Orthogonal decomposition of the sum-symmetry model for square contingency tables with ordinal categories: Use of the exponential sum-symmetry model

Summary In the existing decomposition theorem, the sum-symmetry model holds if and only if both the exponential sum-symmetry and global symmetry models hold. However, this decomposition theorem does not satisfy the asymptotic equivalence for the test statistic. To address the aforementioned gap, this study establishes a decomposition theorem in which the sum-symmetry model holds if and only if both the exponential sum-symmetry and weighted global-sum-symmetry models hold. The proposed decomposition theorem satisfies the asymptotic equivalence for the test statistic. We demonstrate the advantages of the proposed decomposition theorem by applying it to datasets comprising real data and artificial data.

L2 Boundedness of Discrete Double Hilbert Transform Along polynomials

We will show $L^{2}$ boundedness of Discrete Double Hilbert Transform along polynomials satisfying some conditions. Double Hilbert exponential sum along polynomials:$\mu(\xi)$ is Fourier multiplier of discrete double Hilbert transform along polynomials. In chapter 1, we define the reverse Newton diagram. In chapter 2, We make approximation formula for the multiplier of one valuable discrete Hilbert transform by study circle method. In chapter 3, We obtain result that $\mu(\xi)$ is bounded by constants if $|D|\geq2$ or all $(m,n)$ are not on one line passing through the origin. We study property of $1/(qt^{n})$ and use circle method (Propsotion 2.1) to calculate sums. We also envision combinatoric thinking about $\mathbb{N}^{2}$ lattice points in j-k plane for some estimates. Finally, we use geometric property of some inequalities about $(m,n)\in\Lambda$ to prove Theorem 3.3. In chapter 4, We obtain the fact that $\mu(\xi)$ is bounded by sums which are related to $\log_{2}({\xi_{1}-a_{1}\slash {q}})$ and $\log_{2}({\xi_{2}-a_{2}\slash {q}})$ and the boundedness of double Hilbert exponential sum for even polynomials with torsion without conditions in Theorem 3.3. We also use $\mathbb{N}^{2}$ lattice points in j-k plane and Proposition 2.1 which are shown in chapter 2 and some estimates to show that Fourier multiplier of discrete double Hilbert transform is bounded by terms about $\log$ and integral this with torsion is bounded by constants.

Multiplicative functions arising from the study of mutually unbiased bases

We introduce two families of multiplicative functions, which generalize the somewhat unusual function that was serendipitously discovered in 2010 during a study of mutually unbiased bases in the Hilbert space of quantum physics. In addition, we report yet another multiplicative function, which is also suggested by that example; it can be used to express the squarefree part of an integer in terms of an exponential sum.

Generator of PRN's on the norm group

Let $p$ be a prime number, $d\in\mathds{N}$, $\left(\frac{-d}{p}\right)=-1$, $m>2$, and let $E_m$ denotes the set of of residue classes modulo $p^m$ over the ring of Gaussian integers in imaginary quadratic field $\mathds{Q}(\sqrt{-d})$ with norms which are congruented with 1 modulo $p^m$. In present paper we establish the polynomial representations for real and imagimary parts of the powers of generating element $u+iv\sqrt{d}$ of the cyclic group $E_m$. These representations permit to deduce the ``rooted bounds'' for the exponential sum in Turan-Erd\"{o}s-Koksma inequality. The new family of the sequences of pseudo-random numbers that passes the serial test on pseudorandomness was being buit.

A note on two-term exponential sum and the reciprocal of the quartic Gauss sums

The main purpose of this article is by using the properties of the fourth character modulo a prime p and the analytic methods to study the calculating problem of a certain hybrid power mean involving the two-term exponential sums and the reciprocal of quartic Gauss sums, and to give some interesting calculating formulae of them.

A Note on Cube-Full Numbers in Arithmetic Progression

We obtain an asymptotic formula for the cube-full numbers in an arithmetic progression n ≡ l mod q , where q , l = 1 . By extending the construction derived from Dirichlet’s hyperbola method and relying on Kloosterman-type exponential sum method, we improve the very recent error term with x 118 / 4029 < q .