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

Additional file 1: of Awareness of zoonotic diseases and parasite control practices: a survey of dog and cat owners in Qatar

Figure S1. Multiple-choice questionnaire administered to dog and/or cat owners who attended the veterinary medical centres surveyed in the residential centre of Doha. (DOCX 121 kb

Additional file 1 of Walk or be walked by the dog? The attachment role

Supplementary Material

Genetic structure of stone martens in Portugal as assessed by sPCA.

<p>Shown are the first Principal Component (PC) and second PC, and the respective mappings of cluster membership. Lines in both maps represent country borders and main watercourses.</p

Sensitivity and specificity of the immunofluorescent assay in the diagnosis of <i>Leishmania</i> infected dogs.

a<p>[true positives/(true positives+false negatives)] (percentage).</p>b<p>[true negatives/(true negatives+false positives)] (percentage).</p>c<p>Symptomatic and asymptomatic.</p

Cluster membership and genetic structure of red fox individuals in Portugal as inferred by the uncorrelated frequency model in GENELAND and sPCA.

<p>On the left, cluster membership of red fox individuals with a membership probability ≥ 0.9 by the uncorrelated frequency model in GENELAND (the River Tagus is indicated by a dashed arrow); On the middle and right, genetic structure as assessed by sPCA; shown are the first Principal Component (PC) and the respective mapping of cluster membership. Lines in both maps represent country borders and main watercourses.</p

Geographic location of samples in Portugal.

<p>Geographic location of samples of stone martens (left map) and red foxes (right map). Lines represent country borders and main watercourses.</p

Cluster membership of stone marten individuals.

<p>Cluster membership of stone marten individuals with a membership probability ≥ 0.9 in the best run of, respectively from left to right, BAPS (K = 3), TESS with no admixture (K = 3) and GENELAND for the uncorrelated frequency model (K = 3). Stars represent non-assigned individuals (i.e., with membership coefficients < 0.9). Lines represent country borders and main watercourses.</p

Magnetic microspheres flow cytometry characterization and evaluation.

<p>Reactivities of representative symptomatic (A) and asymptomatic (B) sera tested to different defined combinations of <i>Li</i>cTXNPx and rk39 antigens. Results are expressed as the percentage of positive microspheres. The levels of IgG antibodies anti-rK39 (C) and anti-<i>Li</i>cTXNPx (D) were measured in sera of symptomatic (S), asymptomatic (AS1), asymptomatic PCR+ (AS2), <i>Leishmania</i>-negative but presenting other clinical conditions (OP) and <i>Leishmania</i> negative healthy dogs from non-endemic areas (N). Results are expressed as the percentage of positive microspheres.</p