Factor graph and the associated sum-product algorithm have been widely used in probabilistic modeling of the relationship among interdependent (random) variables or parameters.
Motivated by and generalizing this, we consider three types of functions given by sums of products of higher-order Bernoulli functions and we obtain some new identities arising from Fourier series expansions associated with sums of products of higher-order Bernoulli functions.
Several additional resources are also provided, in particular the predicted genomic location of currently unplaced SNPs and associated scaffolds summing up to a total of 72 megabases, that can be useful for the exploitation of the pig genome assembly.
Roundtable statisticians agreed that potential errors associated with summing across multiple species to obtain total folate values should not affect total means but could increase variances.
Finally, sport factors were associated with SUM only in urban girls.
In Section 2, we define some new finite arithmetic sums which are associated with the Dedekind sums, the Hardy-Berndt sums, and Simsek's sum.
One business associate summed up his remembrance of Mr. Baker by saying that "In the 25 years I knew him, I never heard him say anything bad about anyone, and never heard anyone say anything bad about him".
The sums associated with "Avatar" would not be out of place in the projected federal-budget deficit, which was also announced last week (but isn't as much fun to think about).
The final formulae for the overall properties show explicitly the dependence on (i) the properties of the phases, (ii) the radius of the cylindrical fiber and (iii) the lattice sums associated with the square array.
For instance, an explicit expression for the orthogonal polynomials associated to general sums of independent random indicators seems to be unknown.
In (Bailey et al. in J. Phys. A, Math. Theor. 46:1152013 2013, doi 10.1088/1751-8113/46/11/115201) we studied a class of lattice sums that amount to solutions of Poisson's equation, utilizing some striking connections between these sums and Jacobi ϑ-function values, together with high-precision numerical computations and the PSLQ algorithm to find certain polynomials associated with these sums.
