Journal articles on the topic 'Frobenius index'

AbstractThe family of finite local Frobenius non-chain rings of length 5 and nilpotency index 4 is determined, as a by-product all finite local Frobenius non-chain rings with p5 elements (p a prime) and nilpotency index 4 are given. And the number and structure of γ-constacyclic codes over those rings, of length relatively prime to the characteristic of the residue field of the ring, are determined.

Some of the basic facts in the Ocneanu's approach to Jones' index theory are proved. Among other things the biunitarity of connections in paragroups is proved generally by making use of Frobenius reciprocity.

The Frobenius normal form of a matrix is an important tool in analyzing its properties. When a matrix is powered up, the Frobenius normal form of the original matrix and that of its powers need not be the same. In this article, conditions on a matrix $A$ and the power $q$ are provided so that for any invertible matrix $S$, if $S^{-1}A^qS$ is block upper triangular, then so is $S^{-1}AS$ when partitioned conformably. The result is established for general matrices over any field. It is also observed that the contributions of the index of cyclicity to the spectral properties of a matrix hold over any field. The article concludes by applying the block upper triangular powers result to the cone Frobenius normal form of powers of a eventually cone nonnegative matrix.

The Perron-Frobenius root is used to measure the level of interaction in MIMO quantitative feedback design (QFT). A design approach to reduce interaction in uncertain plants via a precompensator is presented. A decentralized stability result applies if the interaction index can be made less than unity by design.

Abstract A one to one correspondence between the elements of a finite local Frobenius non-chain ring of length 5 and nilpotency index 4, and k-tuples of DNA codewords is established. Using this map the structure of DNA codes over these rings is determined, the length of the code is relatively prime to the characteristic of the residue field of the ring.

Using theoretical results about the homogeneous weights for Frobenius rings, we describe the homogeneous weight for the ring family Rk, a recently introduced family of Frobenius rings which have been used extensively in coding theory. We find an associated Gray map for the homogeneous weight using first order Reed-Muller codes and we describe some of the general properties of the images of codes over Rk under this Gray map. We then discuss quasi-twisted codes over Rk and their binary images under the homogeneous Gray map. In this way, we find many optimal binary codes which are self-orthogonal and quasi-cyclic. In particular, we find a substantial number of optimal binary codes that are quasi-cyclic of index 8, 16 and 24, nearly all of which are new additions to the database of quasi-cyclic codes kept by Chen.

AbstractWe prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron–Frobenius theory.We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions use the notion of growth tightness and the work of Dahmani, Guirardel and Osin on rotating families.

Abstract Over finite local Frobenius non-chain rings of length 5 and nilpotency index 4 and when the length of the code is relatively prime to the characteristic of the residue field of the ring, the structure of the dual of γ-constacyclic codes is established and the algebraic characterization of self-dual, reversible γ-constacyclic codes and γ-constacyclic codes with complementary dual are given.

We consider two-to-one interval maps which are expanding except at an indifferent fixed point. Our purpose is to show that the iteration of the Perron–Frobenius operator displays interesting asymptotic behaviour if we stretch the underlying random variables in a suitable way. As an immediate probabilistic consequence we get a distributional limit theorem, closely related to an arc-sine law, or a large deviation type result, depending on whether the return index is less than or equal to 1.

Much information about a finite group is encoded in its character table. Indeed even a small portion of the character table may reveal significant information about the group. By a famous theorem of Jordan, knowing the degree of one faithful irreducible character of a finite group gives an upper bound for the index of its largest normal abelian subgroup.Here we consider b(G), the largest irreducible character degree of the group G. A simple application of Frobenius reciprocity shows that b(G) ≧ |G:A| for any abelian subgroup A of G. In light of this fact and Jordan's theorem, one might seek to bound the index of the largest abelian subgroup of G from above by a function of b(G). If is G is nilpotent, a result of Isaacs and Passman (see [7, Theorem 12.26]) shows that G has an abelian subgroup of index at most b(G)4.

AbstractA ring R is said to be generated by faithful right cyclics (right finitely pseudo-Frobenius), denoted by GFC (FPF), if every faithful cyclic (finitely generated) right R-module generates the category of right R-modules. The class of right GFC rings includes right FPF rings, commutative rings (thus every ring has a GFC subring - its center), strongly regular rings, and continuous regular rings of bounded index. Our main results are: (1) a decomposition of a semi-prime quasi-Baer right GFC ring (e.g., a semiprime right FPF ring) is achieved by considering the set of nilpotent elements and the centrality of idempotnents; (2) a generalization of S. Page's decomposition theorem for a right FPF ring.

A full proof of Ocneanu’s theorem is given that one can produce a rational unitary polyhedral 3-dimensional topological quantum field theory of Turaev-Viro type from a subfactor with finite index and finite depth, and vice versa. The key argument is an equivalence between flatness of a connection in paragroup theory and invariance of a state sum under one of the three local moves of tetrahedra. This was announced by A. Ocneanu and he gave a proof of Frobenius reciprocity and the pentagon relation, which produces a 3-dimensional TQFT via the Turaev-Viro machinery, but he has not published a proof of the converse direction of the equivalence. Details are given here along the lines suggested by him.

Abstract Asymptotic properties of the sequences (a) { P j } j = 1 ∞ $\{P^{j}\}_{j=1}^{\infty}$ and (b) { j − 1 ∑ i = 0 j − 1 P i } j = 1 ∞ $\{ j^{-1} \sum _{i=0}^{j-1} P^{i}\}_{j=1}^{\infty}$ are studied for g ∈ G = {f ∈ L 1(I) : f ≥ 0 and ‖f ‖ = 1}, where P : L 1(I) → L 1(I) is a Markov operator defined by P f := ∫ P y f d p ( y ) $Pf:= \int P_{y}f\, dp(y) $ for f ∈ L 1; {Py } y∈Y is the family of the Frobenius-Perron operators associated with a family {φy } y∈Y of nonsingular Markov maps defined on a subset I ⊆ ℝ d ; and the index y runs over a probability space (Y, Σ(Y), p). Asymptotic properties of the sequences (a) and (b), of the Markov operator P, are closely connected with the asymptotic properties of the sequence of random vectors x j = φ ξ j ( x j − 1 ) $x_{j}=\varphi_{\xi_{j}}(x_{j-1})$ for j = 1,2, . . .,where { ξ j } j = 1 ∞ $\{\xi_{j}\}_{j=1}^{\infty}$ is a sequence of Y-valued independent random elements with common probability distribution p. An operator-theoretic analogue of Rényi’s Condition is introduced for the family {Py } y∈Y of the Frobenius-Perron operators. It is proved that under some additional assumptions this condition implies the L 1- convergence of the sequences (a) and (b) to a unique g 0 ∈ G. The general result is applied to some families {φy } y∈Y of smooth Markov maps in ℝ d .

AbstractThis study has investigated the relationship in the connectivity of motor units in surface electromyogram (sEMG) of biceps brachii muscle. It is hypothesized that with ageing, there is reduction/loss in number of motor units, leading to reduction in the independence between the channels of the recorded muscle activity. Two channels of sEMG were recorded during three levels of isometric muscle contraction: 50 %, 75 % and 100 % maximal voluntary contraction (MVC). 73 subjects (age range 20-70) participated in the experiments. The independence in channel index (ICI) between the two sEMG recording locations was computed using the independent components and Frobenius norm. ANOVA Statistical analysis was performed to test the effect of age (loss of motor units) and level of contraction on ICI. The results show that the ICI among the older cohort was significantly lower compared with the younger adults. This research study has shown that the reduction in number of motor units is reflected by the reduction in the ICI of the sEMG signal.

The presence of fake pictures affects the reliability of visible face images under specific circumstances. This paper presents a novel adversarial neural network designed named as the FTSGAN for infrared and visible image fusion and we utilize FTSGAN model to fuse the face image features of infrared and visible image to improve the effect of face recognition. In FTSGAN model design, the Frobenius norm (F), total variation norm (TV), and structural similarity index measure (SSIM) are employed. The F and TV are used to limit the gray level and the gradient of the image, while the SSIM is used to limit the image structure. The FTSGAN fuses infrared and visible face images that contains bio-information for heterogeneous face recognition tasks. Experiments based on the FTSGAN using hundreds of face images demonstrate its excellent performance. The principal component analysis (PCA) and linear discrimination analysis (LDA) are involved in face recognition. The face recognition performance after fusion improved by 1.9% compared to that before fusion, and the final face recognition rate was 94.4%. This proposed method has better quality, faster rate, and is more robust than the methods that only use visible images for face recognition.

In this paper, a new method of measurement matrix optimization for compressed sensing based on alternating minimization is introduced. The optimal measurement matrix is formulated in terms of minimizing the Frobenius norm of the difference between the Gram matrix of sensing matrix and the target one. The method considers the simultaneous minimization of the mutual coherence indexes including maximum mutual coherence μmax, t-averaged mutual coherence μave and global mutual coherence μall, and solves the problem that minimizing a single index usually results in the deterioration of the others. Firstly, the threshold of the shrinkage function is raised to be higher than the Welch bound and the relaxed Equiangular Tight Frame obtained by applying the new function to the Gram matrix is taken as the initial target Gram matrix, which reduces μave and solves the problem that μmax would be larger caused by the lower threshold in the known shrinkage function. Then a new target Gram matrix is obtained by sequentially applying rank reduction and eigenvalue averaging to the initial one, leading to lower. The analytical solutions of measurement matrix are derived by SVD and an alternating scheme is adopted in the method. Simulation results show that the proposed method simultaneously reduces the above three indexes and outperforms the known algorithms in terms of reconstruction performance.

We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ of $\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})$, where $\unicode[STIX]{x1D6E5}$ is a finite-index subgroup of $\text{SL}(n+1,\mathbb{Z})$. These subsets can be obtained by projecting to the hyperplane $\{(x_{1},\ldots ,x_{n+1})\in \mathbb{R}^{n+1}:x_{n+1}=1\}$ sets of the form $\mathbf{A}=\bigcup _{j=1}^{J}\mathbf{a}_{j}\unicode[STIX]{x1D6E5}$, where for all $j$, $\mathbf{a}_{j}$ is a primitive lattice point in $\mathbb{Z}^{n+1}$. Our method involves applying the equidistribution of expanding horospheres in quotients of $\text{SL}(n+1,\mathbb{R})$ developed by Marklof and Strömbergsson, and more precisely understanding how the full Farey sequence distributes in $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ when embedded on expanding horospheres as done in previous work by Marklof. For each of the Farey sequence subsets, we extend the statistical results by Marklof regarding the full multidimensional Farey sequences, and solutions by Athreya and Ghosh to Diophantine approximation problems of Erdős–Szüsz–Turán and Kesten. We also prove that Marklof’s result on the asymptotic distribution of Frobenius numbers holds for sets of primitive lattice points of the form $\mathbf{A}$.

Background: Gene expression profiling and transcriptomics provide valuable information about the role of genes that are differentially expressed between two or more samples. It is always important and challenging to analyse High-throughput DNA microarray data with a number of missing values under various experimental conditions. </P><P> Objectives: Graphical data visualizations of the expression of all genes in a particular cell provide holistic views of gene expression patterns, which improve our understanding of cellular systems under normal and pathological conditions. However, current visualization methods are sensitive to missing values, which are frequently observed in microarray-based gene expression profiling, potentially affecting the subsequent statistical analyses. Methods: We addressed in this study the problem of missing values with respect to different imputation methods using gene expression biplot (GE biplot), one of the most popular gene visualization techniques. The effects of missing values for mining differentially expressed genes in gene expression data were evaluated using four well-known imputation methods: Robust Singular Value Decomposition (Robust SVD), Column Average (CA), Column Median (CM), and K-nearest Neighbors (KNN). Frobenius norm and absolute distances were used to measure the accuracy of the methods. Results: Three numerical experiments were performed using simulated data (i) and publicly available colon cancer (ii) and leukemia data (iii) to analyze the performance of each method. The results showed that CM and KNN performed better than Robust SVD and CA for identifying the index gene profile in the biplot visualization in both the simulation study and the colon cancer and leukemia microarray datasets. Conclusion: The impact of missing values on the GE biplot was smaller when the data matrix was imputed by KNN than by CM. This study concluded that KNN performed satisfactorily in generating a GE biplot in the presence of missing values in microarray data.

- Victoria J. Baker, Anthony Jackson, Anthropology at home, ASA monographs 25, London: Tavistock Publications, 1987, 221 pages. - Thomas Bargatzky, Martin A. van Bakel, Private politics; A multi-disciplinary approach to ‘Big-Man’ systems, Studies in Human Society, Vol. I, Leiden: E.J. Brill, 1986. x, 220 pp., illustrations, maps, index., Renée R. Hagesteijn, Pieter van de Velde (eds.) - W.E.A. van Beek, Victor W. Turner, The anthropology of experience, (with an epilogue by Clifford Geertz). Urbana and Chicago: University of Illinois Press, 1986., Edward M. Bruner (eds.) - W. van den Broeke, H. Meyer, De Deli Spoorweg Maatschappij; Driekwart eeuw koloniaal spoor, Zutphen: Walburg Pers, met medewerking van F.A.J. Heckler. 1987; 152 blz. - R. Buijtenhuijs, S. Bernus et al., Le fils et le neveu: Jeux et enjeux de la parenté tourarègue, Cambridge University Press, Cambridge/Editions de la Maison des Sciences de l’Homme, Paris, 1986, XI, 343 pp. - R. Buijtenhuijs, Dominique Casajus, La tente dans la solitude: La société et les morts chez les Touaregs Kel Ferwan, Cambridge University Press, Cambridge/Editions de la Maison des Sciences de l’Homme, Paris, 1987, 390 pp. - H.J.M. Claessen, Christine Ward Gailey, Kinship to kingship; Gender hierarchy and state formation in the Tongan Islands. Austin: University of Texas Press (Texas Press Sourcebooks in Anthropology, No. 14.), 1987. 326 pp., figs., index, bibl. - Alfred E. Daniëls, Richard B. Davis, Muang metaphysics, Bangkok: Pandora Press,1984. - Alfred E. Daniëls, Gehan Wijeyewardene, Place and emotion in northern Thai ritual behaviour, Bangkok: Pandora Press, 1986. - P.M.H. Groen, Jacques van Doorn, The process of decolonization 1945-1975; The military experience in comparitive perspective, CASP publications no. 17, Erasmus Universiteit Rotterdam, 1987, 46 pp., Willem J. Hendrix (eds.) - Rosemarijn Hoefte, Luis H. Daal, Antilliaans verhaal. Zutphen: De Walburg Pers, 1988, Ted Schouten (eds.) - W.L. Idema, Claudine Salmon, Literary migrations; Traditional Chinese fiction in Asia (17th-20th centuries), Beijing: International culture publishing corporation, 1987, 11 + vi + 661 pp. - P.E. de Josselin de Jong, Sharon A. Carstens, Cultural identity in Northern Peninsular Malaysia, Athens, Ohio: Ohio University Monographs in international studies, Southeast Asia series no. 63, 1986. 91 pp. - P.E. de Josselin de Jong, Robert Wessing, The soul of ambiguity: The tiger in Southeast Asia. Northern Illinois University, Center for Southeast Asian studies, Special report no. 24, 1986. 148 pp., glossary, bibliography. - G.W. Locher, Martine Segalen, Historical anthropology of the family, Cambridge University Press, 1986, 328 pp. - Bernd Nothofer, Hans Kähler, Enggano-Deutsches Wörterbuch. Aus dem Nachlass herausgegeben und mit einem Deutsch-Enggano-Wörterverzeichnis versehen von Hans Schmidt, Veröffentlichungen des Seminars für Indonesische und Südseesprachen der Universität Hamburg, Band 14, Berlin: Dietrich Reimer Verlag, 1987. XIII + 404 pp. - J.D.M. Platenkamp, Brigitte Renard-Clamagirand, Marobo; Une société ema de Timor. Langues et civillisations de l’Asie du sud-est et du monde insulindien no. 12, Paris: Selaf, 1983, 490 pp. - H.C.G. Schoenaker, Leo Frobenius, Ethnographische Notizen aus den Jahren 1905 und 1906; II: Kuba, Leele, Nord-Kete; III: Luluwa, Süd-Kete, Bena Mai, Pende, Cokwe. Bearb.u.hrsg. von Hildegard Klein. Wiesbaden: Franz Steiner Verlag, 1987; 1988. 223 S., 437 Zeichnungen, 11 fotos, 5 karten; 268 S., 500 Zeichnungen, 15 fotos, 12 karten. - M. Schoffeleers, I.M. Lewis, Religion in context: Cults and charisma, Cambridge: Cambridge University Press, 1986, X + 139 pp. - B. Schuch, Ingrid Liebig-Hundius, Thailands Lehrer zwischen ‘Tradition’ und `Fortschritt’; Eine empirische Untersuchung politisch-sozialer und pädagogischer Einstellungen thailändischer Lehrerstudenten des Jahres 1974. Beiträge zur Südasienforschung, Band 85, Weisbaden: Steiner Verlag, 1984, 342 pp. - Henke Schulte Nordholt, S.J. Tambiah, Thought and social action; An anthropological perspective, Cambridge, Massachusetts: Harvard Univ. Press, 1985, 411 pp. - Nico G. Schulte Nordholt, Shamsul Amri Baharuddin, From British to Bumiputra rule: Local politics and rural development in Peninsular Malaysia, Singapore: Institute of Southeast Asian studies, 1986. 282 pp. - A. Teeuw, I. Syukri, History of the Malay kingdom of Patani - Sejarah Kerajaan Melayu Patani, by Ibrahim Syukri (pseudonym), translated by Conner Bailey and John N. Miksic. Athens, Ohio: Ohio University Center for International studies, monographs in international studies Southeast Asia series number 68, 1985. xx + 90 pp. - Truong Quang, Andrew Vickerman, The fate of the peasantry: Premature `transition to socialism’ in the democratic republic of Vietnam, Monograph No. 28, Yale University, Southeast Asia studies, 1986. 373 pp., incl. bibliography. - Adrian Vickers, H.I.R. Hinzler, Catalogue of Balinese manuscripts in the library of the University of Leiden and other collections in the Netherlands, vol. I: Reproductions of the drawings from the Van der Tuuk collection; vol. II: Descriptions of the Balinese drawings form the Van der Tuuk collection. Leiden: E.J. Brill/Leiden University Press, 1987.

Garsia and Remmel (JCT. A 41 (1986), 246-275) used rook configurations to give a combinatorial interpretation to the $q$-analogue of a formula of Frobenius relating the Stirling numbers of the second kind to the Eulerian polynomials. Later, Remmel and Wachs defined generalized $p,q$-Stirling numbers of the first and second kind in terms of rook placements. Additionally, they extended their definition to give a $p,q$-analogue of rook numbers for arbitrary Ferrers boards. In this paper, we use Remmel and Wach's definition and an extension of Garsia and Remmel's proof to give a combinatorial interpretation to a $p,q$-analogue of a formula of Frobenius relating the $p,q$-Stirling numbers of the second kind to the trivariate distribution of the descent number, major index, and comajor index over $S_n$. We further define a $p,q$-analogue of the hit numbers, and show analytically that for Ferrers boards, the $p,q$-hit numbers are polynomials in $(p,q)$ with nonnegative coefficients.

We define posets of types B, C, and D. These posets encode the matrix forms of certain Lie algebras which lie between the algebras of upper-triangular and diagonal matrices. Our primary concern is the index and spectral theories of such type-B, C, and D Lie poset algebras. For an important restricted class, we develop combinatorial index formulas and, in particular, characterize posets corresponding to Frobenius Lie algebras. In this latter case we show that the spectrum is binary; that is, consists of an equal number of 0's and 1's. Interestingly, type-B, C, and D Lie poset algebras can be related to Reiner's notion of a parset.

A square complex matrix A is eventually nonnegative if there exists a positive integer k_0 such that for all k ≥ k_0, A^k ≥ 0; A is strongly eventually nonnegative if it is eventually nonnegative and has an irreducible nonnegative power. It is proved that a collection of elementary Jordan blocks is a Frobenius Jordan multiset with cyclic index r if and only if it is the multiset of elementary Jordan blocks of a strongly eventually nonnegative matrix with cyclic index r. A positive answer to an open question and a counterexample to a conjecture raised by Zaslavsky and Tam are given. It is also shown that for a square complex matrix A with index at most one, A is irreducible and eventually nonnegative if and only if A is strongly eventually nonnegative.

In this work, we consider the finite ring F_2 +uF_2 +vF_2, u^2 = 1;v^2 = 0, u.v = v.u = 0 which is not Frobenius and chain ring. We studied constacyclic and negacyclic codes over F_2+uF_2+vF_2 of odd length. These codes are compared with codes that had priorly been obtained on the finite field F_2. Moreover, we indicate that the Gray image of a constacyclic and negacyclic code over F_2+uF_2+vF_2 of odd length n is a quasicyclic code of index 4 with length 4n over F_2. In particular, the Gray images are applied to two different rings S_1 = F_2+vF_2, v^2 = 0 and S_2 = F_2+uF_2, u^2 = 1 and negacyclic and constacyclic images of these rings are also discussed.

Abstract In this paper, an exact analytical solution for forced convective heat transfer of nonlinear viscoelastic fluid in isothermal circular micro-channel is presented. The nonlinear Giesekus constitutive equation is used to model the Giesekus fluid heat transfer in micro-channel with constant wall temperature, which is the main innovative aspect of the current study. This constitutive equation is a powerful tool and able to model the fractional viscometric functions, extensional viscosity, and elastic property. The solution of temperature profile and Nusselt number is obtained based on the Frobenius method. The effects of Weissenberg number, mobility factor, slip coefficient, and Navier index on temperature distribution, velocity profile, and Nusselt number are investigated in detail. The results show that the increases in both slip coefficient and Navier index cause the increases in slip velocity and maximum dimensionless temperature at the wall and the micro-channel center, respectively. Moreover, the Nusselt number has an upward trend with increases in slip coefficient and Navier index parameters. The results are indicated that the flow and temperature fields have a complex relation with mobility factor which controls the level of the nonlinearity of the Giesekus model. Additionally, three correlations for Nusselt number of Giesekus flow in micro-channel are presented.

NB: Artiklen er på dansk, kun resuméet er på engelsk.MSS GKS 95 2°, GKS 96 2° and Thott 732° in the RL are among the most remarkable sources in the library’s collections. The nearly 900 folia contained in the volumes represent the literary Nachlass of Erasmus of Rotterdam as it was left by him at his death close to midnight between the 11th and 12th of June 1536. No clue to how this material has come to Denmark has yet come to light. In the present contribution the author gives a brief survey of what is known about the history of the volumes. The aim is to establish a basis for new research into the later history of the manuscripts by looking into the dispositions that were made concerning the papers while still in Basel, and to discuss how the papers were kept by Erasmus himself and managed by his heir, friends and assistants after his death. GKS 95 2° was first recorded in a RL catalogue signed and dated by the librarian, Willum Worm, on the 4th of January 1671. The terminus post quem of accessions recorded in this source which has been quoted as Catalogus Schumacheri cannot, as has previously been assumed, be fixed to the year 1663. GKS 96 2° is first recorded in a RL catalogue in 1729. Thott 732° was included in the RL after the death of count Otto Thott in 1784, and first recorded in the printed Index codicum manuscriptorum, in volume 7 of the Bibliotheca Thottiana from 1795. The presence of this rich autograph material in the RL remained unnoticed by scholars until P.S. Allen published a survey of GKS 95 2°, The Copenhagen Manuscript, as Appendix XIII of Opus Epistolarum in 1913. And it was only when C. Reedijk had published his study Three Erasmus Autographs in the Royal Library at Copenhagen in 1966 (in Festschrift Herman de la Fontaine Verwey) that it became known to Erasmus-scholars that the material in Copenhagen included two more volumes, an addition of more than 1200 manuscript pages in the hand of Erasmus. Erasmus did not give any instructions concerning his literary Nachlass in his last will of 12 February 1536. Nor did he bequeath it to someone. He had sold his library to Jan Laski already in 1525, on the conditions that it remained at his disposal until his death. An analysis of the extant list of books in the library shows that it included manuscripts, and Laski’s correspondence with Bonifacius Amerbach after Erasmus’ death discloses that Laski also acquired manuscripts, not in the list, which he had noticed while staying with Erasmus in Basel, amongst them the famous codex Arcerianus now in Wolfenbüttel. According to the list, the library bought by Laski included one manuscript containing a text of Erasmus, but he did not acquire the Nachlass contained in the volumes in Copenhagen. Laski’s errant life did, in fact, bring him to Denmark in 1553. He and his reformed community had been forced to leave London, and he asked the Danish king for asylum. Most likely he brought his library along, and Erasmus’ books thus may have been in Denmark for a few weeks. Nothing was left there, neither manuscripts nor printed books. The RL has only a single book that may have belonged to Erasmus, but this volume was not included in the list of items acquired by Laski from Erasmus. Erasmus’ library was sent to Laski in Krakow soon after the death of Erasmus. His literary Nachlass remained in Basel. Although it is not stated explicitly in the will, it must have become the property of Bonifacius Amerbach, the main heir, to whom Erasmus bequeathed whatever in his possession had not been given to others. Since the Nachlass is not bequeathed to anybody in the will, it must have been included in the gifts to Amerbach. Its presence in Basel is also attested by two copies of Erasmus’ will, one added to the papers in Thott 732°, another added to the papers in GKS 95 2°. Both were copied from the original will in Erasmus’ own hand which remained with Amerbach. These copies offer an important clue to what happened to the Nachlass after Erasmus’ death. The copy of the will in Thott 732° was made by Nicolaus Episcopius, as is evident both from the handwriting and his signature. The copy in GKS 95 2° is not signed. It was not copied by Gilbertus Cognatus, as maintained by Reedijk. Palaeographical and chronological details do not favour such an identification, and in addition the copy contains faults more grave than one would expect from an experienced copyist and able latinist such as Cognatus. In his Appendix XIII P.S. Allen described GKS 95 2° as “a congeries brought together almost haphazard by some one desirous of gathering and yet hardly heeding what he gathered”, and proposed a theory about how the paper contained in the volume was brought together: “This orderly and yet disordered arrangement is just what might be expected in a collection hurriedly gathered during the necessary dispositions which follow death.” C. Reedijk supported Allen’s theory and found it valid also in relation to MSS GKS 96 2° and Thott 732°. In my opinion both of them stressed the disorder of the papers too much and misinterpreted their context before and after Erasmus’ death. In fact, Erasmus had had the papers in his hands less than half a year before he passed away. As he writes in his letter ‘to the friendly readers’ which he dated on 20 February 1536 (ep. 3100) it was his plan to go through what he described as schedarum mearum confusissimos acervos. His intention was to remove and destroy manuscripts in order to prevent others from publishing them after his death, or even while he was still alive, as had happened more than once before. He also looked for letters that might be suitable for publication. This took place only 8days after he had written his last will and was clearly part of his preparations for death. The rich contents of the manuscripts in Copenhagen seem to indicate that he destroyed little, if anything. The process of going through the papers has given him an opportunity to put them in a slightly better order than previously, and he seems to have used it. The letter to the friendly readers itself was placed on top of a pile with drafts of letters that is still kept together in GKS 95 2°. After Erasmus’ death Amerbach meticulously fulfilled Erasmus’ instructions as formulated in his will. Since there were no instructions concerning the literary Nachlass, he made his own decision and gave the papers now included in Thott 732° to Nicolaus Episcopius who added his own copy of the will – and a note of his ownership which is now heavily damaged and only readable in part. Reedijk interpreted the poor remnants thus: Nicolai Episcopii, succesoris <Io. Frob.> et Justinae <filiae> D <Erasmi mo>numentum Rot<erodami> <………>. Less daring I have been able to decipher only the following with certainty: Nicolai Episcopii, succesor… / numentum hoc. That leaves us with Episcopius’ name in the genitive and monumentum hoc, just enough to assure that it did in fact belong to him and that he valued it highly. GKS 95 2° does not contain any remains of a similar note. As stated above, the copy of the will in this volume is not in the hand of Cognatus. Nor has it been copied by Hieronymus Frobenius, but I believe I have found one small element that might link it to his family or a person in his workshop. In the name Desiderius Erasmus Roterodamus at the very beginning of the will Erasmus has been written with an added i: Erasmius. Really an odd fault at a very visible location, perhaps a kind of Freudian slip committed by someone who knew a person of that name – or indeed was named Erasmius himself, such as Hieronymus’ younger brother, Erasmius Frobenius, Erasmus’ beloved godson. This is not much of a lead, but perhaps sufficient to suggest that the volume belonged to Hieronymus Frobenius, given to him as a gift in memory of Erasmus by Amerbach. The third volume, GKS 96 2°, contains no copy of the will, no notice of an owner, and no traces that may point to one. But perhaps exactly the absence of such traces points to Amerbach himself. If nature had entrusted him with sensibility enough to know that a volume of Erasmus’ autographa would exhilarate his friends and companions as executores testamenti, it is easy to understand why he decided to keep one for himself. And as to the will, Amerbach needed no copy. He possessed the original document, the ultima voluntas as recorded on the 12th of February 1536 by Erasmus himself, propria manu.