Köthe’s upper nil radical for modules

Let M be a left R-module. In this paper a generalization of the notion of an s-system of rings to modules is given. Let N be a submodule of M. Define $\mathcal{S}(N):=\{ {m\in M}:\, \mbox{every } s\mbox{-system containing } m \mbox{ meets}~N \}$ . It is shown that $\mathcal{S}(N)$ is equal to the intersection of all s-prime submodules of M containing N. We define $\mathcal{N}({}_{R}M) = \mathcal{S}(0)$ . This is called (Köthe’s) upper nil radical of M. We show that if R is a commutative ring, then $\mathcal{N}({}_{R}M) = {\mathop{\mathrm{rad}}\nolimits}_{R}(M)$ where ${\mathop{\mathrm{rad}}\nolimits}_{R}(M)$ denotes the prime radical of M. We also show that if R is a left Artinian ring, then ${\mathop{\mathrm{rad}}\nolimits}_{R}(M)=\mathcal{N}({}_{R}M)= {\mathop{\mathrm{Rad}}\nolimits}\, (M)= {\mathop{\mathrm{Jac}}\nolimits}\, (R)M$ where ${\mathop{\mathrm{Rad}}\nolimits}\, (M)$ denotes the Jacobson radical of M and ${\mathop{\mathrm{Jac}}\nolimits}\, (R)$ the Jacobson radical of the ring R. Furthermore, we show that the class of all s-prime modules forms a special class of modules.     

On improving the accuracy of Horner’s and Goertzel’s algorithms

It is known that Goertzels algorithm is much less numerically accurate than the Fast Fourier Transform (FFT) (cf. [2]). In order to improve accuracy we propose modifications of both Goertzels and Horners algorithms based on the divide-and-conquer techniques. The proof of the numerical stability of these two modified algorithms is given. The numerical tests in Matlab demonstrate the computational advantages of the proposed modifications. The appendix contains the proof of numerical stability of Goertzels algorithm of polynomial evaluation. AMS subject classification 65F35, 65G50     

A generalization of Baer’s theorem

In this paper, we studied the supersolvability of the product of two subgroups and got a generalization of Baer’s theorem.     

A generalization of Baer’s Lemma

There is a classical result known as Baer’s Lemma that states that an R-module E is injective if it is injective for R. This means that if a map from a submodule of R, that is, from a left ideal L of R to E can always be extended to R, then a map to E from a submodule A of any R-module B can be extended to B; in other words, E is injective. In this paper, we generalize this result to the category q _ω consisting of the representations of an infinite line quiver. This generalization of Baer’s Lemma is useful in proving that torsion free covers exist for q _ω.      

On a variant of Rado’s selection lemma and its equivalence with the Boolean prime ideal theorem

We establish that, in ZF (i.e., Zermelo–Fraenkel set theory minus the Axiom of Choice AC), the statement RLT: Given a set I and a non-empty set \({\mathcal{F}}\) of non-empty elementary closed subsets of 2 ^I satisfying the fip, if \({\mathcal{F}}\) has a choice function, then \({\bigcap\mathcal{F} \ne \emptyset}\) , which was introduced in Morillon (Arch Math Logic 51(7–8):739–749, 2012), is equivalent to the Boolean Prime Ideal Theorem (see Sect. 1 for terminology). The result provides, on one hand, an affirmative answer to Morillon’s corresponding question in Morillon (2012) and, on the other hand, a negative answer—in the setting of ZFA (i.e., ZF with the axiom of extensionality weakened to permit the existence of atoms)—to the question in Morillon (2012) of whether RLT is equivalent to Rado’s selection lemma.     

A converse of Baer’s theorem

Schur’s classical theorem states that for a group $G$ , if $G/Z(G)$ is finite, then $G'$ is finite. Baer extended this theorem for the factor group $G/Z_n(G)$ , in which $Z_n(G)$ is the $n$ -th term of the upper central series of $G$ . Hekster proved a converse of Baer’s theorem as follows: If $G$ is a finitely generated group such that $\gamma _{n+1}(G)$ is finite, then $G/Z_n(G)$ is finite where $\gamma _{n+1}(G)$ denotes the $(n+1)$ st term of the lower central series of $G$ . In this paper, we generalize this result by obtaining the same conclusion under the weaker hypothesis that $G/Z_n(G)$ is finitely generated. Furthermore, we show that the index of the subgroup $Z_n(G)$ is bounded by a precisely determined function of the order of $\gamma _{n+1}(G)$ . Moreover, we prove that the mentioned theorem of Hekster is also valid under a weaker condition that $Z_{2n}(G)/Z_{n}(G)$ is finitely generated. Although in this case the bound for the order of $\gamma _{n+1}(G)$ is not achieved.     

On Ostrowski’s disk theorem and lower bounds for the smallest eigenvalues and singular values

The paper refines the classical Ostrowski disk theorem and suggests lower bounds for the smallest-in-modulus eigenvalue and the smallest singular value of a square matrix under certain diagonal dominance conditions. A lower bound for the smallest-in-modulus eigenvalue of a product of m ≥ 2 matrices satisfying joint diagonal dominance conditions is obtained. The particular cases of the bounds suggested that correspond to the infinity norm are discussed separately and compared with some known results. Bibliography: 8 titles.     

On the optimality and sharpness of Laguerre’s lower bound on the smallest eigenvalue of a symmetric positive definite matrix

Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix A ∈ R ^m×m play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on Tr(A ^?1) and Tr(A ^?2) have attracted attention recently, because they can be computed in O(m) operations when A is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that can be computed solely from Tr(A ^?1) and Tr(A ^?2) and show that the so called Laguerre’s lower bound is the optimal one in terms of sharpness. Next, we study the gap between the Laguerre bound and the smallest eigenvalue. We characterize the situation in which the gap becomes largest in terms of the eigenvalue distribution of A and show that the gap becomes smallest when {Tr(A ^?1)}²/Tr(A ^?2) approaches 1 or m. These results will be useful, for example, in designing efficient shift strategies for singular value computation algorithms.     

Deligne’s congruence and supersingular reduction of Drinfeld modules

Most rank two Drinfeld modules are known to have infinitely many supersingular primes. But how many supersingular primes of a given degree can a fixed Drinfeld module have? In this paper, a congruence between the Hasse invariant and a certain Eisenstein series is used for obtaining a bound on the number of such supersingular primes. Certain exceptional cases correspond to zeros of certain Eisenstein series with rational j-invariants.     

On the properties of McShane’s functional and their applications

In this paper we derive some remarkable properties of McShane’s functional, defined by means of positive isotonic linear functionals. These properties are then applied to weighted generalized means. A series of consequences among additive and multiplicative type mean inequalities is given, as well as a special consideration of Hölder’s inequality, in view of the new results.     

On the Geometrical and Physical Meaning of Newton’s Solution to Kepler’s Problem

In the short treatise De Motu (1684),which serves as a precursor to the Principia Mathematica (1687),Newton essentially deals with the following two problems.     

Fatou’s lemma and lower epi-limits of integral functionals

In this work, with the introduction in the $\sigma$-finite case of a modulus of equi-integrability, we prove some new extensions of Fatou’s lemma and some of its consequences in the convergence theory of integral functionals. We present the case of a sequence of integral functionals using an analog of Ioffe’s criterion for a sequence of integrands.     

On Pure Subgroups of the Baer–Specker Group and Weak Crawley Groups

In this work we show how endomorphisms of certain pure subgroups of the Baer–Specker group can be extended to endomorphisms of the whole group. This allows us to establish the existence of a large family of essentially indecomposable slender groups with prescribed endomorphism rings. The extension technique is then applied to show that the Baer–Specker group is a countably-free weak Crawley group which is not a Crawley group.     

On an Analog of the Plan’s Formula

In this paper we obtain an analog of the Plan’s formula, which plays an essential role in obtaining a functional relation for classical Riemann zeta-function.We provide examples of rational functions that satisfy a certain symmetry condition and admit a Maclaurin series expansion with coefficients equal to zero or one.