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Erratum to the paper [Konovalov, Alexander; Smoktunowicz, Agata; Vendramin, Leandro. On skew braces and their ideals. Exp. Math. 30 (2021), no. 1, 95–104. DOI: 10.1080/10586458.2018.1492476.].

In 1903, Epstein published his proof of meromorphic continuation and a functional equation for Dirichlet series associated with quadratic forms, now called Epstein zeta-functions. However, already in 1889 (or even earlier) Hurwitz was aware of these results as his mathematical diaries and some unpublished notes (in an almost final form) found in his estate at the ETH Zurich show. In this article we present and analyze Hurwitz's notes and compare his reasoning with Epstein's paper in detail.

Origami is the ancient Japanese art of paper folding. It is possible to fold many intriguing geometrical shapes with paper [M]. In this article, the question we will answer is which shapes are possible to construct and which shapes are impossible to construct using origami. One of the most interesting things we discovered is that it is impossible to construct a cube with twice the volume of a given cube using origami, just as it is impossible to do using a compass and straight edge. As an unexpected surprise, our algebraic characterization of origami is related to David Hilbert's 17th problem. Hilbert's problem is to show that any rational function which is always non-negative is a sum of squares of rational functions [B]. This problem was solved by Artin in 1926 [Ar]. We would like to thank John Tate for noticing the relationship between our present work and Hilbert's 17th problem. This research is the result of a project in the Junior Fellows Program at The University of Texas. The Junior Fellows Program is a program in which a junior undergraduate strives to do original research under the guidance of a faculty mentor. The referee mentioned two references which the reader may find interesting. "Geometric Exercises in Paper Folding" addresses practical problems of paper folding [R]. Arnong many other things, Sundara Row gives constructions for the 5-gon, the 17-gon, and duplicating a cube. His constructions, however, use more general folding techniques than the ones we consider here. Felix Klein cites Row's work in his lectures on selected questions in elementary geometry [K]. In order to understand the rules of origami construction, we will first consider a sheet of everyday notebook paper. Our work with notebook paper will serve as an intuitive model for our definition of origami constructions in the Euclidean plane. There are four natural methods of folding a piece of paper. The methods will serve as the basis of the definition of an origami pair. We construct the line L1, by folding a crease between two different corners of the paper. Another line may be constructed by matching two corners. For example, if corners or and Sy are matched, the crease formed, L2, will be the perpendicular bisector of the segment orSy. Another natural construction is matching one line to another line. For instance, ,8^y, the paper's edge, and L2 are lines. If we lay ,8^y upon L2 and form the crease, then we obtain L3 which is the angle bisector of the two lines. If we start with two parallel lines in this third construction, then we will just get a parallel line half way in between. The fourth and final construction which seems natural is consecutive folding. This is similar to rolling up the sheet of paper only one does not roll it up, he folds it up. More explicitly, start with a piece of paper with two creases on it as in Figure 2. Fold along line Ll and do not unfold the piece of paper. Notice that line L2 lies

We show that the extension of Rock-Paper-Scissors to include Lizard and Spock is the unique such five move fair game up to isomorphism. We also categorize all analogous six move fair games.

Motivated by a paper of Fang (2009), we study the Samuel multiplicity and the structure of essentially semi-regular operators on an infinite-dimensional complex Banach space. First, we generalize Fang’s results concerning Samuel multiplicity from semi-Fredholm operators to essentially semi-regular operators by elementary methods in operator theory. Second, we study the structure of essentially semi-regular operators. More precisely, we present a revised version of Fang’s 4 × 4 upper triangular model with a little modification, and prove it in detail after providing numerous preliminary results, some of which are inspired by Fang’s paper. At last, as some applications, we get the structure of semi-Fredholm operators which revised Fang’s 4 × 4 upper triangular model, from a different viewpoint, and characterize a semi-regular point λ ∈ ℂ in an essentially semi-regular domain.

We show abstract versions for Banach couples of several limiting compact interpolation theorems established by Edmunds and Opic for couples of Lp spaces.

The issue of Bernstein polynomials is enough far from the basic range of Kantorovich’s interests. He devoted only two papers to them in 1930 and 1931 and made a report at the First All-Union Mathematical Congress in Khar’kov in 1930.

What are modules? In Eilenberg [3], they are basically partial deterministic automata without initial or final states. In the paper “Modules” by Bret Tilson, this is taken one level up. Bret, in his seminal work [12], showed that one needs categories to understand wreath products of monoids. Obviously one should then need transformation categories to understand wreath products of transformation monoids. So why modules? Because there are results, like the Presentation Lemma [1, 2, 7, 10, 11, 13], where one needs to look at “quotients” of transformation monoids that lose their algebraic structure. This leads one to modules and wreath products of modules. So, for Bret, a module is like an automaton where the input alphabet is a graph. Such a gadget has a transition category and the derived transformation semigroup from [3] now gets its foundations fixed by becoming a module over a graph. The algebraic viewpoint to all of this is best obtained by reading Bret’s paper. I want to give a geometric overview of what Bret has achieved so that a wider audience can appreciate what he has done. In particular, I believe this gives the best justification for Bret’s change in the definition of division of transformation semigroup from that given in Eilenberg [3] to a tighter notion. For the past decade, or more, a number of people have been looking at automata from a geometric viewpoint. Perhaps Margolis and Meakin, in the case of inverse automata [5], were the first to emphasize this approach by relating inverse automata to Stallings’ notion of graph immersions [8]. Stallings folding, which was independently discovered by Bret [13], plays a key role in the “Modules” paper. I, and other

A left canonical factorization theorem for rational matrix functions relative to the unit circle is presented. The result is a time invariant version of a recent strict LU factorization theorem for certain semi-separable operators, due to Dewilde (2012) [6]. Explicit formulas for the factors are also given. The theorem is proved, first by using the state space method of Dewilde (2012) [6], and next by using an operator theory approach. In both cases the main part of the proof concerns rational matrix functions that are unitary on the unit circle. © 2012 Royal Dutch Mathematical Society (KWG).

The abstract results and applications presented in “Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959, are not correct. Moreover, the class of differential control problems studied in [1] is not H-controllable.

A record mentioned in the paper by Moree (Abh Math Sem Univ Hamburg 77:109–127, 2007) was recently improved on by Akbary and Scholten. However, the record mentioned was not the then record. The then record, due to Gallot (2004), actually slightly improves on that obtained recently by Akbary and Scholten.

Complementing the work of T.-S. Liu and A.C.M. van Rooij we show that the existence of non-zero translation invariant operators between certain function spaces on a locally compact group implies its amenability.

It is proved that the set of those natural numbers which cannot be written as n-Ω(n) is of positive lower density. Here Ω(n) is the number of the prime power divisors of n. This is a refinement of a theorem of F. Luca.

Sandor Szab6 (sszabo7@hotmail.com) was born and grew up in Hungary and graduated from E6tv6s University in Budapest. He has taught at the Technical University in Budapest, has held a variety of visiting positions, and currently is on the faculty of the University of Pecs in Hungary. His main mathematical interests are in algebra and its applications. He has written two monographs, Algebra and Tilingq with S. K. Stein, and Topics in Factorization of Abelian Groups.

We study spectral multipliers for a distinguished Laplacian on certain groups of exponential growth. We obtain a stronger optimal version of the results proved in (CGHM) and (A).

It was proved by Ibukiyama and Skoruppa that the spaces J1,m(l) of Jacobi forms of weight 1, index m and level l vanish if gcd (l,m)=1. We will show that the result holds without the gcd hypothesis.

International audience; In a previous paper, we have described the construction of an automaton from a rational expression which has the property that the automaton built from an expression which is itself computed from a co-deterministic automaton by the state elimination method is co-deterministic. It turned out that the definition on which the construction is based was inappropriate, and thus the proof of the property was flawed. We give here the correct definition of the broken derived terms of an expression which allow to define the automaton and the detailed full proof of the property.

This complement includes stronger versions of the theorems of the paper “A positive answer to the basis problem”, practically with the same proofs.

SynopsisSome disparate ideas in the literature are drawn together. The work of P. J. Olver and his associates on Lagrangians which vanish for arbitrary variations, the so-called null Lagrangians, is viewed as a parallel development to Witten's study of topological field theories. A theorem of Olver, that all hyperjacobians are expressible as divergences, and are thus candidates for the construction of null Lagrangians, is shown to follow directly from the observation that such entities appear in a power series development of the general associative product, and this technique facilitates the construction of multi-dimensional examples.

In section 8.3 of our paper "Duality and Flat Base Change on Formal Schemes" (http://arXiv.org/abs/alg-geom/9708006) some important results concerning localization of, and preservation of coherence by, basic duality functors, were based on the false statement that any closed formal subscheme of an open subscheme of the completion P of a relative projective space is an open subscheme of a closed formal subscheme of P. In this note, the said results are provided with solid foundations.

This note gives the correct form of a result (Theorem 2, p. 478) published recently in [Math. Slovaca 60 (2010), 471–484].

As early as the 16th century, Simon Jacob, a German reckoning master, noticed that the worst case in computing the greatest common divisor of two numbers by the Euclidean algorithm occurs if these numbers are equimultiples of two consecutive members of the Fibonacci sequence.

Around 1900 some noted mathematicians published works developing geometry from its very beginning. They wanted to supplant approaches, based on Euclid's, which handled some basic concepts awkwardly...

We note that Theorem 5.4 of the paper ``Characteristic cycles of perverse sheaves and Milnor fibers" (by P. Nang and K. Takeuchi, published online in Math. Zeitschrift, October 15, 2004) holds for "any" non-zero complex number $a$., Comment: 1page