Explanation to the Problem of the Naturalness measurements

The sensitivity parameter is widely used in measuring the severity of fine-tuning, while many examples show it doesn't work under certain circumstances. The validity of the sensitivity is in question. We argue that the dimensional effect is the reason why it fails in those scenarios. To guarantee the sensitivity parameter correctly reflects the severity of fine-tuning, it should be avoided to use under these specific circumstances.Comment: 7 page

Problems of the Sensitivity Parameter and Its Relation to the Time-varying Fundamental Couplings Problems

The sensitivity parameter is widely used for quantifying fine tuning. However, examples show it fails to give correct results under certain circumstances. We argue that these problems only occur when calculating the sensitivity of a dimensionful mass parameter at one energy scale to the variation of a dimensionless coupling constant at another energy scale. Thus, by mechanisms such as dynamical symmetry breaking etc, the high sensitivity of the energy scale parameter (\Lambda) to the dimensionless coupling constant can affect the reliability of the sensitivity parameter through the renormalization invariant factor of the dimensionful parameter. Theoretically, These phenomena are similar to the problems associated with the time-varying coupling constant discovered recently. We argue that, the reliability of the sensitivity parameter can be improved if it is used properly.Comment: 6 page

kkth power residue chains of global fields

In 1974, Vegh proved that if $k$ is a prime and $m$ a positive integer, there is an $m$ term permutation chain of $k$th power residue for infinitely many primes [E.Vegh, $k$th power residue chains, J.Number Theory, 9(1977), 179-181]. In fact, his proof showed that $1,2,2^2,...,2^{m-1}$ is an $m$ term permutation chain of $k$th power residue for infinitely many primes. In this paper, we prove that for any "possible" $m$ term sequence $r_1,r_2,...,r_m$, there are infinitely many primes $p$ making it an $m$ term permutation chain of $k$th power residue modulo $p$, where $k$ is an arbitrary positive integer [See Theorem 1.2]. From our result, we see that Vegh's theorem holds for any positive integer $k$, not only for prime numbers. In fact, we prove our result in more generality where the integer ring $\Z$ is replaced by any $S$-integer ring of global fields (i.e. algebraic number fields or algebraic function fields over finite fields).Comment: 4 page

On a uniformly distributed phenomenon in matrix groups

We show that a classical uniformly distributed phenomenon for an element and its inverse in ($\mathbb{Z}/n\mathbb{Z})^{*}$ also exists in $\textrm{GL}_{n}(\mathbb{F}_{p})$. A $\textrm{GL}_{n}(\mathbb{F}_{p})$ analogy of the uniform distribution on modular hyperbolas has also been considered.Comment: 11 page

Cross-domain Semantic Parsing via Paraphrasing

Existing studies on semantic parsing mainly focus on the in-domain setting. We formulate cross-domain semantic parsing as a domain adaptation problem: train a semantic parser on some source domains and then adapt it to the target domain. Due to the diversity of logical forms in different domains, this problem presents unique and intriguing challenges. By converting logical forms into canonical utterances in natural language, we reduce semantic parsing to paraphrasing, and develop an attentive sequence-to-sequence paraphrase model that is general and flexible to adapt to different domains. We discover two problems, small micro variance and large macro variance, of pre-trained word embeddings that hinder their direct use in neural networks, and propose standardization techniques as a remedy. On the popular Overnight dataset, which contains eight domains, we show that both cross-domain training and standardized pre-trained word embeddings can bring significant improvement.Comment: 12 pages, 2 figures, accepted by EMNLP201

The genus fields of Artin-Schreier extensions

Let $q$ be a power of a prime number $p$. Let $k=\mathbb{F}_{q}(t)$ be the rational function field with constant field $\mathbb{F}_{q}$. Let $K=k(\alpha)$ be an Artin-Schreier extension of $k$. In this paper, we explicitly describe the ambiguous ideal classes and the genus field of $K$ . Using these results we study the $p$-part of the ideal class group of the integral closure of $\mathbb{F}_{q}[t]$ in $K$. And we also give an analogy of R$\acute{e}$dei-Reichardt's formulae for $K$.Comment: 9 pages, Corrected typo

Weak KAM theorem for Hamilton-Jacobi equations

In this paper, we generalize weak KAM theorem from positive Lagrangian systems to "proper" Hamilton-Jacobi equations. We introduce an implicitly defined solution semigroup of evolutionary Hamilton-Jacobi equations. By exploring the properties of the solution semigroup, we prove the convergence of solution semigroup and existence of weak KAM solutions for stationary equations: \begin{equation*} H(x, u, d_x u)=0. \end{equation*}Comment: 31 page

On the exact degree of multi-cyclic extension of Fq(t)\mathbb{F}_{q}(t)

Let $q$ be a power of a prime number $p$, $k=\mathbb{F}_{q}(t)$ be the rational function field over finite field $\mathbb{F}_{q}$ and $K/k$ be a multi-cyclic extension of prime degree. In this paper we will give an exact formula for the degree of $K$ over $k$ by considering both Kummer and Artin-Schreier cases.Comment: Notice that this paper will not be published. We put it on the arXiv, since we hope that some character sums estimations over $\mathbb{F}_{q}[t]$ in Sections 2 and 3 of this paper may be useful for someone in the futur

On Menon-Sury's identity with several Dirichlet characters

The Menon-Sury's identity is as follows: \begin{equation*} \sum_{\substack{1 \leq a, b_1, b_2, \ldots, b_r \leq n\\\mathrm{gcd}(a,n)=1}} \mathrm{gcd}(a-1,b_1, b_2, \ldots, b_r,n)=\varphi(n) \sigma_r(n), \end{equation*} where $\varphi$ is Euler's totient function and $\sigma_r(n)=\sum_{d\mid n}{d^r}$. Recently, Li, Hu and Kim \cite{L-K} extended the above identity to a multi-variable case with a Dirichlet character, that is, they proved \begin{equation*} \sum_{\substack{a\in\Bbb Z_n^\ast \\ b_1, \ldots, b_r\in\Bbb Z_n}} \mathrm{gcd}(a-1,b_1, b_2, \ldots, b_r,n)\chi(a)=\varphi(n)\sigma_r{\left(\frac{n}{d}\right)}, \end{equation*} where $\chi$ is a Dirichlet character modulo $n$ and $d$ is the conductor of $\chi$. In this paper, we explicitly compute the sum \begin{equation*}\sum_{\substack{a_1, \ldots, a_s\in\Bbb Z_n^\ast \\ b_1, ..., b_r\in\Bbb Z_n}}\gcd(a_1-1, \ldots, a_s-1,b_1, \ldots, b_r, n)\chi_{1}(a_1) \cdots \chi_{s}(a_s).\end{equation*} where $\chi_{i} (1\leq i\leq s)$ are Dirichlet characters mod $n$ with conductor $d_i$. A special but common case of our main result reads like this : \begin{equation*}\sum_{\substack{a_1, \ldots, a_s\in\Bbb Z_n^\ast \\ b_1, ..., b_r\in\Bbb Z_n}}\gcd(a_1-1, \ldots, a_s-1,b_1, \ldots, b_r, n)\chi_{1}(a_1) \cdots \chi_{s}(a_s)=\varphi(n)\sigma_{s+r-1}\left(\frac{n}{d}\right)\end{equation*} if $d$ and $n$ have exactly the same prime factors, where $d={\rm lcm}(d_1,\ldots,d_s)$ is the least common multiple of $d_1,\ldots,d_s$. Our result generalizes the above Menon-Sury's identity and Li-Hu-Kim's identity.Comment: 12 page

Inter-Block Permuted Turbo Codes

The structure and size of the interleaver used in a turbo code critically affect the distance spectrum and the covariance property of a component decoder's information input and soft output. This paper introduces a new class of interleavers, the inter-block permutation (IBP) interleavers, that can be build on any existing "good" block-wise interleaver by simply adding an IBP stage. The IBP interleavers reduce the above-mentioned correlation and increase the effective interleaving size. The increased effective interleaving size improves the distance spectrum while the reduced covariance enhances the iterative decoder's performance. Moreover, the structure of the IBP(-interleaved) turbo codes (IBPTC) is naturally fit for high rate applications that necessitate parallel decoding. We present some useful bounds and constraints associated with the IBPTC that can be used as design guidelines. The corresponding codeword weight upper bounds for weight-2 and weight-4 input sequences are derived. Based on some of the design guidelines, we propose a simple IBP algorithm and show that the associated IBPTC yields 0.3 to 1.2 dB performance gain, or equivalently, an IBPTC renders the same performance with a much reduced interleaving delay. The EXIT and covariance behaviors provide another numerical proof of the superiority of the proposed IBPTC.Comment: 44 pages, 17 figure