A novel approach to fractional calculus: utilizing fractional integrals and derivatives of the Dirac delta function

While the definition of a fractional integral may be codified by Riemann and Liouville, an agreed-upon fractional derivative has eluded discovery for many years. This is likely a result of integral definitions including numerous constants of integration in their results. An elimination of constants of integration opens the door to an operator that reconciles all known fractional derivatives and shows surprising results in areas unobserved before, including the appearance of the Riemann Zeta Function and fractional Laplace and Fourier Transforms. A new class of functions, known as Zero Functions and closely related to the Dirac Delta Function, are necessary for one to perform elementary operations of functions without using constants. The operator also allows for a generalization of the Volterra integral equation, and provides a method of solving for Riemann's "complimentary" function introduced during his research on fractional derivatives

Literature review of average trace element concentrations in bovine kidney (mg/kg WW).

<p>¹ Geometric mean,</p><p>² Presumably <i>Bos indicus</i> crossbred cattle based upon location or mentioned as such.</p

Trace element concentrations in zebu (<i>Bos indicus</i>) bull (<i>n</i> = 28) plasma sampled at Jimma, Ethiopia with median, first quartile (Q1) and third quartile (Q3) as summary statistics.

<p>Adequate range for cattle <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0085300#pone.0085300-Puls1" target="_blank">[14]</a>.</p><p>¹ upper threshold value indicating a deficiency risk in <i>Bos taurus</i> cattle <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0085300#pone.0085300-Suttle1" target="_blank">[7]</a>.</p><p>^a Co: lower boundary of normal Co concentrations in <i>Bos taurus</i> cattle <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0085300#pone.0085300-Puls1" target="_blank">[14]</a>.</p><p>^b Mo: lower boundary of Mo concentrations in <i>Bos taurus</i> cattle considered elevated <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0085300#pone.0085300-Puls1" target="_blank">[14]</a>.</p

Literature review of average trace element concentrations in bovine liver (mg/kg WW).

<p>¹ Geometric mean,</p><p>² Presumably <i>Bos indicus</i> cattle based upon location or mentioned as such.</p

Trace element concentrations (mg/kg DW) in zebu (<i>Bos indicus</i>) bull (<i>n</i> = 60) tissues sampled at Jimma, Ethiopia with median, first quartile (Q1) and third quartile (Q3) as summary statistics.

<p>DW = dry weight.</p><p>Adequate range for cattle <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0085300#pone.0085300-Puls1" target="_blank">[14]</a>.</p><p>^a,b Medians sharing a same letter do not differ significantly from each other (p<0.050).</p

Map presenting the cities of origin of zebu (<i>Bos indicus</i>) bulls (<i>n</i> = 60) sampled at Jimma, South-West Ethiopia.

<p>The red lines depict the woredas, an administrative unit in Ethiopia, to which the cities belong = Bonga (Gimbo), Gera (Gera), Seka (Seka Chekorsa), Agaro (Goma), Jimma, Serbo (Kersa) and Dedo (Dedo).</p

PRISMA 2020 checklist.

BackgroundCystic echinococcosis (CE) is a neglected zoonotic disease that is caused by Echinococcus granulosus sensu lato (s.l.), the life cycle of which involves multiple hosts. We conducted a systematic review (SR) on E. granulosus s.l. in the Greater Horn of Africa (GHA), to provide a picture of its recent epidemiology across all hosts.MethodsFor this SR, conducted in accordance with the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) statement, five electronic databases, as well experts in the region were consulted to retrieve records published between 2000 and 2022, reporting the presence of E. granulosus s.l. infections in any natural host in the GHA (Djibouti, Eritrea, Ethiopia, Kenya, Sudan, Somalia, South Sudan, Tanzania and Uganda).Principal findingsA total of 247 records were retained, describing the presence of E. granulosus s.l. throughout the GHA, except for Djibouti. Only few population surveys on human CE were conducted in the area, with the prevalence ranging between 0.3 and 11.3%. In animals, the reported prevalence ranged up to 61.6% in camels, 88.4% in cattle; 65.2% in goats, 9.9% in pigs, 67.8% in sheep and 94.5% in dogs. In addition, E. granulosus s.l. was also reported in wildlife. A total of five species were reported in the different hosts, namely E. granulosus sensu stricto (G1, G3, GOmo), E. canadensis (G6/7), E. ortleppi (G5), E. felidis, and E. equinus (G4).ConclusionsWe confirm that E. granulosus s.l. is prevalent throughout the GHA. Nevertheless, despite our efforts to screen grey literature, an accurate assessment of the epidemiology in GHA remains challenging, due to the lack of combined host, in-depth risk factor and behavioural studies, as well as the wide diversity in subpopulations studied and diagnostic tools used. Interdisciplinary and transboundary partnerships would be essential for the design of effective control strategies, tuned to the GHA setting.</div

Literature review of trace element concentrations in bovine heart (mg/kg WW).

<p>¹ Presumably <i>Bos indicus×Bos taurus</i> crossbred cattle based on location.</p

Summary of organ distribution, clinical manifestation, treatment, and treatment outcome of 76 human CE cases reported in the GHA between 2000 and 2022.

Summary of organ distribution, clinical manifestation, treatment, and treatment outcome of 76 human CE cases reported in the GHA between 2000 and 2022.</p

PRISMA 2020 flow diagram for new systematic reviews which included searches of databases, registers and other sources.

PRISMA 2020 flow diagram for new systematic reviews which included searches of databases, registers and other sources.</p