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We observe structure in the sequences of quotients and remainders of the Euclidean algorithm with two families of inputs. Analyzing the remainders, we obtain new algorithms for computing modular inverses and representing prime numbers by the binary quadratic form . The Euclidean algorithm is commenced with inputs from one of the families, and the first remainder less than a predetermined size produces the modular inverse or representation.
Due to its fractal nature, much about the area of the Mandelbrot set remains to be understood. While a series formula has been derived by Ewing and Schober (1992) to calculate the area of by considering its complement inside the Riemann sphere, to date the exact value of this area remains unknown. This paper presents new improved upper bounds for the area based on a parallel computing algorithm and for the 2-adic valuation of the series coefficients in terms of the sum-of-digits function.
We utilize a theorem of B. Feigin and S. Loktev to give explicit bases for the global Weyl modules for the map algebras of the form of highest weight . These bases are given in terms of specific elements of the universal enveloping algebra, , acting on the highest weight vector.
In 2010, Joyce et al. defined the leverage centrality of a graph as a means to analyze connections within the brain. In this paper we investigate this property from a mathematical perspective and determine the leverage centrality for knight’s graphs, path powers, and Cartesian products.
and the finite set . The moment map of the -action on the cotangent bundle maps each conormal bundle closure onto the closure of a single nilpotent -orbit, . We use combinatorial techniques to describe .
The pressure in the renal interstitium is an important factor for normal kidney function. Here we develop a computational model of the rat kidney and use it to investigate the relationship between arterial blood pressure and interstitial fluid pressure. In addition, we investigate how tissue flexibility influences this relationship. Due to the complexity of the model, the large number of parameters, and the inherent uncertainty of the experimental data, we utilize Monte Carlo sampling to study the model’s behavior under a wide range of parameter values and to compute first- and total-order sensitivity indices. Characteristically, at elevated arterial blood pressure, the model predicts cases with increased or reduced interstitial pressure. The transition between the two cases is controlled mostly by the compliance of the blood vessels located before the afferent arterioles.
Lagrange’s four squares theorem states that any positive integer can be expressed as the sum of four integer squares. We investigate the analogous question for quaternion rings, focusing on squares of elements of quaternion rings with integer coefficients. We determine the minimum necessary number of squares for infinitely many quaternion rings, and give global upper and lower bounds.
We investigate the symmetric spaces associated to the family of semidihedral groups of order . We begin this study by analyzing the structure of the automorphism group and by determining which automorphims are involutions. We then determine the symmetric spaces corresponding to each involution and the orbits of the fixed-point groups on these spaces.
We consider a generalization of Eulerian numbers which count the number of placements of rooks on an chessboard where there are exactly rooks in each row and each column, and exactly rooks below the main diagonal. The standard Eulerian numbers correspond to the case . We show that for any the resulting numbers are symmetric and give generating functions of these numbers for small values of .
For a fixed graph , a graph is -linked if any injection can be extended to an -subdivision in . The concept of -linked generalizes several well-known graph theory concepts such as -connected, -linked, and -ordered. In 2012, Ferrara et al. proved a sharp (or degree-sum) bound for a graph to be -linked. In particular, they proved that any graph with vertices and is -linked, where is a parameter maximized over certain partitions of . However, they do not discuss the calculation of in their work. In this paper, we prove the exact value of in the cases when is a path, a cycle, a union of stars, a complete graph, and a complete bipartite graph. Several of these results lead to new degree-sum conditions for particular graph classes while others provide alternate proofs of previously known degree-sum conditions.
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