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Scientific papers, as pivotal tools for academic communication, should be articulated with clarity and precision to ensure the effective conveyance of scholarly ideas and to prevent reader confusion. Yet, many such papers conspicuously lack in-depth research, and their core content is often ambiguously presented. This pattern poses a significant impediment to the progressive evolution of science and technology. While numerous researchers have recognized this widespread challenge, a holistic theoretical or methodological solution remains elusive in the academic realm. To bridge this gap, we introduce the INTEGrity vERification (INTEGER) task. This task aids researchers in assessing the integrity of their papers by verifying the clarity of each knowledge unit. To implement this task on text, we propose a multi-task learning model that utilizes the Tucker decomposition and span-level attention mechanism to identify terms and their integrity precisely. More specifically, to provide insights into the INTEGER task and validate the effectiveness of the proposed model, we collect 8076 sentences and construct three new datasets containing various types of terms and descriptions in different domains. Extensive experimental results show that our proposed model has an average performance improvement of 1.1% F1 over the three datasets compared to a series of state-of-the-art baseline methods. [ABSTRACT FROM AUTHOR]

1 student-> 1/4 Each group gets 1 pancake 1 group-> 1 72 pancake 1 pancake for 4 students 1/4+1/4+1/4+1/4+1/4+1/4 = 6/4 =11/2 * x x x 1 pancake xxx*2 <1 1/2 1; <1 1/2 DURING (Working On it) • 20 minutes-4 ~ Pancakes j { pancakes) ! Pancakes Students Student 112 Class 2: 6 pancakes for 24 8 32 8/32 3456789 students-> (6/24)/(6/6) = 1/4 4 16 4/16 4 \ 8 12 16 20 24 28 32 36 Class 3: 10 pancakes 30 Each student gets 1/4 of one They ali get 1/4 of a pancake each because students-> (10/30) / (10/10) = 2 8 2/8 pancake. - divide 1 pancake into 5 pieces (for 5 children), so each child gets 1/5 of 1 pancake, so for 4 pancakes, one child gets 1/5 of l pancake x 4 pancakes or 4/5 pancake Solution 1 Solution 2 \ 6/4 =1 1/2 ~ Solution 2» te,\,/rr~ /r]EN /R-* 1/5 four times Class 1 Class 2 / Class 3 [ xx Hxx 1(xx)(xx) # 4/5 of 1 pancake 8 pancakes B groups 6 pancakes 4 groups. [Extracted from the article]

This article is a follow-up contribution that extends the conference paper (Nxumalo, in: Laarman, Sokolova (eds)Model Checking Software - 27th International Symposium, 2021) which presented the spotlight abstraction method to enable an efficient model checking of schedulability properties of real-time operating systems. Our approach is based on iterative spotlight abstraction whereby the real-time task queue is divided into a so-called spotlight and a shade. Initially, an abstract state space model is generated from the spotlight, which contains a small number of tasks that appear at the front of the queue and the shade contains the remaining tasks. The schedulability of the spotlight tasks is model checked, whereas the behavior of the shade is summarized, and the result is saved for later re-use. If this result is still inconclusive, more tasks are iteratively brought from the shade into the spotlight, with which the model checker can proceed. These steps are repeated until a decisive schedulability result is obtained. In this article, we enrich the encoding of real-time systems into timed automata by separating operations over clock variables from operations based on other variables such as Boolean and integers. We enforce to maintain a reasonable size of the spotlight model by applying counter abstraction to the tasks that were previously confirmed schedulable in the previous iterations. We compare our approach against Timestool and RTLib, an Uppaal template, in the analysis of the schedulability of real-time tasks for the First-In-First-Out and Shortest-Deadline-First scheduling policies. Empirical results show that our approach can handle more tasks than Timestool and RTLib. [ABSTRACT FROM AUTHOR]

This paper is a corrigendum to the article "On the ideal theorem for number fields". The main result of this paper proves to be untrue and is replaced by an estimate of a weighted sum with an improved error term. [ABSTRACT FROM AUTHOR]

Nowadays, researchers are very interested in studying various Diophantine equations due to their importance in Cryptography, Chemistry, Knot Theory, Astronomy, Geometry, Trigonometry, Biology, Algebra, Electrical Engineering, Economics, and Astrology. The present paper is about the non-negative integer solution of the exponential Diophantine equation nx + 43y = z², where are non-negative integers, is a positive integer with nx + 43y = z², where n = 2 (mod 129) and n + 1 and is not a perfect square. The authors use the famous Catalan conjecture for this purpose. Results of the present paper indicate that 2, 3, 0, and 3 are the only required values of and respectively, that satisfy the exponential Diophantine equation, where are non-negative integers, is a positive integer with nx + 43y = z², where n = 2 (mod 129) and n + 1 and is not a perfect square. The present technique of this paper proposes a new approach to solving the Diophantine equations, which is the main scientific contribution of this study, and it is very beneficial, especially for researchers, scholars, academicians, and people interested in the same field. [ABSTRACT FROM AUTHOR]

We show that the results of the paper mentioned in the title are known or follow from the known results. [ABSTRACT FROM AUTHOR]

Integer variables are among the neo-logistic programming issues that most frequently pertain to system reliability improvement. If 'k' out of 'n' elements are present in each technology, then the configuration can be employed with 'k' out of 'n' systems. Only particular objective function structures and restrictions may be used in a heuristic technique to precisely address a dependability optimization problem. As more constraints are placed on it, the more useful it becomes; it is worthless for boosting dependability in a large system. This paper analyzes existing research on integrated reliability models with redundancy and explores system reliability optimization before proposing new recommendations. [ABSTRACT FROM AUTHOR]

Having recently come across a 1970 paper of R. L. Graham about cube-numbering and its generalizations, we found that one proof uses an inequality about the summatory function of the sum of binary digits of integers. Graham gave a very elegant and somehow unexpected proof of this inequality. We propose a more "pedestrian" --and somehow more standard-- proof of this inequality, as well as questions about possible generalizations. [ABSTRACT FROM AUTHOR]

RESEARCH SUMMARY- EQUIVALENCE FOR FRACTIONS, DECIMALS, AND PERCENTS Randall (2005) identifies a big idea for equivalence: "Every fraction/ratio can be represented by an infinite number of different but equivalent fractions/ratios" and "Decimal numbers can be named in an infinite number of different but equivalent forms" (p. 14). Fraction/Percent Strip 1 1 .44 Highlight 6/8 on the fraction/percent strip and 3/4 on the strip. lili 1 1 lili 11 1 1 - Compare the lengths for equivalence. Equivalent Fractions, Decimals, and Percents According to Fosnot and Dolk (2002), decimals and percents are really just specific equivalents of fractional relations. [Extracted from the article]

The article provide a range of conceptual approaches to the development of fraction-based understanding including paper folding, the use of number lines and using multiple fraction representations to build a foundation of success in mathematics. Topics include earliest experiences students have when learning fractions are hands-on activities involving paper folding and the coloring of 2D shapes; and useful to work with the length model first and then transition students to the number line.

This paper presents a modified version of Leibman's group-theoretic generalizations of the difference calculus for polynomial maps from nonempty commutative semigroups to groups, and proves that it has many desirable formal properties when the target group is locally nilpotent and also when the semigroup is the set of nonnegative integers. We will apply it to solve Waring's problem for general discrete Heisenberg groups in a sequel to this paper. [ABSTRACT FROM AUTHOR]