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Arrow categories establish a categorical and algebraic description of L -fuzzy relations, i.e., relations that use membership values from an arbitrary but fixed complete Heyting algebra L . With other words arrow categories describe the fixed-basis case. In this paper we are interested in the variable-basis case, i.e., the case where relations between different objects may use different membership values. We will investigate the structure of the collection of lattices of membership values within a given Dedekind category. This will lead to a complete characterization of the variable-basis case in this context.

We determine the shapes of pure cubic fields and show that they fall into two families based on whether the field is wildly or tamely ramified (of Type I or Type II in the sense of Dedekind). We show that the shapes of Type I fields are rectangular and that they are equidistributed, in a regularized sense, when ordered by discriminant, in the one-dimensional space of all rectangular lattices. We do the same for Type II fields, which are however no longer rectangular. We obtain as a corollary of the determination of these shapes that the shape of a pure cubic field is a complete invariant determining the field within the family of all cubic fields., 14 pages. To appear in the Proceedings of the AMS. Postprint version. Comments welcome!

In order to provide a common framework for classical, fuzzy and L-fuzzy databases we assume that databases are set-valued, i.e., that every value of an attribute is a set in some suitable set theory. Tables of a database are modeled using the theory of Dedekind categories and its extensions. These theories establish an abstract approach to binary, classical (crisp) or fuzzy relations. Based on this model we investigate dependencies in databases in a general setting.

In 2008, Han rediscovered an expansion of powers of Dedekind η function attributed to Nekrasov and Okounkov (which was actually first proved the same year by Westbury) by using a famous identity of Macdonald in the framework of type A ˜ affine root systems. In this paper, we obtain new combinatorial expansions of powers of η, in terms of partition hook lengths, by using the Macdonald identity in type C ˜ and a new bijection between vectors with integral coordinates and a subset of t-cores for integer partitions. As applications, we derive a symplectic hook formula and an unexpected relation between the Macdonald identities in types C ˜ , B ˜ , and BC ˜ . We also generalize these expansions through the Littlewood decomposition and deduce in particular many new weighted generating functions for subsets of integer partitions and refinements of hook formulas.

M.P. Olson, Proc. Am. Math. Soc. 28, 537–544 (1971) showed that the system of effect operators of the Hilbert space can be ordered by the so-called spectral order such that the system of effect operators is a complete lattice. Using his ideas, we introduce a partial order, called the Olson order, on the set of bounded observables of a complete lattice effect algebra. We show that the set of bounded observables is a Dedekind complete lattice.

A classical result of Claborn states that every abelian group is the class group of a commutative Dedekind domain. Among noncommutative Dedekind prime rings, apart from PI rings, the simple Dedekind domains form a second important class. We show that every abelian group is the class group of a noncommutative simple Dedekind domain. This solves an open problem stated by Levy and Robson in their recent monograph on hereditary Noetherian prime rings., Comment: 17 pages; final version (no significant changes)

The Boolean algebra of fragments of a positive abstract Uryson operator recently was described in M. Pliev (Positivity, doi: 10.1007/s11117-016-0401-9 , 2016). Using this result, we prove a theorem of domination for AM-compact positive abstract Uryson operators from a Dedekind complete vector lattice E to a Banach lattice F with an order continuous norm.

Let Kn be a number field of degree n over ℚ. By D(x,Kn) denote the number of all nonzero integral ideals in Kn with norm ≤ x. The Dedekind zeta function ζKn(s) is a meromorphic function with a simple pole at s = 1 and with residue, say, Λn. Define

We examine Paul Halmos' comments on category theory, Dedekind cuts, devil worship, logic, and Robinson's infinitesimals. Halmos' scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is "certainty" and "architecture" yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to teach us the opposite lession, namely that the castle is floating in midair. Halmos' realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians' concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson's framework is "unnecessary" but Henson and Keisler argue that Robinson's framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos' criticisms. Keywords: Archimedean axiom; bridge between discrete and continuous mathematics; hyperreals; incomparable quantities; indispensability; infinity; mathematical realism; Robinson., 15 pages, to appear in Logica Universalis

Let φ be the Euler's totient function and σ k ( n ) = ∑ d | n d k , that is, the sum of the kth powers of the divisors of n. B. Sury showed that ∑ t 1 ∈ U ( Z n ) , t 2 , … , t r ∈ Z n gcd ( t 1 − 1 , t 2 , … , t r , n ) = φ ( n ) σ r − 1 ( n ) , where U ( Z n ) is the group of units in the ring of residual classes modulo n. Here, this identity is extended to residually finite Dedekind domains.

It is well known that the equivalence of injectivity and divisibility characterizes the Dedekind domains. In this note, we shed more light on this problem and show that a similar property characterizes Krull domains and π-domains.

May soon be invented quantum computer and some known cryptosystems (such as RSA) will be threatened breaking. This paper is aimed at establishing necessary conditions for the maximum possible algebraic generalization of the classical RSA algorithm. We substitute ideals of a Dedekind ring for integers. Ideals in Dedekind rings allow the unique decomposition into a product of maximal ideals, but may not be the principal ideals. Also we define Euler’s ϕ-function for ideal of a Dedekind ring and describe some properties of this function. We hope that our proposed method will help to develop algorithms for encryption, which is hard to crack using a quantum computer.

It is well known that the sum of two linear continuous narrow operators in the spaces Lp with 1 < p < ∞ is not necessarily a narrow operator. However, the sum of a narrow operator and a compact linear continuous operator is a narrow operator. In a recent paper, Pliev and Popov originated the investigation of nonlinear narrow operators and, in particular, of orthogonally additive operators. As our main result, we prove that the sum of a narrow orthogonally additive operator and a finite-rank laterally-to-norm continuous orthogonally additive operator acting from an atomless Dedekind complete vector lattice into a Banach space is a narrow operator.

Structural theorems for spaces of modular forms with respect to congruence subgroups are proved. The Dedekind η-function plays an important role in our study.

Kamp's theorem established the expressive completeness of the temporal modalities Until and Since for the First-Order Monadic Logic of Order (FOMLO) over real and natural time flows. Over natural time, a single future modality (Until) is sufficient to express all future FOMLO formulas. These are formulas whose truth value at any moment is determined by what happens from that moment on. Yet this fails to extend to real time domains: here no finite basis of future modalities can express all future FOMLO formulas. In this paper we show that finiteness can be recovered if we slightly soften the requirement that future formulas must be totally past-independent: we allow formulas to depend just on the arbitrarily recent past, and maintain the requirement that they be independent of the rest - actually - of most of the past. We call them 'almost future' formulas, and show that there is a finite basis of almost future modalities which is expressively complete (over all Dedekind complete time flows) for the almost future fragment of FOMLO.

Some notions in mathematics can be considered relative. Relative is a term used to denote when the variation in the position of an observer implies variation in properties or measures on the observed object. We know, from Skolem theorem, that there are first-order models where the set of real numbers is countable and some where it is not. This fact depends on the position of the observer and on the instrument/language the obserevr uses as well, i.e., it depends on whether he/she is inside the model or not and in this particular case the use of first-order logic. In this article, we assume that computation is based on finiteness rather than natural numbers and discuss Turing machines computable morphisms defined on top of the sole notion finiteness. We explore the relativity of finiteness in models provided by toposes where the Axiom of Choice (AC) does not hold, since Tarski proved that if AC holds then all finiteness notions are equivalent. Our toposes do not have natural numbers object (NNO) either, since in a topos with a NNO these finiteness notions are equivalent to Peano finiteness going back to computation on top of Natural Numbers. The main contribution of this article is to show that although from inside every topos, with the properties previously stated, the computation model is standard, from outside some of these toposes, unexpected properties on the computation arise, e.g., infinitely long programs, finite computations containing infinitely long ones, infinitely branching computations. We mainly consider Dedekind and Kuratowski notions of finiteness in this article., Comment: In Proceedings DCM 2015, arXiv:1603.00536

In this article, we will give a new proof of the reciprocity law for Dedekind sums, as well as a proof of the transformation formula for the Dedekind [Formula: see text]-function using the Chinese Remainder Theorem.

The “Up-and-down” theorem which describes the structure of the Boolean algebra of fragments of a linear positive operator is the well known result in operator theory. We prove an analog of this theorem for a positive abstract Uryson operator defined on a vector lattice and taking values in a Dedekind complete vector lattice. This result is used to prove a theorem of domination for order narrow positive abstract Uryson operators from a vector lattice E to a Banach lattice F with an order continuous norm.

In this paper we study a model of intuitionistic higher-order logic which we call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, \emph{the Muchnik reals}, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists w\,\forall x\,A(x,wx)$ and a \emph{bounding principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists z\,\forall x\,\exists y\,(y\le_{\mathrm{T}}(x,z)\land A(x,y))$ where $x,y,z$ range over Muchnik reals, $w$ ranges over functions from Muchnik reals to Muchnik reals, and $A(x,y)$ is a formula not containing $w$ or $z$. For the convenience of the reader, we explain all of the essential background material on intuitionism, sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems, Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an English translation of Muchnik's 1963 paper on Muchnik degrees., 44 pages

In this paper, we extend the theorem of Ore regarding factorization of polynomials over p-adic numbers to henselian valued fields of arbitrary rank thereby generalizing the main results of Khanduja and Kumar (J Pure Appl Algebra 216:2648–2656, 2012) and Cohen et al. (Mathematika 47:173–196, 2000). As an application, we derive the analogue of Dedekind’s Theorem regarding splitting of rational primes in algebraic number fields as well as of its converse for general valued fields extending similar results proved for discrete valued fields in Khanduja and Kumar (Int J Number Theory 4:1019–1025, 2008). The generalized version of Ore’s Theorem leads to an extension of a result of Weintraub dealing with a generalization of Eisenstein Irreducibility Criterion (cf. Weintraub in Proc Am Math Soc 141:1159–1160, 2013). We also give a reformulation of Hensel’s Lemma for polynomials with coefficients in henselian valued fields which is used in the proof of the extended Ore’s Theorem and was proved in Khanduja and Kumar (J Algebra Appl 12:1250125, 2013) in the particular case of complete rank one valued fields.

Non-periodic groups without free Abelian subgroups of rank 2 with non-Dedekind norm of Abelian noncyclic subgroups are studied.

In this paper, we find some properties of Euler&rsquo, s function and Dedekind&rsquo, s function. We also generalize these results, from an algebraic point of view, for extended Euler&rsquo, s function and extended Dedekind&rsquo, s function, in algebraic number fields. Additionally, some known inequalities involving Euler&rsquo, s function, we generalize them for extended Euler&rsquo, s function, working in a ring of integers of algebraic number fields.

In a previous effort [arXiv:1708.05492] we have created a framework that explains why topological structures naturally arise within a scientific theory; namely, they capture the requirements of experimental verification. This is particularly interesting because topological structures are at the foundation of geometrical structures, which play a fundamental role within modern mathematical physics. In this paper we will show a set of necessary and sufficient conditions under which those topological structures lead to real quantities and manifolds, which are a typical requirement for geometry. These conditions will provide a physically meaningful procedure that is the physical counter-part of the use of Dedekind cuts in mathematics. We then show that those conditions are unlikely to be met at Planck scale, leading to a breakdown of the concept of ordering. This would indicate that the mathematical structures required to describe space-time at that scale, while still topological, may not be geometrical., Comment: 16 pages, Accepted by Physica Scripta

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.

Despite the great success of Weierstrass, Dedekind and Cantor in constructing the continuum from arithmetical materials, a number of thinkers of the late nineteenth and early twentieth centuries remained opposed, in varying degrees, to the idea of explicating the continuum concept entirely in discrete terms. These include the mathematicians du Bois-Reymond, Veronese, Poincare, Brouwer and Weyl, and the philosophers Brentano and Peirce.

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.

We extend the even weight modular forms of modular invariant approach to general integral weight modular forms. We find that the modular forms of integral weights and level $N$ can be arranged into irreducible representations of the homogeneous finite modular group $\Gamma'_N$ which is the double covering of $\Gamma_N$. The lowest weight 1 modular forms of level 3 are constructed in terms of Dedekind eta-function, and they transform as a doublet of $\Gamma'_3 \cong T'$. The modular forms of weights 2, 3, 4, 5 and 6 are presented. We build a model of lepton masses and mixing based on $T'$ modular symmetry., Comment: 19 pages

 This article shows how fundamental higher-order theories of mathematical structures of computer science (e.g. natural numbers [Dedekind 1888] and Actors [Hewitt et. al. 1973]) are categorical meaning that they can be axiomatized up to a unique isomorphism thereby removing any ambiguity in the mathematical structures being axiomatized. Having these mathematical structures precisely defined can make systems more secure because there are fewer ambiguities and holes for cyberattackers to exploit. For example, there are no infinite elements in models for natural numbers to be exploited. On the other hand, the 1 st-order theories and computational systems which are not strongly-typed necessarily provide opportunities for cyberattack. Cyberattackers have severely damaged national, corporate, and individual security as well causing hundreds of billions of dollars of economic damage. [Sobers 2019] A significant cause of the damage is that current engineering practices are not sufficiently grounded in theoretical principles. In the last two decades, little new theoretical work has been done that practically impacts large engineering projects with the result that computer systems engineering education is insufficient in providing theoretical grounding. If the current cybersecurity situation is not quickly remedied, it will soon become much worse because of the projected development of Scalable Intelligent Systems by 2025 [Hewitt 2019]. Kurt Godel strongly advocated that the Turing Machine is the preeminent universal model of computation. A Turing machine formalizes an algorithm in which computation proceeds without external interaction. However, computing is now highly interactive, which this article proves is beyond the capability of a Turing Machine. Instead of the Turing Machine model, this article presents an axiomatization of a strongly-typed universal model of digital computation (including implementation of Scalable Intelligent Systems) up to a unique isomorphism. Strongly-typed Actors provide the foundation for tremendous improvements in cyberdefense.

Pr\"{u}fer domains and subclasses of integral domains such as Dedekind domains admit characterizations by means of the properties of their ideal lattices. Interestingly, a Leavitt path algebra $L$, in spite of being non-commutative and possessing plenty of zero divisors, seems to have its ideal lattices possess the characterizing properties of these special domains. In [8] it was shown that the ideals of $L$ satisfy the distributive law, a property of Pr\"{u}fer domains and that $L$ is a multiplication ring, a property of Dedekind domains. In this paper, we first show that $L$ satisfies two more characterizing properties of Pr\"{u}fer domains which are the ideal versions of two theorems in Elementary Number Theory, namely, for positive integers $a,b,c$, $\gcd(a,b)\cdot\operatorname{lcm}(a,b)=a\cdot b$ and $a\cdot \operatorname{gcd}(b,c)=\operatorname{gcd}(ab,ac)$. We also show that $L$ satisfies a characterizing property of almost Dedekind domains in terms of the ideals whose radicals are prime ideals. Finally, we give necessary and sufficient conditions under which $L$ satisfies another important characterizing property of almost Dedekind domains, namely the cancellative property of its non-zero ideals., Comment: 18 pages

In Sect. 4.3 we introduced ten Dedekind j-numeric types for an arbitrary modality j. Namely, we have the j-local lower and upper reals, improper and proper intervals, and real numbers, as well as their unbounded counterparts. They are denoted

Let $p$ be a prime number. Mazur proved that a geometrically maximal unramified abelian covering of $X_0(p)$ over $\mathbb Q$ is given by the Shimura covering $X_2(p) \to X_0(p)$, that is, a unique subcovering of $X_1(p) \to X_0(p)$ of degree $N_p := (p-1)/\gcd(p-1, 12)$. In this short paper, we show that a geometrically maximal abelian covering $X_2'(p) \to X_0(p)$ of $X_0(p)$ over $\mathbb Q$ unramified outside cusps is cyclic of degree $2N_p$. The main ingredient for the construction of $X_2'(p)$ is the generalized Dedelind eta functions., Comment: 12 pages

We prove that ifEis a Dedekind complete atomless Riesz space andXis a Banach space, then the sum of two laterally continuous orthogonally additive operators fromEtoX, one of which is strictly narrow and the other one is hereditarily strictly narrow with finite variation (in particular, has finite rank), is strictly narrow. Similar results were previously obtained for narrow operators by different authors; however, no theorem of the kind was known for strictly narrow operators.

In this chapter, we develop the theory of semistar operations on commutative rings and apply the theory to study many important classes of integral domains, such as the Dedekind domains, Prufer domains, UFDs, Krull domains, and PVMDs, along with natural generalizations of these classes of domains to rings with zerodivisors. We postpone the use of local methods until Chapter 3. All rings in Chapters 2– 4 are assumed commutative with identity.

As a natural generalization of the Euler-Mascheroni constant $\gamma$, Y. Ihara introduced the Euler-Kronecker constant $\gamma_K$ attached to any number field $K$. In this paper, we prove that a certain bound on $\gamma_K$ in a tower of number fields $\mathcal{K}$ implies the generalized Brauer-Siegel conjecture for $\mathcal{K}$ as formulated by Tsfasman and Vl\v{a}du\c{t}. Moreover, we use known bounds on $\gamma_K$ for cyclotomic fields to obtain a finer estimate for the number of zeros of the Dedekind zeta-function $\zeta_K(s)$ in the critical strip., Comment: 16 pages, to appear in the Proceedings of the American Mathematical Society

Various representations of the Dedekind completions of Riesz spaces of continuous functions are known in the literature. The aim of this review paper is to collect together some of these results and to connect the various constructions to each other.

The theory Ordinals can serve as a replacement for the theory ZFC because: • Ordinals are a very well understood mathematical structure. • There is only one model of Ordinals up to a unique isomorphism, which decides every proposition of the theory Ordinals in the model. • The theory Ordinals is much more powerful than ZFC. Standard mathematics that has been carried out in ZFC can more easily be done in Ordinals. Axioms of ZFC are in effect theorems of Ordinals. The type Ordinal is larger than any set in ZFC because no set can be placed in one-to-one correspondence with instances of the type Ordinal. • The theory Ordinals is algorithmically inexhaustible, i.e., it is impossible to computationally enumerate theorems of the theory thereby reinforcing the intuition behind [Franzen, 2004]. Contrary to [Church 1934], the conclusion in this article is to abandon the assumption that theorems of a theory must be computationally enumerable while retaining the requirement that proof checking must be computationally decidable. • There are no “monsters” [Lakatos 1976] in models of Ordinals such as the ones in models of 1st-order ZFC. Consequently unlike ZFC, the theory Ordinals is not subject to cyberattacks using “monsters” in models such as the ones that plague 1st-order ZFC. • The theory Ordinals formally proves its own consistency contra [Godel 1931] • Longstanding issues such as the nonexistence of large cardinals and the nonexistence of nonstandard models [Cohen 1966] are resolved in the theory Ordinals. The theory Ordinals is based on intensional types as opposed to extensional sets of ZFC. Using intensional types together with strongly-typed ordinal induction is key to proving that there is just one model of the theory Ordinals up to a unique isomorphism.

This article is motivated by the open question of whether every integrally closed domain is an intersection of finitely many Prufer overrings. We survey the work of Dan Anderson and David Anderson on this and related questions, and we show that an integrally closed domain that is a finitely generated algebra over a Dedekind domain or a field is a finite intersection of Dedekind overrings. We also discuss how recent work on the intersections of valuation rings has implications for this question.

We give a definition of a class of Dedekind domains which includes the rings of integers of global fields and give a proof that all rings in this class have finite ideal class group. We also prove that this class coincides with the class of rings of integers of global fields., Comment: 11 pages

In this paper we study the extension of a cumulative distribution function (defined from a probability measure) on a separable linearly ordered topological space to its Dedekind-MacNeille completion (in this case, also compactification). We show that the pseudo-inverse of a distribution function on a separable linearly ordered topological space is naturally defined from [ 0 , 1 ] to the Dedekind-MacNeille completion, so we can prove some results that lead us to use the pseudo-inverse to generate a random sample in the space, just as in the classical case. Finally, we prove that there is an equivalence between probability measures and distribution functions defined on the Dedekind-MacNeille completion of a separable linearly ordered topological space. In particular, there is such an equivalence for a compact separable linearly ordered topological space.

Let [Formula: see text] be an algebraic number field with [Formula: see text] an algebraic integer having minimal polynomial [Formula: see text] over the field [Formula: see text] of rational numbers and [Formula: see text] be the ring of algebraic integers of [Formula: see text]. For a fixed prime number [Formula: see text], let [Formula: see text] be the factorization of [Formula: see text] modulo [Formula: see text] as a product of powers of distinct irreducible polynomials over [Formula: see text] with [Formula: see text] monic. In 1878, Dedekind proved a significant result known as Dedekind Criterion which says that the prime number [Formula: see text] does not divide the index [Formula: see text] if and only if [Formula: see text] is coprime with [Formula: see text] where [Formula: see text]. This criterion has been widely used and generalized. In this paper, a simple proof of Generalized Dedekind Criterion [S. K. Khanduja and M. Kumar, On Dedekind criterion and simple extensions of valuation rings, Comm. Algebra 38 (2010) 684–696] using elementary valuation theory is given.

Let R be a Dedekind domain and G a split reductive group, i.e. a Chevalley–Demazure group scheme, of rank ≥ 2 {\geq 2} . We prove that G ⁢ ( R ⁢ [ x 1 , … , x n ] ) = G ⁢ ( R ) ⁢ E ⁢ ( R ⁢ [ x 1 , … , x n ] ) for any ⁢ n ≥ 1 . G(R[x_{1},\ldots,x_{n}])=G(R)E(R[x_{1},\ldots,x_{n}])\quad\text{for any}\ n% \geq 1. In particular, this extends to orthogonal groups the corresponding results of A. Suslin and F. Grunewald, J. Mennicke and L. Vaserstein for G = SL N , Sp 2 ⁢ N {G=\mathrm{SL}_{N},\mathrm{Sp}_{2N}} . We also deduce some corollaries of the above result for regular rings R of higher dimension and discrete Hodge algebras over R.

We classify all indecomposable quasi comultiplication modules over pullback of two Dedekind domains. We extend the de nitions and the results of comultiplication modules over pullback rings to a more general quasi comultiplication modules case.

We employ the slice spectral sequence, the motivic Steenrod algebra, and Voevodsky's solutions of the Milnor and Bloch-Kato conjectures to calculate the hermitian $K$-groups of rings of integers in number fields. Moreover, we relate the orders of these groups to special values of Dedekind $\zeta$-functions for totally real abelian number fields. Our methods apply more readily to the examples of algebraic $K$-theory and higher Witt-theory, and give a complete set of invariants for quadratic forms over rings of integers in number fields., Comment: 64 pages