This paper presents the symmetries of differential equations associated with one-variable and Bivariate Hermite polynomials by proposing a representation of Lie algebra for these differential operators. Applying the Baker-Campbell-Hausdorff formula to these algebras, results in new relations and generating functions in one-variable and Bivariate Hermite polynomials. A general form of representation for other orthogonal polynomials such as Laguerre polynomials is introduced.

The present paper introduces a method of basis transformation of vector fields that is specifically applicable to polynomials space and differential equations with certain polynomials solutions such as Hermite, Laguerre and Legendre polynomials. The method based on separated transformation of vector space basis by a set of operators that are equivalent to the formal basis transformation and connected to it by linear combination with projection operators. Applying the Forbenius covariants yields a general method that incorporates the Rodrigues formula as a special case in polynomial space. Using the Lie algebra modules, specifically , on polynomial space results in isomorphic algebras whose Cartan sub-algebras are Hermite, Laguerre and Legendre differential operators. Commutation relations of these algebras and Baker-Campbell-Hausdorff formula gives new formulas for special polynomials

The purpose of this paper is to compare daily sunshine duration (SD) measured by Campbell-Stokes sunshine recorder (CS) with three automatic instruments (Kipp & Zonen pyranometers (CM6B), pyrheliometer (CHP1) and CSD3 sunshine recorders). The comparisons are essential, since in recent decades automatic sunshine duration sensors have been systematically replacing the manual measurements affecting therefore the continuity of the CS time series. The comparisons were applied to the daily SD data of three synoptic stations (Larnaca A/P, Pafos A/P and Athalassa) where parallel measurements of manual and automatic instruments are carried out during the periods 2009-2012 and 2012-2015. The CSD3 measurements were obtained from two automatic weather stations (AWS) (Mennogia and Pentakomo) which are closed to the synoptic station of Larnaca A/P. The comparisons cover the daily and monthly sums, their statistical distributions and their relationships. The daily differences of the pairs of the sunshine recorders installed at the same location are mostly within the range of -1.0 to +1.0 h. At Larnaca, in the period 2009-2012, the pyranometric method (Gl) is higher than the CS in the summer months, while during the winter months it is lower than CS. Pafos and Athalassa showed higher values of CS comparing to Gl and the pyrheliometric method (Pr), respectively, almost throughout the year. Generally, the annual average daily differences between the different pairs range between 0.3 and 0.5 h. Regression equations were determined allowing the estimation of daily SD under different sky conditions. The closest relations between different pairs were obtained in ‘clear sky’ and ‘all sky’ conditions, as it is indicated by the high coefficient of determinations. R2 reached the values of 0.92 and 0.96 under ‘all sky’ conditions, and between 0.80 and 0.91 under ‘clear’ sky conditions, respectively. An increase in cloud cover causes an increase in the SD differences recorded by the relevant devices. Comparing statistically the different pairs using the Mann-Whitney nonparametric test which is the alternative to the Two Sample T parametric test, it was found that the medians are not statistically different. However, significant differences were only detected in the case of the pair of CS and Pr at Athalassa (2012-2015) using both tests. In the rest of the cases the differences are not statistically significant at the 5% level.

In this speculative analysis, interdimensionality is introduced as the (co)existence of universes embedded into larger ones. These interdimensional universes may be isolated or intertwined, suggesting a variety of interdimensional intrinsic phenomena that can only be understood in terms of the outer, extrinsic reality.

Suppose set

I want to find an extension of the Hausdorff Measure and Fractal Measure which allows me to compute a unique, satisfying average for functions defined on non-fractal, measurable sets in the sense of Carathedory which has no gauge function.

(RADA)4 nanoscaffolds are excellent candidates for use as peptide delivery vehicles: they are relatively easy to synthesize with custom bio-functionality, and assemble in situ to allow a focal point of release. This enables (RADA)4 to be utilized in multiple release strategies by embedding a variety of bioactive molecules in an all-in-one ‘construct’. One novel strategy focuses on the local, on-demand release of peptides triggered via proteolysis of tethered peptide sequences. However, the spatial-temporal morphology of self-assembling nanoscaffolds may greatly influence the ability for enzymes to both diffuse into as well as actively cleave substrates. Fine structure and its impact on overall affect on peptide release is poorly understood. In addition, fractal networks observed in nanoscaffolds are linked to the fractal nature of diffusion in these systems. Therefore, matrix morphology and fractal dimension of virgin (RADA)4 and mixtures of (RADA)4 and matrix metalloproteinase 2 (MMP-2) cleavable substrate modified (RADA)4 were characterized over time. Sites of high (GPQG+IASQ, CP1) and low (GPQG+PAGQ, CP2) cleavage activity were chosen. Fine structure was visualized using established according to established methods. After 2 hrs of incubation, nanofiber networks showed an established fractal nature, however nanofibers continued to bundle in all cases as incubation times increased. It was observed that despite extensive nanofiber bundling after 24 hrs of incubation time, the CP1 and CP2 nanoscaffolds were susceptible to MMP-2 cleavage. The properties of these engineered nanoscaffolds characterized herein illustrate that they are an excellent candidate as an enzymatically initiated peptide delivery platform.

Suppose for n∈ ℕ, set A⊆ℝⁿ and function f∶A→ℝⁿ. If set A, using the Hausdorff outer measure, is measurable in the sense of Carathèodory; we want to find an extension of the expected value, w.r.t the Hausdorff measure, that's unique, finite and "natural" (defined on §2.3 & §2.4) for all f in a prevalent subset of ℝᴬ. The issue is current extensions of the expected value are finite for all functions in only a shy subset of ℝᴬ. The reason this issue wasn't resolved is mathematicians have not thought of the problem, focusing on application rather than generalization. Despite the lack of potential use, we'll attempt to solve the problem by defining a choice function---this shall choose a unique set of equivalent sequences of sets (Fₖ***) , where the set-theoretic limit of Fₖ*** is the graph of f; the measure Hʰ is the ℎ-Hausdorff measure, such for each k∈ℕ, 0 < Hʰ(Fₖ***) < +∞; and (fₖ*) is a sequence of functions where {(x,fₖ*(x))∶x∈ dom(Fₖ***)}=Fₖ***. Thus, the extended expected value of or E**[f,Fₖ***] is: ∀(ε>0)∃(N∈ℕ)∀(k∈N)(k≥N⇒1/Hʰ(dom(Fₖ***))∫dom(Fₖ*** )fₖ* dHʰ - E**[f,Fₖ***]<ε ) which should be unique, finite, and "natural" (defined on §2.3 & §2.4) for all f in a prevalent subset of ℝᴬ. Note we guessed the choice function using computer programming but we don’t use mathematical proofs due to the lack of expertise in the subject matter. Despite this, the biggest use of this research is the extension of the expected value is finite for "almost all" functions: this is easier to use in application when finding the "average" of functions covering an infinite expanse of space.