<abstract xmlns="http://www.w3.org/1999/xhtml">
<bold>Summary</bold>. In this paper, we discuss the properties that hold in finite dimensional vector spaces and related spaces. In the Mizar language [1], [2], variables are strictly typed, and their type conversion requires a complicated process. Our purpose is to formalize that some properties of finite dimensional vector spaces are preserved in type transformations, and to contain the complexity of type transformations into this paper. Specifically, we show that properties such as algebraic structure, subsets, finite sequences and their sums, linear combination, linear independence, and affine independence are preserved in type conversions among <monospace>TOP-REAL(n)</monospace>, <monospace>REAL-NS(n)</monospace>, and <monospace>n-VectSp over F Real</monospace>. We referred to [4], [9], and [8] in the formalization.
</abstract>

Real Vector Space and Related Notions

<abstract xmlns="http://www.w3.org/1999/xhtml">
<bold>Summary</bold>. In this article we further develop field theory in Mizar [1], [2]: we prove existence and uniqueness of splitting fields. We define the splitting field of a polynomial <italic>p</italic> ∈ <italic>F</italic> [<italic>X</italic>] as the smallest field extension of <italic>F</italic>, in which <italic>p</italic> splits into linear factors. From this follows, that for a splitting field <italic>E</italic> of <italic>p</italic> we have <italic>E</italic> = <italic>F</italic> (<italic>A</italic>) where <italic>A</italic> is the set of <italic>p</italic>’s roots. Splitting fields are unique, however, only up to isomorphisms; to be more precise up to <italic>F</italic> -isomorphims i.e. isomorphisms <italic>i</italic> with <italic>i|F</italic> = Id<italic>F</italic>. We prove that two splitting fields of <italic>p</italic> ∈ <italic>F</italic> [<italic>X</italic>] are <italic>F</italic> -isomorphic using the well-known technique [4], [3] of extending isomorphisms from <italic>F</italic>1 → <italic>F</italic>2 to <italic>F</italic>1(<italic>a</italic>) → <italic>F</italic>2(<italic>b</italic>) for <italic>a</italic> and <italic>b</italic> being algebraic over <italic>F</italic>1 and <italic>F</italic>2, respectively.
</abstract>

Splitting Fields

<abstract xmlns="http://www.w3.org/1999/xhtml">
<bold>Summary</bold>. The bin packing problem is a fundamental and important optimization problem in theoretical computer science [4], [6]. An instance is a sequence of items, each being of positive size at most one. The task is to place all the items into bins so that the total size of items in each bin is at most one and the number of bins that contain at least one item is minimum.
Approximation algorithms have been intensively studied. Algorithm NextFit would be the simplest one. The algorithm repeatedly does the following: If the first unprocessed item in the sequence can be placed, in terms of size, additionally to the bin into which the algorithm has placed an item the last time, place the item into that bin; otherwise place the item into an empty bin. Johnson [5] proved that the number of the resulting bins by algorithm NextFit is less than twice the number of the fewest bins that are needed to contain all items.
In this article, we formalize in Mizar [1], [2] the bin packing problem as follows: An instance is a sequence of positive real numbers that are each at most one. The task is to find a function that maps the indices of the sequence to positive integers such that the sum of the subsequence for each of the inverse images is at most one and the size of the image is minimum. We then formalize algorithm NextFit, its feasibility, its approximation guarantee, and the tightness of the approximation guarantee.
</abstract>

Algorithm NextFit for the Bin Packing Problem

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  Pioneering Journal in the field of formalized and mechanically checked mathematics. Excellent articles on latest research in formal methods and computer-checked mathematics. Full-length original research articles in English. Archival issues available at the journal's home page fm.mizar.org/.   To build a uniform and extensible computer-checked repository of mathematical knowledge.  Reconstruction of mathematics based on mechanization of mathematical reasoning.  To check a wide range of mathematical theorems using a computer in an automatic fashion.  General mathematics: especially topology, lattice theory, functional analysis.  Applications: computer-aided verification of software and hardware, computer-aided teaching of mathematics.   Why subscribe and read   Pioneering source of high quality research in formalized and mechanically checked mathematics.  The oldest journal in the field, a medium for the publication of research reporting pioneering findings of broad mathematical significance that are likely to advance the direction and thinking of the field.  Long-term experience in publishing original articles on mechanizing mathematics and various aspects of automated reasoning, since 1990.   Why submit   Fair and constructive peer review by international experts in the field (three referees are designated for the assessment of each manuscript) - double-blind review process.  New publishing technology: articles automatically translated from a formal language into English and automatically typeset, available in both forms - electronic and printed.  Articles abstracted by online indexing services and databases.   Rejection Rate   59% of submitted articles.  The rejection percentage is described in detail in the following paper, pp.239-240: dx.doi.org/10.1007/978-3-540-73086-6_20 Archiving  Sciendo archives the contents of this journal in Portico - digital long-term preservation service of scholarly books, journals and collections.  Plagiarism Policy  The editorial board is participating in a growing community of Similarity Check System's users in order to ensure that the content published is original and trustworthy. Similarity Check is a medium that allows for comprehensive manuscripts screening, aimed to eliminate plagiarism and provide a high standard and quality peer-review process. Back Issues fm.mizar.org/   Other Files  Addenda 2008Formalized Mathematics Bibliography File.Formalized Mathematics External Bibliography File.  "In years: 2016-2017, the English version of Formalized Mathematics was financed under agreements 548/P-DUN/2016 and 548/1/P-DUN/2016 with the funds from the Polish Minister of Science and Higher Education for the dissemination of science." 

Formalized Mathematics

Volume 29 (2021): Issue 3 (October 2021)

Artikel

Real Vector Space and Related Notions

Splitting Fields

Algorithm NextFit for the Bin Packing Problem

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