Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

Apparatus and method for obtaining the reciprocal of a number and the quotient of two numbers

Abstract: Apparatus and method for obtaining the reciprocal of a number and the quotient of two numbers is disclosed. The dividend and divisor, after left justification of the most significant ''''one'''' of each, are supplied to an array of combinatorial logic, the output of which is a group of polynomials having positive and negative terms. Arithmetic means are provided for subtracting the negative terms of the polynomials from the positive terms thereof to obtain the reciprocal of the divisor. This reciprocal may thereafter be multiplied by the dividend by well-known multiplication means to form the desired quotient. The apparatus and method perform the described arithmetic functions according to a flow-through scheme, where a flow-through scheme is defined as a scheme not requiring iterative techniques.

Large classes of permutation polynomials over $$\mathbb {F}_{q^2}$$Fq2

Abstract: Permutation polynomials (PPs) of the form $$(x^{q} -x + c)^{\frac{q^2 -1}{3}+1} +x$$(xq-x+c)q2-13+1+x over $$\mathbb {F}_{q^2}$$Fq2 were presented by Li et al. (Finite Fields Appl 22:16---23, 2013). More recently, we have constructed PPs of the form $$(x^{q} + bx + c)^{\frac{q^2 -1}{d}+1} -bx$$(xq+bx+c)q2-1d+1-bx over $$\mathbb {F}_{q^2}$$Fq2, where $$d=2, 3, 4, 6$$d=2,3,4,6 (Yuan and Zheng in Finite Fields Appl 35:215---230, 2015). In this paper we concentrate our efforts on the PPs of more general form $$\begin{aligned} f(x)=(ax^{q} +bx +c)^r \phi \big ((ax^{q} +bx +c)^{(q^2 -1)/d}\big ) +ux^{q} +vx ~{\text {over}}\; \mathbb {F}_{q^2}, \end{aligned}$$f(x)=(axq+bx+c)rź((axq+bx+c)(q2-1)/d)+uxq+vxoverFq2,where $$a,b,c,u,v \in \mathbb {F}_{q^2}$$a,b,c,u,vźFq2, $$r \in \mathbb {Z}^{+}$$rźZ+, $$\phi (x)\in \mathbb {F}_{q^2}[x]$$ź(x)źFq2[x] and d is an arbitrary positive divisor of $$q^2-1$$q2-1. The key step is the construction of a commutative diagram with specific properties, which is the basis of the Akbary---Ghioca---Wang (AGW) criterion. By employing the AGW criterion two times, we reduce the problem of determining whether f(x) permutes $$\mathbb {F}_{q^2}$$Fq2 to that of verifying whether two more polynomials permute two subsets of $$\mathbb {F}_{q^2}$$Fq2. As a consequence, we find a series of simple conditions for f(x) to be a PP of $$\mathbb {F}_{q^2}$$Fq2. These results unify and generalize some known classes of PPs.

The Circle Problem of Gauss and the Divisor Problem of Dirichlet—Still Unsolved

Abstract: Let r2(n) denote the number of representations of the positive integer n as a sum of two squares, and let d(n) denote the number of positive divisors of n Gauss and Dirichlet were evidently the fi

High-radix divider

Abstract: A dividend or partial remainder is stored in a partial remainder register. An output of the partial remainder register is shifted to the left by a radix which uses the power of 2 and is larger than 2. A divisor is stored in a divisor register. Comparison constants obtained by subjecting an output of the divisor register to predetermined operations are stored in comparison constant registers. Subtracters respectively receive outputs of the comparison constant registers as one input, receive upper bits of a bit number representing a precision required for conversion and included in the output of the partial remainder shifter as another input, and compare the magnitudes of the two inputs with each other to derive partial quotients. A selector shifts and selects an output of the divisor register according to the signs of the remainder quotients output from the subtracters to create a factor having a value equal to the integer multiple of the divisor. An adder/subtracter receives an output of the selector and an output of the partial remainder shifter and the addition or subtraction of the adder/subtracter is selectively specified by the sign bit of an output of the partial remainder shifter to derive a partial remainder.

The algebra of generating functions for multiple divisor sums and applications to multiple zeta values

Abstract: We study the algebra $${{\mathrm{{\mathcal {MD}}}}}$$ of generating functions for multiple divisor sums and its connections to multiple zeta values. The generating functions for multiple divisor sums are formal power series in q with coefficients in $${\mathbb {Q}}$$ arising from the calculation of the Fourier expansion of multiple Eisenstein series. We show that the algebra $${{\mathrm{{\mathcal {MD}}}}}$$ is a filtered algebra equipped with a derivation and use this derivation to prove linear relations in $${{\mathrm{{\mathcal {MD}}}}}$$ . The (quasi-)modular forms for the full modular group $${{\mathrm{SL}}}_2({\mathbb {Z}})$$ constitute a subalgebra of $${{\mathrm{{\mathcal {MD}}}}}$$ , and this also yields linear relations in $${{\mathrm{{\mathcal {MD}}}}}$$ . Generating functions of multiple divisor sums can be seen as a q-analogue of multiple zeta values. Studying a certain map from this algebra into the real numbers we will derive a new explanation for relations between multiple zeta values, including those of length 2, coming from modular forms.