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The change of variable formula, or Ito's rule, is studied in a Dedekind complete vector lattice E with weak order unit E. Using the functional calculus we prove that for a Holder continuous semimartingale X t = X a + M t + B t , t ∈ J , and a twice continuously differentiable function f, the formula (0.1) f ( X t ) = f ( X a ) + ∫ 0 t f ′ ( X s ) d M s + ∫ 0 t f ′ ( X s ) d B s + 1 2 ∫ 0 t f ″ ( X s ) d 〈 M 〉 s , 0 ≤ s ≤ t ∈ J holds. The first integral in the formula is an Ito integral with reference to the local martingale M and the second and third integrals are Dobrakov-type integrals of a vector valued function with reference to a vector valued measure. Using the formula, we prove Levy's characterization of Brownian motion as being a continuous martingale with compensator tE. The proof of this result yields a concrete description of abstract Brownian motion defined in vector lattices.

We show that an Urysohn lattice pre-homomorphism defined on a normal sublattice D of a vector lattice E can be extended to the whole space E and the extended operator is an Urysohn lattice homomorphism. We introduce a new class of nonlinear operators which called φ -operators and describe some of their properties. Finally we investigate a structure of positive orthogonally additive operators dominated by an Urysohn lattice homomorphism defined on a vector lattice E and taking value in Dedekind complete vector lattice F .

The aim of this work is to introduce a purely order-based Kantorovich–Wright type integration of scalar functions with respect to a vector measure defined on a δ-ring and taking values in a Dedekind σ-complete vector lattice and use it for obtaining general representation theorems for Dedekind complete vector lattices.

We establish a duality between flat affine group schemes and rigid tensor categories equipped with a neutral fiber functor (called Tannakian lattice), both defined over a Dedekind ring. We use this duality and the known Tannakian duality due to Saavedra to study morphisms between flat affine group schemes. Next, we apply our new duality to the category of stratified sheaves on a smooth scheme over a Dedekind ring R to define the relative differential fundamental group scheme of the given scheme and compare the fibers of this group scheme with the fundamental group scheme of the fibers. When R is a complete DVR of equal characteristic we show that this category is Tannakian in the sense of Saavedra., Comment: 41 pages, final version, accepted for publication in Mathematische Zeitschrift

In this article the whole set of n-1 rank Monotone Boolean Functions (MBFs) is divided into equivalence classes. It shows how the Dedekind number D(n) can be calculated by using this partition. Five formulas were found to calculate this number as well as the algebraic properties of MBF blocks.

Using Dedekind categories as an algebraic structure for (binary) set-theoretic relations without complements, we present purely algebraic definitions of “to be bipartite” and “to possess no odd cycles” and prove that both notions coincide in case that the reflexive–transitive closure of the relation in question is symmetric. This generalises D. Kőnig's well-known theorem from undirected graphs to abstract relations and, hence, to models such as L-relations that are different from set-theoretic relations. One direction of this generalisation holds for general relations. The other direction requires the symmetry of the reflexive–transitive closure as premise and is shown by specifying a bipartition for the given relation in form of a pair of disjoint relational vectors. For set-theoretic relations this immediately leads to a relational program for computing a bipartition. We also investigate the structure of a certain set of bipartitions that is generated by a given bipartition, explore the set of all bipartitions of a bipartite relation with respect to minimality and maximality of elements in view of the component-wise inclusion, show a relationship between bipartitions and bichromatic partitions and also discuss how the algebraic proofs can be mechanised using theorem proving tools.

Using a one-to-one correspondence between observables and their spectral resolutions, we introduce the sum of any two bounded observables of a $$\sigma $$ -MV-effect algebra. This sum is commutative, associative with neutral element. Under the Olson order of observables, the set of bounded observables is a partially ordered semigroup, and the set of sharp observables is even a Dedekind $$\sigma $$ -complete $$\ell $$ -group with strong unit.

Using a one-to-one correspondence between observables and their spectral resolutions, we introduce the sum of any two bounded observables of a $\sigma$-MV-effect algebra. This sum is commutative, associative and with neutral element. Under the Olson order of observables, the set of bounded observables is a partially ordered semigroup, and the set of sharp observables is even a Dedekind $\sigma$-complete $\ell$-group with strong unit.

Let gcd ⁡ ( k , j ) be the greatest common divisor of the integers k and j . We establish some asymptotic formulas for weighted averages of the gcd-sum functions, that is ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r f ( gcd ⁡ ( k , j ) ) with f = id , ϕ , ϕ s , ψ and ψ s for any fixed positive integers r and s , where ϕ , ϕ s , ψ and ψ s are the Euler, the Jordan, the Dedekind and the generalized Dedekind function, respectively, and also prove the mean square formulas of the error term of the gcd-sum functions ∑ k ≤ x 1 k r + 1 ⋅ ∑ j = 1 k j r ϕ ( gcd ⁡ ( k , j ) ) and ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r ψ ( gcd ⁡ ( k , j ) ) .

In the study, two relations between the Dedekind's η-function and θ-function were established by the using characteristic values ( ϵ, ϵ') = (0, 0), (0, 1) (mod2) for θ- function according to 1/8 coefficient of period pair (u,τ) and the transformations among the theta functions according to the periods have been given and a Jacobian style elliptic functions has been set up the theta function by the help of a defined function.

Der folgende Artikel prasentiert einige Originaldokumente, die im Zusammenhang mit R. Dedekinds Berufung (1858) an das Eidgenossische Polytechnikum in Zurich stehen, und kommentiert diese.

This text is based on an invited talk at the Dedekind Symposium at Braunschweig in October 2016. It summarizes views from my recent commented edition of Dedekinds two books on the foundations of mathematics., Revised version

B. Sury proved the following Menon-type identity, ∑ a ∈ U ( Z n ) , b 1 , ⋯ , b r ∈ Z n gcd ⁡ ( a − 1 , b 1 , ⋯ , b r , n ) = φ ( n ) σ r ( n ) , where U ( Z n ) is the group of units of the ring for residual classes modulo n, φ is the Euler's totient function and σ r ( n ) is the sum of r-th powers of positive divisors of n with r being a non-negative integer. Recently, C. Miguel extended this identity from Z to any residually finite Dedekind domain. In this note, we will give a further extension of Miguel's result to the case with many tuples of group of units. For the case of Z , our result reads as follows ∑ a 1 , ⋯ , a s ∈ U ( Z n ) , b 1 , ⋯ , b r ∈ Z n gcd ⁡ ( a 1 − 1 , ⋯ , a s − 1 , b 1 , ⋯ , b r , n ) = φ ( n ) ∏ i = 1 m ( φ ( p i k i ) s − 1 p i k i r − p i k i ( s + r − 1 ) + σ s + r − 1 ( p i k i ) ) , where n = p 1 k 1 ⋯ p m k m is the prime factorization of n.

International audience; In this paper, I present and analyse Dedekind's and Weber's editorial work which led to the edition of Riemann's Gesammelte Werke in 1876. With several examples, I suggest that this editorial work is to be understood as a mathematical activity in and of itself and provide evidence for it.

We derive a new special case C(q) of a general continued fraction recorded by Ramanujan in his Lost Notebook. We give a representation of the continued fraction C(q) as a quotient of Dedekind eta-function and then use it to prove modular identities connecting C(q) with each of the continued fractions $$C(-q)$$ , $$C(q^{2})$$ , $$C(q^{3})$$ , $$C(q^{5})$$ , $$C(q^{7})$$ , $$C(q^{11})$$ , $$C(q^{13})$$ and $$C(q^{17})$$ . We also prove general theorems for the explicit evaluation of the continued fraction C(q) by using Ramanujan’s class invariants.

We study the flatness and the projectivity of Hopf algebras, defined over a Dedekind ring, over their Hopf subalgebras. We give a criterion for the faithful flatness and use it to show the faithful flatness of an arbitrary flat Hopf algebra upon its finite normal Hopf subalgebras. For the projectivity over Hopf subalgebras of a Hopf algebra we need some finiteness condition in terms of the module of integrals. In particular we show that the module of integrals is projective of rank one.

The purpose of this paper is two-fold: first, to outline a purely order-based integral of the type of the Kantorovich–Wright integral of scalar functions with respect to a vector measure defined on a δ-ring and taking values in a K σ-space (that is, a Dedekind σ-complete vector lattice) and, secondly, prove new theorems on the representation of Dedekind complete vector lattices and quasi-Banach lattices in the form of lattices of functions integrable or “weakly” integrable with respect to an appropriate vector measure. In particular, it is shown that, in studying quasi-Banach lattices, when the duality method does not apply, the Kantorovich–Wright integral is more flexible than the Bartle–Dunford–Schwartz integral.

We give several equivalent conditions characterizing the case when Krb-AM(E,F) is Dedekind σ-complete. Moreover, we describe the case when the space of all regular b-AM-compact operators from E to F is complete under the b-AM-norm.Keywords: Banach lattices, b-AM-compact operator, discrete space

Tightness is a notion that arose in an attempt to understand the reverse reflection problem: given a monoreflection of a category onto a subcategory, determine which subobjects of an object in the subcategory reflect to it — those which do are termed tight. Thus tightness can be seen as a strong density property. We present an analysis of λ-tightness, tightness with respect to the localic Lindel¨of reflection. Leading to this analysis, we prove that the normal, or Dedekind-MacNeille, completion of a regular σ-frame A is a frame. Moreover, the embedding of A in its normal completion is the Bruns-Lakser injective hull of A in the category of meet semilattices and semilattice homomorphisms.Since every regular σ-frame is the cozero part of a regular Lindel¨of frame, this result points towards λ-tightness. For any regular Lindel¨of frame L, the normal completion of Coz L embeds in L as the sublocale generated by Coz L. Although this completion is clearly contained in every sublocale having the same coz...

Let R be a semilocal Dedekind domain, and let K be the field of fractions of R. Let G be a reductive semisimple simply connected R-group scheme such that every semisimple normal R-subgroup scheme of G contains a split R-torus $$ {\mathbb{G}}_{m,R} $$ . It is proved that the kernel of the map $$ {H}_{\overset{\prime }{e}t}^1\left(R,\kern0.5em G\right)\to {H}_{\overset{\prime }{e}t}^1\left(K,\kern0.5em G\right) $$ induced by the inclusion of R into K is trivial. This result partially extends the Nisnevich theorem.

The paper examines different kinds of p-convexity of a function g which are sufficient for the existence of a linear functional such that in Theorem 1.13 of Simons in his monograph ‘From Hahn–Banach to monotonicity’, published in Springer lecture notes (2008). We replace sublinearity of p with convexity, the field with Dedekind vector lattice and present -convexity which is also necessary. In Theorem 4.7 we also generalize a result of MM. Neumann from 1991 published in Czech. Mathem. Journal Vol 41 on the Mazur–Orlicz theorem.

In a family of $S_{d+1}$-fields ($d=2,3,4$), we obtain the true upper and lower bound of the residues of Dedekind zeta functions except for a density zero set. For $S_5$-fields, we need to assume the strong Artin conjecture. We also show that there exists an infinite family of number fields with the upper and lower bound, resp., The definition of $L(s,\rho)$ is given in the introduction. The proof of Proposition 4.3 is revised

In this paper, Dedekind’s theorem on the linear independence of homomorphisms of commutative semirings is studied. Furthermore, this theorem is extended to the case of linear independence of compos...

We study the existence of unimodular matrices with a prescribed submatrix over a ring R . In particular, when either R is Hermite and Dedekind finite or R has stable range one, we give necessary and sufficient conditions for the existence of these unimodular completions.

In this paper, using Cohen's and Tangedal and Young's theory on the p -adic Hurwitz zeta functions, we construct the analytic Dedekind sums on the p -adic complex plane C p . We show that these Dedekind sums interpolate Carlitz's higher order Dedekind sums p -adically. From Apostol's reciprocity law for the generalized Dedekind sums, we also prove a reciprocity relation for the special values of these p -adic Dedekind sums. Finally, the parallel results for the analytic Dedekind sums on the p -adic complex plane associated with Euler polynomials have also been given.

We consider a family of approximations of the Dedekind zeta function ζK(s) of a number field K/Q. Weighted L^2-norms of the difference of two such approximations of ζK(s) are computed. We work with a weight which is a compactly supported smooth function. Mean square estimates for the difference of approximations of ζK(s) can be obtained from such weighted L^2-norms. Some results on the location of zeros of a family of approximations of Dedekind zeta functions are also derived. These results extend results of Gonek and Montgomery on families of approximations of the Riemann zeta-function.

This paper contains examples of shuffle relations among multiple Dedekind zeta values. Dedekind zeta values were defined by the author in his paper "Multiple Dedekind zeta functions". Here we concentrate on the cases of real or imaginary quadratic fields with up to double iteration. We give examples of integral shuffle relation in terms of iterated integrals over membranes and of infinite sum shuffle relation, sometimes called stuffle relation. Using both types of shuffles for the product $\zeta_{K,C}(2)\zeta_{K,C}(2)$, we find relations among multiple Dedekind zeta values for real and for imaginary quadratic fields., Comment: Defines more general multiple Dedekind zeta values; contains 43 diagrams; 25 pages

A new lower bound on the size of product sets of rational numbers is obtained. An upper estimate for the multiplicative energy of two sets of rational numbers is also found.

We consider the lattice-ordered groups Inv ( R ) and Div ( R ) of invertible and divisorial fractional ideals of a completely integrally closed Prufer domain. We prove that Div ( R ) is the completion of the group Inv ( R ) , and we show there is a faithfully flat extension S of R such that S is a completely integrally closed Bezout domain with Div ( R ) ≅ Inv ( S ) . Among the class of completely integrally closed Prufer domains, we focus on the one-dimensional Prufer domains. This class includes Dedekind domains, the latter being the one-dimensional Prufer domains whose maximal ideals are finitely generated. However, numerous interesting examples show that the class of one-dimensional Prufer domains includes domains that differ quite significantly from Dedekind domains by a number of measures, both group-theoretic (involving Inv ( R ) and Div ( R ) ) and topological (involving the maximal spectrum of R ). We examine these invariants in connection with factorization properties of the ideals of one-dimensional Prufer domains, putting special emphasis on the class of almost Dedekind domains, those domains for which every localization at a maximal ideal is a rank one discrete valuation domain, as well as the class of SP-domains, those domains for which every proper ideal is a product of radical ideals. For this last class of domains, we show that if in addition the ring has nonzero Jacobson radical, then the lattice-ordered groups Inv ( R ) and Div ( R ) are determined entirely by the topology of the maximal spectrum of R , and that the Cantor–Bendixson derivatives of the maximal spectrum reflect the distribution of sharp and dull maximal ideals.

Let R be a ring with identity and let M be a unitary left module over R. In this paper, we study direct summand (direct sum) of essentially small quasi-Dedekind module (essentially small quasi-Dedekind modules). Also, give the definition of essentially small quasi-Dedekind relative to a module with some examples. We give some of their basic properties and some examples that illustrate these properties.

In this paper we generalize and refine a partition inequality of Bessenrodt–Ono. We introduce and study the m shifted inequality $$\begin{aligned} p(a) \, p(b) \ge p(a+b+m-1) \end{aligned}$$where p(n) is the nth partition number, and $$a,b,m \in \mathbb {N}_0$$ with a, b positive. The inequality was first studied by Bessenrodt–Ono for $$m=1$$. We finally suggest another generalization involving polynomials, which dictate the vanishing properties of the Fourier coefficients of powers of the Dedekind $$\eta $$-function.

We prove an asymptotic formula for the second moment of a product of two Dirichlet L-functions on the critical line, which has a power saving in the error term and which is uniform with respect to the involved Dirichlet characters. As special cases we give uniform asymptotic formulae for the fourth moment of individual Dirichlet L-functions and for the second moment of Dedekind zeta functions of quadratic number fields on the critical line., 35 pages

Among all of Frege’s contemporaries, Richard Dedekind is arguably the thinker closest to him in terms of their general backgrounds and core projects. This essay provides a reexamination of Frege’s critical reactions to Dedekind, in Grundgesetze and some related texts. The reexamination includes documenting their interaction in some detail and putting it into a broader context, both philosophically and systematically. It also involves separating Frege’s less compelling criticisms of Dedekind from those that have deeper, more lasting significance. The essay ends with a suggestion for how to reconcile Fregean and Dedekindian forms of logicism, based on distinguishing two distinct but complementary kinds of abstraction principles.

International audience; Unbounded entailment relations, introduced by Paul Lorenzen (1951), are a slight variant of a notion which plays a fundamental rôle in logic (see Scott 1974) and in algebra (see Lombardi and Quitté 2015). We call “systems of ideals” their single-conclusion counterpart. If they preserve the order of a commutative ordered monoid G and are equivariant w.r.t. its law, we call them “equivariant systems of ideals for G”: they describe all morphisms from G to meet-semilattice-ordered monoids generated by (the image of) G. Taking an article by Lorenzen (1953) as a starting point, we also describe all morphisms from a commutative ordered group G to lattice-ordered groups generated by G through unbounded entailment relations that preserve its order, are equivariant, and satisfy a regularity property invented by Lorenzen (1950); we call them “regular entailment relations”. In particular, the free lattice-ordered group generated by G is described through the finest regular entailment relation for G, and we provide an explicit description for it; it is order-reflecting if and only if the morphism is injective, so that the Lorenzen-Clifford-Dieudonné theorem fits into our framework. Lorenzen’s research in algebra starts as an inquiry into the system of Dedekind ideals for the divisibility group of an integral domain R, and specifically into Wolfgang Krull’s “Fundamentalsatz” that R may be represented as an intersection of valuation rings if and only if R is integrally closed: his constructive substitute for this representation is the “regularisation” of the system of Dedekind ideals, i.e. the lattice-ordered group generated by it when doing as if its elements are comparable.

It is often remarked that first-order Peano Arithmetic is non-categorical but deductively well-behaved, while second-order Peano Arithmetic is categorical but deductively ill-behaved. This suggests that, when it comes to axiomatizations of mathematical theories, expressive power and deductive power may be orthogonal, mutually exclusive desiderata. In this paper, I turn to Hintikka's (Philos Top 17(2):69-90, 1989) distinction between descriptive and deductive approaches in the foundations of mathematics to discuss the implications of this observation for the first-order logic versus second-order logic divide. The descriptive approach is illustrated by Dedekind's discovery' of the need for second-order concepts to ensure categoricity in his axiomatization of arithmetic; the deductive approach is illustrated by Frege's Begriffsschrift project. I argue that, rather than suggesting that any use of logic in the foundations of mathematics is doomed to failure given the impossibility of combining the descriptive approach with the deductive approach, what this apparent predicament in fact indicates is that the first-order versus second-order divide may be too crude to investigate what an adequate axiomatization of arithmetic should look like. I also conclude that, insofar as there are different, equally legitimate projects one may engage in when working on the foundations of mathematics, there is no such thing as the One True Logic for this purpose; different logical systems may be adequate for different projects.

We prove an explicit upper bound for the k-th prime ideal with fixed Artin symbol, under the assumption of the validity of the Riemann hypothesis for the Dedekind zeta functions., Comment: We have improved the introduction and made clearer some computations. arXiv admin note: text overlap with arXiv:1709.07609

Dedekind's proof of the Cantor–Bernstein theorem is based on his chain theory, not on Cantor's well-ordering principle. A careful analysis of the proof extracts an argument structure that can be seen in the many other proofs that have been given since. I contend there is essentially one proof that comes in two variants due to Dedekind and Zermelo , respectively. This paper is a case study in analysing proofs of a single theorem within a given methodological framework, here Zermelo–Fraenkel set theory (ZF). It uses tools from proof theory, but focuses on heuristic ideas that shape proofs and on logical strategies that help to construct them. It is rooted in a perspective on Beweistheorie that predates its close connection and almost exclusive attention to the goals of Hilbert's finitist consistency programme. This earlier perspective can be brought to life (only) with the support of powerful computational tools. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

We give a signed fundamental domain for the action on $\mathbb{C}^*\times \mathbb{R}_+^{n-2}$ of the totally positive units $E(k)_+$ of a number field $k$ of degree $n$ and having exactly one pair of complex embeddings. This signed fundamental domain, built of $k$-rational simplicial cones, is as convenient as a true fundamental domain for the purpose of studying Dedekind zeta functions. However, while there is no general construction of a true fundamental domain, we construct a signed fundamental domain from any set of fundamental units of $k$., 37 pages, 2 figures. arXiv admin note: text overlap with arXiv:1303.3989 by other authors

We consider some congruences involving arithmetical functions. For example, we study the congruences nψ(n) ≡ 2 (mod ϕ(n)), nϕ(n) ≡ 2 (mod ψ(n)), ψ(n)d(n) − 2 ≡ 0 (mod n), where ϕ(n), ψ(n), d(n) denote Euler's totient, Dedekind's function, and the number of divisors of n, respectively. Two duals of the Lehmer congruence n − 1 ≡ 0 (mod ϕ(n)) are also considered.

We extend the notions of Dedekind complete and σ-Dedekind complete Banach lattices to Banach C(K)-modules. As our main result we prove for these modules an analogue of Lozanovsky’s well-known characterization of Banach lattices with order continuous norm.

Let $\mathfrak D$ be a residually finite Dedekind domain, $a\in \mathfrak D$ be a nonzero element and $\mathfrak n$ be a nonzero ideal of $\mathfrak D$. In this paper we describe the dynamics of the map $x\mapsto ax$ over the quotient ring $\mathfrak D/\mathfrak n$. We further present some applications of our main result., Comment: 15 pages; comments welcome

Families of polynomials associated to arithmetic functions [Formula: see text] are studied. The case [Formula: see text], the divisor sum, dictates the non-vanishing of the Fourier coefficients of powers of the Dedekind eta function. The polynomials [Formula: see text] are defined by [Formula: see text]-term recurrence relations. For the case that [Formula: see text] is a polynomial of degree [Formula: see text], we prove that at most a [Formula: see text] term recurrence relation is needed. For the special case [Formula: see text], we obtain explicit formulas and results.

Set Theory was first developed by Cantor and Dedekind to handle infinite collections. This chapter looks at their theory of countably and uncountably infinite sets. Around 1900, mathematicians were motivated by encountering some perplexing results for infinite sets to provide Set Theory with a more solid foundation. We’ll briefly explore the standard axiomatization of Set Theory, along with its relevance to Peano Arithmetic, and we’ll also look at an application of Set Theory to the theory of computation.