Resonance between the Representation Function and Exponential Functions over Arithemetic Progression

Let r n denote the number of representations of a positive integer n as a sum of two squares, i.e., n = x 1 2 + x 2 2 , where x 1 and x 2 are integers. We study the behavior of the exponential sum twisted by r n over the arithmetic progressions ∑ n ∼ X n ≡ l mod q r n e α n β , where 0 ≠ α ∈ ℝ , 0 < β < 1 , e x = e 2 π i x , and n ∼ X means X < n ≤ 2 X . Here, X > 1 is a large parameter, 1 ≤ l ≤ q are integers, and l , q = 1 . We obtain the upper bounds in different situations.

PARTITIONS OF WITH IDENTICAL REPRESENTATION FUNCTION

For a given set $S\subseteq \mathbb {Z}_m$ and $\overline {n}\in \mathbb {Z}_m$ , let $R_S(\overline {n})$ denote the number of solutions of the equation $\overline {n}=\overline {s}+\overline {s'}$ with ordered pairs $(\overline {s},\overline {s'})\in S^2$ . We determine the structure of $A,B\subseteq \mathbb {Z}_m$ with $|(A\cup B)\setminus (A\cap B)|=m-2$ such that $R_{A}(\overline {n})=R_{B}(\overline {n})$ for all $\overline {n}\in \mathbb {Z}_m$ , where m is an even integer.

Probabilistic Representation Approach for Multiple Types of Epistemic Uncertainties Based on Cubic Normal Transformation

Many non-probabilistic approaches have been widely regarded as mathematical tools for the representation of epistemic uncertainties. However, their heavy computational burden and low computational efficiency hinder their applications in practical engineering problems. In this article, a unified probabilistic representation approach for multiple types of epistemic uncertainties is proposed based on the cubic normal transformation method. The epistemic uncertainties can be represented using an interval approach, triangular fuzzy approach, or evidence theory. The uncertain intervals of four statistical moments, which contain mean, variance, skewness, and kurtosis, are calculated using the sampling analysis method. Subsequently, the probabilistic cubic normal distribution functions are conducted for sampling points of four statistical moments of epistemic uncertainties. Finally, a calculation procedure for the construction of probabilistic representation functions is proposed, and these epistemic uncertainties are represented with belief and plausibility continuous probabilistic measure functions. Two numerical examples and one engineering example demonstrate that the proposed approach can act as an accurate probabilistic representation function with high computational efficiency.

Zur (In)Korrektheit der erbrechtlichen Terminologie in der polnischen und deutschen Rechtssprache anhand ausgewählter Beispiele. Eine kritische Studie

The linguistic picture of the world as the way of perception of the world determined by the culture and reflected in the language, as well as communicative function of language may constitute a basis for research of two questions: 1) whether - and to what extent - a term used in communication act reflects the intention of the speaker and corresponds to the reflection of the picture of this term in his or her mind; 2) whether the term expressed by the speaker evokes the same - or at least similar - picture in the recipient’s mind. The importance of this matter becomes particularly evident in respect of legal language within the scope of which the utterance precision shall be the main goal of legislators. This article is an attempt at finding the answer whether the terms of Polish and German languages of inheritance law always express the content they shall reflect or whether there are such terms in both languages which suggest on the linguistic basis another content than that supposed by the legislators. The article presents the results of the analysis of the chosen terms describing institutions of Polish and German inheritance law in consideration of the representation function of the terms and the linguistic picture of the world reflected in them.

The effects of a visible-annotation tool for sequential knowledge construction on discourse patterns and collaborative outcomes

This study aimed to explore learners’ discourse patterns and outcomes while using a visible-annotation tool as a collaborative representation tool. The tool used in this study introduced two types of sharing activities before the problem-solving phase to support sequential knowledge construction. Forty participants were randomly assigned to one of three groups according to two variables: type of sharing activities (meaning sharing activity (M) and opinion sharing activity (O)), and type of representation function to guide sharing activities (word-based function (W) and sentence-based function (S)). All three groups performed sharing activities during the same period. After completing these, the participants carried out a lesson-planning task in pairs during the problem-solving phase. All annotations across three learning phases were categorised to investigate discourse patterns. The findings revealed that Group MWOS, provided with M based on W and O based on S, had the most effective knowledge construction process, showing sequential discourse patterns. In addition, differences in discourse patterns among groups positively influenced the level of collaborative outcomes.