Frequency permutation arrays

Motivated by recent interest in permutation arrays, we introduce and investigate the more general concept of frequency permutation arrays (FPAs). An FPA of length n=m lambda and distance d is a set T of multipermutations on a multiset of m symbols, each repeated with frequency lambda, such that the Hamming distance between any distinct x,y in T is at least d. Such arrays have potential applications in powerline communication. In this paper, we establish basic properties of FPAs, and provide direct constructions for FPAs using a range of combinatorial objects, including polynomials over finite fields, combinatorial designs, and codes. We also provide recursive constructions, and give bounds for the maximum size of such arrays.Comment: To appear in Journal of Combinatorial Design

On the structure of non-full-rank perfect codes

The Krotov combining construction of perfect 1-error-correcting binary codes from 2000 and a theorem of Heden saying that every non-full-rank perfect 1-error-correcting binary code can be constructed by this combining construction is generalized to the $q$-ary case. Simply, every non-full-rank perfect code $C$ is the union of a well-defined family of $\mu$-components $K_\mu$, where $\mu$ belongs to an "outer" perfect code $C^*$, and these components are at distance three from each other. Components from distinct codes can thus freely be combined to obtain new perfect codes. The Phelps general product construction of perfect binary code from 1984 is generalized to obtain $\mu$-components, and new lower bounds on the number of perfect 1-error-correcting $q$-ary codes are presented.Comment: 8 page

Around the Hossz\'u-Gluskin theorem for nn-ary groups

We survey results related to the important Hossz\'u-Gluskin Theorem on $n$-ary groups adding also several new results and comments. The aim of this paper is to write all such results in uniform and compressive forms. Therefore some proofs of new results are only sketched or omitted if their completing seems to be not too difficult for readers. In particular, we show as the Hossz\'u-Gluskin Theorem can be used for evaluation how many different $n$-ary groups (up to isomorphism) exist on some small sets. Moreover, we sketch as the mentioned theorem can be also used for investigation of $\mathcal{Q}$-independent subsets of semiabelian $n$-ary groups for some special families $\mathcal{Q}$ of mappings

Ontology as Transcendental Philosophy

How does the critical Kant view ontology? There is no shared scholarly answer to this question. Norbert Hinske sees in the Critique of Pure Reason a “farewell to ontology,” albeit one that took Kant long to bid (Hinske 2009). Karl Ameriks has found evidence in Kant’s metaphysics lectures from the critical period that he “was unwilling to break away fully from traditional ontology” (Ameriks 1992: 272). Gualtiero Lorini argues that a decisive break with the tradition of ontology is essential to Kant’s critical reform of metaphysics, as is reflected in his shift from “ontology” to “transcendental philosophy,” two notions that Lorini takes to be related by mere “analogy” (Lorini 2015). I agree with Lorini that a thorough reform of ontology is a pivotal part of Kant’s critical plan for metaphysics and that ontology somehow “survives within the critical philosophy” (Lorini 2015: 76). To make this case, however, I deem it important to identify “ontology” and “transcendental philosophy” in the sense of extensional equivalence. While we can detect this identification in Kant’s writings, only from his metaphysics lectures can we get a full sense of its historical and philosophical significance. In this chapter I focus on how it represents a definitive turn from as well as notable continuity with traditional treatments of ontology, particularly the Wolffian one

A Guide to Ground in Kant's Lectures on Metaphysics

While scholars have extensively discussed Kant’s treatment of the Principle of Sufficient Ground in the Antinomies chapter of the Critique of Pure Reason, and, more recently, his relation to German rationalist debates about it, relatively little has been said about the exact notion of ground that figures in the PSG. My aim in this chapter is to explain Kant’s discussion of ground in the lectures and to relate it, where appropriate, to his published discussions of ground

T-functions revisited: New criteria for bijectivity/transitivity

The paper presents new criteria for bijectivity/transitivity of T-functions and fast knapsack-like algorithm of evaluation of a T-function. Our approach is based on non-Archimedean ergodic theory: Both the criteria and algorithm use van der Put series to represent 1-Lipschitz $p$-adic functions and to study measure-preservation/ergodicity of these

State-of-the art data normalization methods improve NMR-based metabolomic analysis

Extracting biomedical information from large metabolomic datasets by multivariate data analysis is of considerable complexity. Common challenges include among others screening for differentially produced metabolites, estimation of fold changes, and sample classification. Prior to these analysis steps, it is important to minimize contributions from unwanted biases and experimental variance. This is the goal of data preprocessing. In this work, different data normalization methods were compared systematically employing two different datasets generated by means of nuclear magnetic resonance (NMR) spectroscopy. To this end, two different types of normalization methods were used, one aiming to remove unwanted sample-to-sample variation while the other adjusts the variance of the different metabolites by variable scaling and variance stabilization methods. The impact of all methods tested on sample classification was evaluated on urinary NMR fingerprints obtained from healthy volunteers and patients suffering from autosomal polycystic kidney disease (ADPKD). Performance in terms of screening for differentially produced metabolites was investigated on a dataset following a Latin-square design, where varied amounts of 8 different metabolites were spiked into a human urine matrix while keeping the total spike-in amount constant. In addition, specific tests were conducted to systematically investigate the influence of the different preprocessing methods on the structure of the analyzed data. In conclusion, preprocessing methods originally developed for DNA microarray analysis, in particular, Quantile and Cubic-Spline Normalization, performed best in reducing bias, accurately detecting fold changes, and classifying samples

Wild chimeras: Enthusiasm and intellectual virtue in Kant

Kant typically is not identified with the tradition of virtue epistemology. Although he may not be a virtue epistemologist in a strict sense, I suggest that intellectual virtues and vices play a key role in his epistemology. Specifically, Kant identifies a serious intellectual vice that threatens to undermine reason, namely enthusiasm (Schwärmerei). Enthusiasts become so enamored with their own thinking that they refuse to subject reason to self‐critique. The particular danger of enthusiasm is that reason colludes in its own destruction: Enthusiasm occurs when self‐conceit and reason\u27s desire to transcend its boundaries mutually reinforce each other. I conclude by sketching an account of Kantian intellectual virtue that is consistent with Kantian moral virtue

Kant on the role of the imagination (and images) in the transition from intuition to experience

In this chapter I will argue against both of these interpretations and will begin to develop an alternate account of imagination in experience. Against those who minimize imagination’s role, I will highlight the distinctive contribution of the imagination to experience. In particular, I will foreground the specific role that the imagination plays in making possible the distinct mental act, intermediate between intuition and experience, that Kant calls “perception [Wahrnehmung]” as the “empirical consciousness [Bewußtsein]” of appearances (cf. B207). Because perception involves images essentially (cf. A120), and because Kant understands experience itself to be a “synthesis of perceptions” (cf. B218), this strongly suggests (against minimalists) that experience, too, will incorporate images into the manner in which it allows us to cognize physical objects. By highlighting the contribution of imagination prior to experience, my own account, therefore, overlaps in part with the readings that seek instead to maximize the role of imagination. Against maximalists, however, I will argue that imagination contributes only in (and after) the transition from intuition to perception, rather than already being at work in the stage of intuition itself. More specifically, I will argue that Kant takes the activity of imagination to make perception possible by acting on already-formed intuitions in order to bring about the consciousness of them, rather than to bring the intuitions about in the first place. I will also argue that this synthesis of intuitions should be kept distinct from the activity of understanding