoch. The divergence time estimation presented in this study will facilitate future research on Cynanchum, aid in species differentiation, and facilitate diverse investigations into this economically and ecologically important genus.

Shared clusters between phylogenetic trees for genomic segments of Rotavirus A with distinct genotype constellations

Rotavirus A (RVA) possesses a genome of 11-segmented, double-stranded RNAs, each of which is classified into genotypes. A variety of genotype constellations have been generated through reassortment, but reassortment does not appear to occur randomly. Here genomic sequences for 281 RVA strains with distinct genotype constellations retrieved from the International Nucleotide Sequence Database were analyzed to identify shared clusters between phylogenetic trees for genomic segments. The largest numbers of shared clusters were observed between the genomic segments encoding viral protein 1 (VP1) and VP2 as well as VP2 and VP3, suggesting that not only VP1 but also VP3 may interact with VP2 inside the core. Larger numbers were also associated with the segment encoding non-structural protein 5/6 (NSP5/6), which may be a hub for bundling genomic segments. Although VP7 and VP4 segments were associated with smaller numbers, reassortments between these segments as well as VP7 and VP6 segments appeared to be restricted due to interactions in constructing the virion. For VP4 segment, the number with NSP1 segment was significantly greater than those with other segments, possibly reflecting that specificities in receptor binding and interferon antagonism may define host range restriction. Overall, no cluster was shared by all genomic segments, supporting that RVA constituted a single species.

WEED SPECIES COMPOSITION IN TERRITORIES OF DIFFERENT TYPES IN VINNYTSIA REGION

The article highlights the results of studying the taxonomic composition, floristic features and occurrence of weed species in different types of territories in Vinnytsia region – 159 species of weeds belonging to 116 genera from 32 families were found in the investigated lands of different types – segetal (fields) and ruderal (field margins, field roads, highways). The largest number of species was documented on field margins (123 species), and the smallest number – on roadsides of highways (80 species). The number of identified species on fields and field roads is similar to that on field margins (117-111 species).The values of P. Jaccard’s floristic similarity coefficient indicate a high similarity of the species composition of fields, their roadsides, and field roads (65.2–69.6%) compared to the roadsides of highways (47.0–55.0%). This trend indicates a close interrelation of weed species composition of the main habitat types that make up each agroecosystem (fields, field margins and field roads), which is the reason for the need for regular monitoring of weeds not only in fields, but also in their margins and surrounding field roads.It was found that the distribution of species by families is irregular. The first places by the number of species for all comparison positions are occupied by the families Asteraceae Dumort. and Poaceae Barnhart., with the family Asteraceae significantly exceeding other families by the number of recorded species.It was established that most of the registered weed species belong to the dominant families. It was found out that 59 weed species were registered on all studied types of habitats. A comparative analysis of the obtained data showed that the species are represented in different types of localities to different degrees.The weed species composition in Vinnytsia region is characterized by the unity and stability of the taxonomic structure over a long period of time and regardless of the type of habitat. High indicators of similarity lead to a comprehensive study of weed species compositions of the main habitat types that make up each agroecosystem (fields, field margins, and field roads).

Optimal L(3,2,1)-labeling of trees

Given a graph G, an L ( 3 , 2 , 1 ) -labeling of G is an assignment f of non-negative integers (labels) to the vertices of G such that | f ( u ) − f ( v ) | ≥ 4 − i if dist ( u , v ) = i (i = 1, 2, 3). For a non-negative integer k, a k- L ( 3 , 2 , 1 ) -labeling is an L ( 3 , 2 , 1 ) -labeling such that no label is greater than k. The L ( 3 , 2 , 1 ) -labeling number of G, denoted by λ 3 , 2 , 1 ( G ) , is the smallest number k such that G has a k- L ( 3 , 2 , 1 ) -labeling. Chia proved that the L ( 3 , 2 , 1 ) -labeling number of a tree T with maximum degree Δ can have one of three values: 2 Δ + 1 , 2 Δ + 2 and 2 Δ + 3 . This paper gives some sufficient conditions for λ 3 , 2 , 1 ( T ) ≥ 2 Δ + 2 and λ 3 , 2 , 1 ( T ) = 2 Δ + 3 , respectively. As a result, the L ( 3 , 2 , 1 ) -labeling numbers of complete m-ary trees, spiders and banana trees are completely determined.

Linear support for the prime number sequence and the first and second Hardy-Littlewood conjectures

Servais and Grün used results about linear support for the prime number sequence to obtain upper bounds on the smallest prime in odd perfect numbers. This was extended by Cohen and Hendy who proved that for every $n \in \mathbb {N}$ there exists an integer $b_n$ such that $p_i \ge p_1 + ni - b_n$ for any sequence $(p_i)$ of odd primes, and found minimal values of $b_n$ for $3 \le n \le 5$, and conjectured minimal values for $6 \le n \le 10$. We give a new proof of the existence of $b_n$ and the values for $3 \le n \le 5$. We also show that if we assume that the second Hardy-Littlewood conjecture, $\pi (a+b) \le \pi (a) + \pi (b)$ for $a, b \ge 2$, is true, then we can in a finite number of steps determine numbers, $T_n$, that give quite close bounds for the values of $b_n$, namely $T_n \le b_n \le T_n + n - 2$, and determine the values of $T_n$ for $6 \le n \le 20$. We also consider the question of whether the values of $b_n$ can be replaced by smaller numbers if we assume that $p_1 > 3$. We will show that if we assume that the first Hardy-Littlewood conjecture is true, then we can determine the minimum such values, $a_n$, for $3 \le n \le 5$. We also determine some lower bounds for $a_n$ for $n \ge 6$. It is well-known that the two Hardy-Littlewood conjectures are mutually exclusive, but we never use both conjectures simultaneously. These results give us the upper bound $p_1<\frac {n}{2^n-1}t+b_n$ on the smallest prime in odd perfect numbers, where $t$ is the number of distinct primes dividing the odd perfect number. This bound was also found by Cohen and Hendy, but we improve the constants $b_n$.

Trends in botanical exploration in Nigeria forecast over 1000 yet undescribed vascular plant species.

Taxonomists are primary actors of biodiversity assessment. At the same time, there is awareness by the taxonomic community at large that the field is going through a crisis, sometimes referred to as the 'taxonomic impediment'. Coupled with the ongoing biodiversity crisis, or 6th mass extinction, this biodiversity impedance puts at risk the target set in the Convention on Biological Diversity's (CBD) Global Biodiversity Framework vision 2050, which calls for urgent action to '… put biodiversity on a path to recovery by 2030 for the benefit of planet and people'. This risk is particularly pronounced in tropical African countries where taxonomic studies are done on an ad hoc basis. In this study, our aim is to investigate the historical trends in botanical exploration of vascular plants in Nigeria and forecast the near-term (50-year) description of presently unknown species, which we use to discuss scenarios of taxonomic effort that may be necessary for a comprehensive biodiversity assessment in the country. The study is based on a dataset from the World Checklist of Vascular Plants, containing all vascular plant species reported to occur in Nigeria. We fit nested Bayesian time series regressions to estimate the long-term trend in the rate of description of vascular plant species in Nigeria. From these models, we use an ensemble forecast to estimate the number of species descriptions by the year 2070, and then evaluate the description rates per taxonomist required to meet this estimate under different totals of active taxonomists. We find a striking difference in species description between Nigerian botanists and their foreign counterparts, with the former contributing relatively small numbers. Additionally, only a fraction of the authors involved in describing Nigeria's vascular plants are of indigenous origin. Our study reveals that the number of new species described annually exhibits a long-term increasing trend, with an average of 19.5 species described per year. However, after taking into account year-to-year variability and the number of taxonomists active in a given year, the long-term trend in species descriptions credibly declines over time. While the number of authors involved in describing species has generally increased over time, it has remained stable since the 1950s. Predictions for the number of new species descriptions by 2070 vary by model, with an ensemble prediction estimating 1140 species descriptions, but ranging from 1004 to 2239 between individual models. The study estimates that current levels of taxonomic activity should lead to a 20 % increase in known species of vascular plants in Nigeria over the next 50 years, which is still probably an underestimate of the true, unknown species richness. Urgent action is needed to address the taxonomic impediment so that local taxonomic studies in tropical African countries can achieve the CBD's Global Biodiversity Framework vision 2050. Here, we outline some key pathways to achieving this goal.