We show that certain quadratic base change $L$-functions for $\mathrm{Gl}\left(2\right)$ are non-negative at their center of symmetry.

Let $F/k$ be a finite abelian extension of number fields with $k$ imaginary quadratic. Let $O_F$ be the ring of integers of $F$ and $n\ge 2$ a rational integer. We construct a submodule in the higher odd-degree algebraic $K$-groups of $O_F$ using corresponding Gross’s special elements. We show that this submodule is of finite index and prove that this index can be computed using the higher “twisted” class number of $F$, which is the cardinal of the finite algebraic $K$-group $K_{2n-2}\left(O_F\right)$.

CONTENTSIntroduction.......................................................................................61. Definition of certain special forms...........................................62. Statement of results...................................................................83. Proof of Theorem 2.....................................................................94. Preliminaries for Theorem 1.....................................................105. Further preliminaries for Theorem...

The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense $\mathrm{d}x\left(t\right)=\mathrm{d}{A}_0\left(t\right)·x\left(t\right)+\mathrm{d}{f}_0\left(t\right)$, $x\left(t_0\right)=c_0$ $(t\in I)$ with a unique solution $x_0$ is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems $\mathrm{d}x\left(t\right)=\mathrm{d}{A}_k\left(t\right)·x\left(t\right)+\mathrm{d}{f}_k\left(t\right)$, $x\left(t_k\right)=c_k$ $(k=1,2,\cdots )$ to have a unique solution $x_k$ for any sufficiently large $k$ such that $x_k\left(t\right)\to x_0\left(t\right)$ uniformly on $I$. Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems....

We consider digit expansions ${\sum }_{j=0}^{\ell -1}{\Phi }^j\left(d_j\right)$ with an endomorphism $\Phi$ of an Abelian group. In such a numeral system, the $w$-NAF condition (each block of $w$ consecutive digits contains at most one nonzero) is shown to minimise the Hamming weight over all expansions with the same digit set if and only if it fulfills the subadditivity condition (the sum of every two expansions of weight $1$ admits an optimal $w$-NAF). This result is then applied to imaginary quadratic bases, which are used for scalar...

This thesis is concerned with the problem of determining a measure of algebraic independence for a particular m-tuple θ₁,...,${\theta }_m$ of complex numbers. Specifically, let K be a number field and let f₁(z),...,$f_m\left(z\right)$ be elements of K[[z]] algebraically independent over K(z) satisfying equations of the form(*) $f_j\left(z^b\right)={\sum }_{i=1}^m{f}_i\left(z\right)a_{ij}\left(z\right)+b_j\left(z\right)$ (j = i,...,m)for b ≥ 2, $a_{ij}\left(z\right)$, $b_j\left(z\right)$ in K(z). Suppose finally that α ∈ K is such that 0 < |α| < 1, the $f_j(z$) converge at z = α and the $a_{ij}\left(z\right)$, $b_j\left(z\right)$ are analytic at $z=\alpha ,{\alpha }^b,{\alpha }^{b²},...$ Then the ${\theta }_i=f_i\left(\alpha \right)$ are algebraically independent...

Let $\alpha \&gt;1$ be irrational. Several authors studied the numbers $${\ell }^m\left(\alpha \right)=inf\left\{\right|y|:y\in {\Lambda }_m,y\ne 0\},$$ where $m$ is a positive integer and ${\Lambda }_m$ denotes the set of all real numbers of the form $y={ϵ}_0{\alpha }^n+{ϵ}_1{\alpha }^{n-1}+\cdots +{ϵ}_{n-1}\alpha +{ϵ}_n$ with restricted integer coefficients $|{ϵ}_i|\le m$. The value of ${\ell }^1\left(\alpha \right)$ was determined for many particular Pisot numbers and ${\ell }^m\left(\alpha \right)$ for the golden number. In this paper the value of ${\ell }^m\left(\alpha \right)$ is determined for irrational numbers $\alpha$, satisfying ${\alpha }^2=a\alpha \pm 1$ with a positive integer $a$.

Let $𝕂$ be a quadratic field over the rational field and $a_{𝕂}\left(n\right)$ be the number of nonzero integral ideals with norm $n$. We establish Erdős-Kac type theorems weighted by $a_{𝕂}{\left(n\right)}^l$ and $a_{𝕂}{\left(n^2\right)}^l$ of quadratic field in short intervals with $l\in {\mathbb{Z}}^{+}$. We also get asymptotic formulae for the average behavior of $a_{𝕂}{\left(n\right)}^l$ and $a_{𝕂}{\left(n^2\right)}^l$ in short intervals.

Let $d$ be a square-free positive integer and $h\left(d\right)$ be the class number of the real quadratic field $\mathbb{Q}\left(\sqrt{d}\right).$ We give an explicit lower bound for $h(n^2+r)$, where $r=1,4$. Ankeny and Chowla proved that if $g>1$ is a natural number and $d=n^{2g}+1$ is a square-free integer, then $g\mid h\left(d\right)$ whenever $n>4$. Applying our lower bounds, we show that there does not exist any natural number $n>1$ such that $h(n^{2g}+1)=g$. We also obtain a similar result for the family $\mathbb{Q}\left(\sqrt{n^{2g}+4}\right)$. As another application, we deduce some criteria for a class group of prime power order to be...

Let $𝔻$ denote the unit disk $\{z:|z|<1\}$ in the complex plane $ℂ$. In this paper, we study a family of polynomials $P$ with only one zero lying outside $\overline{𝔻}$.  We establish  criteria for $P$ to satisfy implying that each of $P$ and $P^{\text{'}}$  has exactly one critical point outside $\overline{𝔻}$.