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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 However, already in 1889 (or even earlier) Hurwitz was aware of these results as his mathematical diaries and some unpublished notes (in an almost final form) found in his estate at the ETH Zurich show.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

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).

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.

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).

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

We observe that a strictly singular operator is not necessarily condensing, so that the invariant subspace problem for strictly singular operators remains open.

That Euler was quite aware of the subtleties of assigning a sum to a divergent series is amply demonstrated in his paper De seriebus divergentibus which appeared in Novi commentarii academiae scientiarum Petropolitanae 5 (1754/55), 205–237 (= Opera Omnia (1) 14, 585–617) in the year 1760. The first half of this paper contains a detailed exposition of Euler's views which should be more readily accessible to the mathematical community.The authors present here a translation from Latin of the summary and first twelve sections of Euler's paper with some explanatory comments. The remainder of the paper, treating Wallis' hypergeometric series and other technical matter, is described briefly. Appended is a short bibliography of works concerning Euler which are available to the English-speaking reader.

Let A be an m × n matrix of complex numbers. Let Aτ and A* denote the transpose and conjugate transpose, respectively, of A. We say A is diagonal if A contains only zeros in all positions (i, j) with i ≠ j. In a recently published paper [4], M.H. Pearl established the following fact: There exist real orthogonal matrices O1 and O2 (O1 m-square, O2 n-square) such that O1AO2 is diagonal, if and only if both AA* and A*A are real. It is the purpose of this paper to show that a theorem substantially stronger than this result of Pearl's is included in the real case of a theorem of N.A. Wiegmann [2]. (For other papers related to Wiegmann's, see [l; 3].)

It is pointed out that the proof of the basic result of Rahmanov's paper has a serious gap. It is documented by original sources that a statement he relied on in the proof contains a misprint, and it is shown by a counterexample that this statement (with the misprint) is, in fact, false. A somewhat weaker statement is proved true.

The proof of Theorem 1 in the paper “On Krull’s conjecture concerning valuation rings” (vol. 4 (1952) of this journal) is not correct. We want to give here a corrected proof of the theorem: From p. 30, l. 14 to p. 31, l. 7 should be changed as follows.

Several formulas for translation-invariant operator ideals overLp-spaces of compact groups are proved by tensor product methods.

It is shown that a result by Moeckel holds not only for admissible subgroups of SL (2, ℤ), but also for arbitrary subgroups of finite index.

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A subset A A of a topological space is said to be semi-open if there exists an open set U U such that U ⊆ A ⊆ Cl ⁡ ( U ) U \subseteq A \subseteq \operatorname {Cl} (U) where Cl ⁡ ( U ) \operatorname {Cl} (U) denotes the closure of U U . The class of semi-open sets of a given topological space ( X , T ) (X,\mathcal {T}) is denoted S .O . ( X , T ) {\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) . In the paper Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41, Norman Levine states in Theorem 10 that if T \mathcal {T} and T ∗ {\mathcal {T}^ \ast } are two topologies for a set X X such that S .O . ( X , T ) ⊆ S .O . ( X , T ∗ ) {\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) \subseteq {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast }) , then T ⊆ T ∗ \mathcal {T} \subseteq {\mathcal {T}^ \ast } . In a corollary to this theorem, Levine states that if S .O . ( X , T ) = S .O . ( X , T ∗ ) {\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) = {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast }) , then T = T ∗ \mathcal {T} = {\mathcal {T}^ \ast } . An example is given which shows the above-mentioned theorem and its corollary are false. This paper shows how different topologies on a set which determine the same class of semi-open subsets can arise from functions, and points out some of the implications of two topologies being related in this manner.

We give another formulation of Theorem 1 of Ribenboim's paper, Bull. Austral. Math. Soc. 2 (1970), 81–88. Our condition on a module to be totally ordered such that this order extends a given strict order seems to be more useful for applications than condition (2) of Ribenboim. This is shown for example in Corollaries 3 and It of our paper. Corollary 4 is due to Ribenboim. Corollary 3 generalizes the corresponding well known result on abelian groups.

where the oi are the usual generators of Bn. (See [1; 4].) A textile manufacturer asked the following question: Can one decide whether or not a braid in An is also in Qn? (It is possible to "weave" a braid in Qn while keeping its ends "tied together.") Shepperd [6] solved the above decision problem using the generators and relations of the braid group and subgroups. In this paper we give another distinct solution working in terms of automorphisms of free groups. In particular, we use formulas developed in [3; 4]. The author wishes to thank his teacher, and friend, Professor Wilhelm Magnus who has contributed more than anyone else towards the author's knowledge of mathematics.