Latin Trades Research Articles

Preventive maintance plan for SKF Latin Trade SAS equipment

This research presents the design and implementation of the preventive and predictive maintenance plan developed for the seven polishing machines of the Swedish multinational SKF, of the LSB (Large Size Bearing) line. Based on Reliability Centered Maintenance (RCM), which allows identifying the most relevant equipment for the process through a criticality analysis adopting strategies in order to eliminate the occurrence of failures. The methodology implemented in this research allowed the generation of maintenance tasks that facilitated the management and visualization of the information of each asset. The company provides maintenance services for large bearings used in cement plants, foundries and industrial material transport machines, seeking to optimize production methods and the application of this theoretical and analytical procedure allowed verifying that the conditioning services offered by the company are more efficient, demonstrating with figures the retribution of the plan.

Authority and Power in the Medieval Church, c. 1000–c. 1500 ed. by Thomas W. Smith

Reviewed by: Authority and Power in the Medieval Church, c. 1000–c. 1500 ed. by Thomas W. Smith Beth C. Spacey Smith, Thomas W., ed., Authority and Power in the Medieval Church, c. 1000–c. 1500, Brepols, Turnhout, 2020; hardback; pp. 412; 19 b/w illustrations, 2 b/w tables; R.R.P. €100.00; ISBN 9782503585291. This is a superb and substantial collection. Thomas Smith has successfully marshalled twenty-two chapters in this interdisciplinary exploration of power and authority in the medieval Church, curating a varied yet coherent compilation. While some of the chapters are more explicitly linked to the book's central themes than others, the collection nonetheless succeeds in demonstrating how enormously variable Church power and authority was in form, justification, and expression over time and space. A key issue that emerges across the contributions concerns the Church's ability to move beyond claims to authority to act upon those claims, and how often those mechanisms of power were dependent upon local cooperation. The range of disciplines and approaches represented in this collection is commendable and highlights that much important research continues to be undertaken in what remains a vigorous field, and that many fruitful avenues of inquiry remain. Part I explores ideas of papal authority. Ian Robinson examines how eleventh-century ecclesiasts drew on fifth- and ninth-century ideas of judicial supremacy in their own discussions of papal primacy. Benedict Wiedemann explores how letters from the papal curia of Innocent III drew upon different methods to legitimize the Pope's authority as regent of the Kingdom of Sicily during the minority of Frederick II, while Rebecca Rist addresses the question of whether the popes of the eleventh to thirteenth centuries were anti-Judaic or anti-Semitic. Laura Cleaver examines how the genealogical diagrams contained in four thirteenth-century English manuscripts portrayed papal power and authority through the positioning of depictions of popes and their contemporaries. Part II looks to the authority of papal representatives. Smith uses the case study of thirteenth-century Languedoc to examine how the absence of local support could affect papal legates' ability to wield power, while Gábor Barabás outlines the varied fortunes of legates in thirteenth-century Hungary, from cooperation to incarceration. Philippa Mesiano uses the case of thirteenth-century papal nuncio Master Rostand Masson to consider how papal representatives exercised power [End Page 262] in their diplomatic destinations. Finally, Jean Dunbabin identifies how papal representatives could respond pragmatically to the local diplomatic landscape, by exploring a case of dual secular-ecclesial rulership in late thirteenth-century Sicily. Part III considers relations between the papacy and the Eastern Churches. The chapter by the late Bernard Hamilton examines papal attitudes towards the separated Eastern Churches in the twelfth and thirteenth centuries and the papacy's prioritization of 'corporate' over 'individual' conversion. James Hill analyses the entanglement of papal power and its ability to achieve unity with and reform of the Byzantine and Armenian Churches respectively, while Mike Carr argues that the Avignon papacy did wield authority in its regulation of Latin trade with Muslims, as evidenced by trade licences. Part IV examines the cultural history of ecclesiastical authority and power. Matthew Ross traces cultural life at the medieval curia through the examination of the papal chapel. Jan Vanderburie analyses the symbolism of the Oignies treasures associated with Jacques de Vitry, arguing that Jacques deliberately drew on specific imagery in his self-fashioning. Catherine Lawless uses two images from the Florentine Franciscan context to explore the relationship between the power of female mysticism and the authority of friars, the latter of which recognized the devotional power of the former and acted as mediators with the realm of male ecclesiastical power. Kirsty Day demonstrates how examples of royal and noble women associated with female Franciscan communities in Bohemia and the Polish duchies challenge the narrative of women religious as defiers of male ecclesiastical authority. Part V discusses ecclesiastical communities. Nicholas Vincent considers how secular authorities exercised power over bishops, examining the conventions for the ordering of bishops in the witness lists of Henry II's charters. Maroula Perisanidi considers how Eustathios of Thessalonike offers an alternative model of spiritual and moral authority in...

Silk Velvets Identified as Byzantine

This paper will examine the possibility of whether warp-looped pile velvets, made of silk, were woven during the Byzantine Empire. This study is a continuation of my research for “Velvet and Patronage: the Origin and Historical Background of Ottoman and Italian Velvets.” Research has been conducted under these five themes: 1. Byzantine silk industry; 2. The terminology of velvet in Greek; 3. Velvets in Byzantine written sources; 4. Historical background: the relation between Byzantines and Latin powers, Turks and the Middle East; 5. Latin trade in the Black Sea and 6. Velvet production in Anatolia in the Byzantine period.

Orthogonal Trades in Complete Sets of MOLS

Let $B_p$ be the Latin square given by the addition table for the integers modulo an odd prime $p$. Here we consider the properties of Latin trades in $B_p$ which preserve orthogonality with one of the $p-1$ MOLS given by the finite field construction. We show that for certain choices of the orthogonal mate, there is a lower bound logarithmic in $p$ for the number of times each symbol occurs in such a trade, with an overall lower bound of $(\log{p})^2/\log\log{p}$ for the size of such a trade. Such trades imply the existence of orthomorphisms of the cyclic group which differ from a linear orthomorphism by a small amount. We also show that any transversal in $B_p$ hits the main diagonal either $p$ or at most $p-\log_2{p}-1$ times. Finally, if $p\equiv 1\pmod{6}$ we show the existence of a Latin square which is orthogonal to $B_p$ and which contains a $2\times 2$ subsquare.

On the distances between Latin squares and the smallest defining set size

We show that for each Latin square L of order n≥2, there exists a Latin square L′≠L of order n such that L and L′ differ in at most 8n cells. Equivalently, each Latin square of order n contains a Latin trade of size at most 8n. We also show that the size of the smallest defining set in a Latin square is Ω(n3/2).

On the Distances between Latin Squares and the Smallest Defining Set Size

AbstractIn this note, we show that for each Latin square L of order , there exists a Latin square of order n such that L and differ in at most cells. Equivalently, each Latin square of order n contains a Latin trade of size at most . We also show that the size of the smallest defining set in a Latin square is .

The Spectrum for $3$-Way $k$-Homogeneous Latin Trades

A $\mu$-way $k$-homogeneous Latin trade was defined by Bagheri Gh, Donovan, Mahmoodian (2012), where the existence of $3$-way $k$-homogeneous Latin trades was specifically investigated. We investigate the existence of a certain class of $\mu$-way $k$-homogeneous Latin trades with an idempotent like property. We present a number of constructions for $\mu$-way $k$-homogeneous Latin trades with this property, and show that these can be used to fill in the spectrum of $3$-way $k$-homogeneous Latin trades for all but $196$ possible exceptions.

On the simultaneous edge coloring of graphs

A μ-simultaneous edge coloring of graph G is a set of μ proper edge colorings of G with a same color set such that for each vertex, the sets of colors appearing on the edges incident to that vertex are the same in each coloring and no edge receives the same color in any two colorings. The μ-simultaneous edge coloring of bipartite graphs has a close relation with μ-way Latin trades. Mahdian et al. (2000) conjectured that every bridgeless bipartite graph is 2-simultaneous edge colorable. Luo et al. (2004) showed that every bipartite graphic sequence S with all its elements greater than one, has a realization that admits a 2-simultaneous edge coloring. In this paper, the μ-simultaneous edge coloring of graphs is studied. Moreover, the properties of the extremal counterexample to the above conjecture are investigated. Also, a relation between 2-simultaneous edge coloring of a graph and a cycle double cover with certain properties is shown and using this relation, some results about 2-simultaneous edge colorable graphs are obtained.

Growth rate of canonical and minimal group embeddings of spherical latin trades

A spherical latin trade is a partial latin square associated with a face 2-colourable triangulation of the sphere. A latin trade W embeds into an abelian group G if the Cayley table of G contains a partial latin square isotopic to W. A trade W embeds minimally into G if W embeds into G but not into any proper subgroup of G. The canonical groupAW⁎ of a spherical latin trade W is a finite abelian group that W embeds into, and possesses a “universal property”: every abelian group that W embeds into minimally is isomorphic to a quotient group of AW⁎.We study simple graph operations that create larger spherical latin trades from smaller ones. In particular, we analyse the effect of these operations on the canonical groups of the trades. We then apply our results to answer several open problems about canonical and minimal groups recently posed by Cavenagh and Wanless and by Blackburn and McCourt.A family of trades is constructed that exhibits the fastest known asymptotic growth for |AW⁎| with respect to the size |W|. It achieves |AW⁎|=r|W|(1+o(1)), where r≈1.329. Another family is shown to embed minimally into groups of exponentially increasing order; previously exponential growth had only been observed for canonical groups.Questions about the rank (size of a smallest generating set) of canonical and minimal groups are also resolved. The rank of any minimal group is shown to be at most logarithmic in the size of the trade, resulting in an example where the canonical group grows much faster than any minimal group, with regards to both order and rank.

On the existence of [formula omitted]-way [formula omitted]-homogeneous Latin trades

A μ-way Latin trade of volume s is a collection of μ partial Latin squares T1,T2,…,Tμ, containing exactly the same s filled cells, such that, if cell (i,j) is filled, it contains a different entry in each of the μ partial Latin squares, and such that row i in each of the μ partial Latin squares contains, set-wise, the same symbols, and column j likewise. It is called a μ-wayk-homogeneous Latin trade if, in each row and each column, Tr, for 1≤r≤μ, contains exactly k elements, and each element appears in Tr exactly k times. It is also denoted as a (μ,k,m) Latin trade, where m is the size of the partial Latin squares.We introduce some general constructions for μ-way k-homogeneous Latin trades, and specifically show that, for all k≤m, 6≤k≤13, and k=15, and for all k≤m, k=4,5 (except for four specific values), a 3-way k-homogeneous Latin trade of volume km exists. We also show that there is no (3,4,6) Latin trade and there is no (3,4,7) Latin trade. Finally, we present general results on the existence of 3-way k-homogeneous Latin trades for some modulo classes of m.

TRANSVERSALS, INDIVISIBLE PLEXES AND PARTITIONS OF LATIN SQUARES

A k-plex in a latin square of order n is a selection of kn entries in which each row, column and symbol occurs exactly k times. A 1-plex is also called a transversal. A k-plex is indivisible if it contains no c-plex for 0 1. For n ∈ {2, 6}, the existence of a latin square of order n with a partition into 1- plexes was shown by Bose, Shrikhande and Parker [1]. A natural extension of the result to k-plexes is the following. We prove that if k is a proper divisor of n, then there exists a latin square of order n composed of parallel indivisible k-plexes. Define κ(n) to be the largest integer k such that some latin square of order n contains an indivisible k-plex. A conjecture by Rodney [3, p143] says that every latin square contains a 2-plex, which implies that κ(n) 2. We show that for all n > 2, there exists a latin square of order n with two parallel indivisible ⌊ n 2 ⌋-plexes. A corollary is that κ(n) ≥ ⌊ n 2 ⌋ for all n > 2. We report on extensive computations of the indivisible partitions of the latin squares of order n ≤ 9. Up to order 8 we count the number of indivisible partitions of every type. Due to Bose, Shrikhande and Parker [1], and Finney [10] for n = 6, we know that there exists a latin square for each order n > 2 which possesses a k-plex for all 0 ≤ k ≤ n. Wanless [12] showed that there are latin squares of every even order which contain no odd plexes, where an odd plex is a k-plex such that k is odd. It has been conjectured by Rodney [4], and by Wanless [12], that every latin square has a set of ⌊ n 2 ⌋ parallel 2-plexes which implies that every latin square of odd order has a k-plex for each 0 ≤ k ≤ n. We show that among the latin squares of even order there are many other possibilities concerning the existence of odd k-plexes. 1 We prove that for all even n > 2, there exists a latin square of order n which has no k-plex for any odd k 3, there exists a latin square of order n with at least one entry not in any transversal. Such a latin square is called a confirmed bachelor and illustrates a restriction on the transversals in a latin square. In our study of transversals we consider bachelor latin squares in greater detail and other types of restrictions that might occur. We give a concise alternative proof of the result by Wanless and Webb. A main result is that for all even n ≥ 10, except perhaps if n is a power of 2, there exists a latin square of order n that possesses a transversal, but every transversal coincides on a single entry. A theorem by Wanless, which was prompted by our data, shows that there exist arbitrarily large latin squares of odd order in which the proportion of entries not in a transversal is asymptotic to one ninth. We report on computations of parallel transversals in the latin squares of order 9. Thus we prove that the above mentioned conjectures by Rodney and Wanless are true for order 9, and that every latin square of order 9 has at least 3 parallel transversals. Our computations suggest that the constructions which prove our theorems illustrate rare behaviour in large latin squares. The computations also give further evidence in support of a conjecture by Ryser [11] (see [3, p143]) that every latin square of odd order has a transversal. Many of the theorems in this thesis rely on a simple but powerful lemma [6, 9]. The lemma gives a necessary condition for the existence of k-plexes. To show the existence of k-plexes we use constructive techniques. It remains open to establish a sufficient condition for the existence of k-plexes in latin squares. The main results of this thesis can be found in [2, 5–8]. Keywords : latin square, transversal, plex, duplex, triplex, quadruplex, indivisible plex, indivisible partition, orthogonal partition, bachelor square, mutually orthogo- nal latin squares, partial latin square, protoplex, homogeneous partial latin square, latin trade, homogeneous latin trade, rainbow factor, graph factorisation, quasi- group, complete mapping, orthomorphism.

Multi-latin squares

A multi-latin square of order n and index k is an n × n array of multisets, each of cardinality k , such that each symbol from a fixed set of size n occurs k times in each row and k times in each column. A multi-latin square of index k is also referred to as a k -latin square. A 1 -latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square. In this note we show that any partially filled-in k -latin square of order m embeds in a k -latin square of order n , for each n ≥ 2 m , thus generalizing Evans’ Theorem. Exploiting this result, we show that there exist non-separable k -latin squares of order n for each n ≥ k + 2 . We also show that for each n ≥ 1 , there exists some finite value g ( n ) such that for all k ≥ g ( n ) , every k -latin square of order n is separable. We discuss the connection between k -latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and k -latin trades. We also enumerate and classify k -latin squares of small orders.

Geometrical structure and construction of latin trades

Abstract Latin trades can be interpreted geometrically as oriented surfaces. Those of genus zero are called spherical. The paper contains a construction that shows how all latin trades can be obtained from the spherical ones by a cut-and-paste method that raises the genus

Latin trades in groups defined on planar triangulations

For a finite triangulation of the plane with faces properly coloured white and black, let A be the abelian group constructed by labelling the vertices with commuting indeterminates and adding relations which say that the labels around each white triangle add to the identity. We show that A has free rank exactly two. Let A* be the torsion subgroup of A, and B* the corresponding group for the black triangles. We show that A* and B* have the same order, and conjecture that they are isomorphic. For each spherical latin trade W, we show there is a unique disjoint mate B such that (W,B) is a connected and separated bitrade. The bitrade (W,B) is associated with a two-colourable planar triangulation and we show that W can be embedded in A*, thereby proving a conjecture due to Cavenagh and Drapal. The proof involves constructing a (0,1) presentation matrix whose permanent and determinant agree up to sign. The Smith Normal Form of this matrix determines A*, so there is an efficient algorithm to construct the embedding. Contrasting with the spherical case, for each genus g>0 we construct a latin trade which is not embeddable in any group and another that is embeddable in a cyclic group. We construct a sequence of spherical latin trades which cannot be embedded in any family of abelian groups whose torsion ranks are bounded. Also, we show that any trade that can be embedded in a finitely generated abelian group can be embedded in a finite abelian group. As a corollary, no trade can be embedded in a free abelian group.

Latin bitrades derived from groups

A Latin bitrade is a pair of partial Latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. In [A. Drápal, On geometrical structure and construction of Latin trades, Advances in Geometry (in press)] it is shown that a Latin bitrade may be thought of as three derangements of the same set, whose product is the identity and whose cycles pairwise have at most one point in common. By letting a group act on itself by right translation, we show how some Latin bitrades may be derived directly from groups. Properties of Latin bitrades such as homogeneity, minimality (via thinness) and orthogonality may also be encoded succinctly within the group structure. We apply the construction to some well-known groups, constructing previously unknown Latin bitrades. In particular, we show the existence of minimal, k -homogeneous Latin bitrades for each odd k ≥ 3 . In some cases these are the smallest known such examples.

Latin trades on three or four rows

Latin trades are closely related to the problem of critical sets in Latin squares. We denote the cardinality of the smallest critical set in any Latin square of order n by scs ( n ) . A consideration of Latin trades which consist of just two columns, two rows, or two elements establishes that scs ( n ) ⩾ n - 1 . We conjecture that a consideration of Latin trades on four rows may establish that scs ( n ) ⩾ 2 n - 4 . We look at various attempts to prove a conjecture of Cavenagh about such trades. The conjecture is proven computationally for values of n less than or equal to 9. In particular, we look at Latin squares based on the group table of Z n for small n and trades in three consecutive rows of such Latin squares.

Minimal homogeneous latin trades

A latin trade is a subset of a latin square which may be replaced with a disjoint mate to obtain a new latin square. A d-homogeneous latin trade is one which intersects each row, each column and each entry of the latin square either 0 or d times. In this paper we give a construction for minimal d-homogeneous latin trades of size dm, for every integer d ⩾ 3 , and m ⩾ 1.75 d 2 + 3 . We also improve this bound for small values of d. Our proof relies on the construction of cyclic sequences whose adjacent sums are distinct.

3-Homogeneous latin trades

Let T be a partial latin square and L be a latin square with T ⊆ L . We say that T is a latin trade if there exists a partial latin square T ′ with T ′ ∩ T = ∅ such that ( L ⧹ T ) ∪ T ′ is a latin square. A k-homogeneous latin trade is one which intersects each row, each column and each entry either 0 or k times. In this paper, we construct 3-homogeneous latin trades from hexagonal packings of the plane with circles. We show that 3-homogeneous latin trades of size 3 m exist for each m ⩾ 3 . This paper discusses existence results for latin trades and provides a glueing construction which is subsequently used to construct all latin trades of finite order greater than three.

Constructing and deconstructing latin trades

This paper discusses existence results for latin trades and provides a Glueing Construction which is subsequently used to construct all latin trades of finite order greater than three.

On the possible volumes of μ -way latin trades

A μ-way latin trade of volume s is a set of μ partial latin rectangles (of inconsequential size) containing exactly the same s filled cells, such that if cell (i, j) is filled, it contains a different entry in each of the μ partial latin rectangles, and such that row i in each of the μ partial latin rectangles contains, set-wise, the same symbols and column j, likewise. In this paper we show that all μ-way latin trades with sufficiently large volumes exist, and state some theorems on the non-existence of μ-way latin trades of certain volumes. We also find the set of possible volumes (that is, the volume spectrum) of μ-way latin trades for μ = 4 and 5. (The case μ = 2 was dealt with by Fu, and the case μ = 3 by the present authors.)