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Gӧdel’s incompleteness theorem asserts that if formal arithmetic is consistent then there exists an arithmetic statement such that neither the statement nor its negation can be deduced from the axioms of formal arithmetic. Previously [3], it was proved that formal arithmetic remains incomplete if, instead of the set of all arithmetic statements, we consider any set of some class of “almost all” statements (the class of so-called strongly generic subsets). This result is strengthened as follows: formal arithmetic is incomplete for any generic subset of arithmetic statements (i.e., a subset of asymptotic density 1).

Recently, Hasankhani et al. proved that any Felbin-fuzzy inner product space can be imbedded in a complete Felbin-fuzzy inner product space or Felbin-fuzzy Hilbert space. In this paper, it is showed a general result that any classical Hilbert space is a Felbin-fuzzy Hilbert space, so it shows that all results in classical Hilbert spaces are immediate consequences of the corresponding results for Felbin-fuzzy Hilbert spaces. Moreover by an example, it is showed that the spectrum of the category of Felbin-fuzzy Hilbert spaces is broader than the category of classical Hilbert spaces. Finally the authors are able to state a transformation theorem from an ascending family of crisp inner product into Felbin-fuzzy inner product that shows how to formulate the recent results.

The objective of this paper is to discuss existence, uniqueness and regularity issues of minimizers of one dimensional variational problems in Hilbert spaces. We obtain existence of C2 local minimizers and prove that the value function of an optimal control problem solves corresponding Hamilton-Jacobi equation in a viscosity sense.

Hilbert C ⁎ -modules are the analogues of Hilbert spaces where a C ⁎ -algebra plays the role of the scalar field. With the advent of Kasparovʼs celebrated KK -theory they became a standard tool in the theory of operator algebras. While the elementary properties of Hilbert C ⁎ -modules can be derived basically in parallel to Hilbert space theory the lack of an analogue of the Projection Theorem soon leads to serious obstructions and difficulties. In particular the theory of unbounded operators is notoriously more complicated due to the additional axiom of regularity which is not easy to check. In this paper we present a new criterion for regularity in terms of the Hilbert space localizations of an unbounded operator. We discuss several examples which show that the criterion can easily be checked and that it leads to nontrivial regularity results.

The development of the theory of extensions of Hilbert C*-modules was initiated by Bakic and Guljas. An easy observation shows that, if the underlying C*-algebra extension is commutative and the Hilbert C*-modules are projective of finite type, then the algebraic properties of the corresponding Busby invariant enable one to identify extensions with isometric mappings of the corresponding vector bundles. For the case in which the Hilbert C*-modules are free of rank one, we evaluate the set of extensions in topological terms.

In this paper, we will study the class of Hilbert algebras with supremum, i.e., Hilbert algebras where the associated order is a join-semilattice. First, we will give a simplified topological duality for Hilbert algebras using sober topological spaces with a basis of open-compact sets satisfying an additional condition. Next, we will extend this duality to Hilbert algebras with supremum. We shall prove that the ordered set of all ideals of a Hilbert algebra with supremum has a lattice structure. We will also see that in this lattice, it is possible to define an implication, but the resulting structure is neither a Heyting algebra nor an implicative semilattice. Finally, we will give a dual description of the lattice of ideals of a Hilbert algebra with supremum.

Deligne proved that the Hilbert series of all Artin monoids are rational functions. We give an algorithm to compute the Hilbert series of the braid monoids [Formula: see text]. We also show that the Hilbert series of the positive words in [Formula: see text] with a given prefix are rational functions.

The aim of this article is to establish some interesting inequalities involving arithmetic functions.

In this paper we give some new results for g-frames in Hilbert C∗-modules and then we introduce a bounded A-linear operator L, by means of this operator L we character the properties of the g-frames and g-Riesz basis in Hilbert C∗-modules. Finally, we establish some important equalities and inequalities for frames and g-frames in Hilbert C∗-modules.

Inspired by examples of small Hilbert matrices, the author proves a property of symmetric totally positive Cauchy matrices, called AT-property, and consequences for the Hilbert matrix.

In this paper we introduce a special kind of ordered topological spaces, called Hilbert spaces. We prove that the category of Hilbert algebras with semi-homomorphisms is dually equivalent to the category of Hilbert spaces with certain relations. We restrict this result to give a duality for the category of Hilbert algebras with homomorphisms. We apply these results to prove that the lattice of the deductive systems of a Hilbert algebra and the lattice of open subsets of its dual Hilbert space, are isomorphic. We explore how this duality is related to the duality given in [6] for finite Hilbert algebras, and with the topological duality developed in [7] for Tarski algebras.

Bounds for the Castelnuovo-Mumford regularity and Hilbert coefficients are given in terms of the arithmetic degree (if the ring is reduced) or in terms of the defining degrees. From this it follows that there exists only a finite number of Hilbert functions associated to reduced algebras over an algebraically closed field with a given arithmetic degree and dimension. A good bound is also given for the Castelnuovo-Mumford regularity of initial ideals which depends neither on term orders nor on the coordinates, and holds for any field., Comment: To appear in Trans. Amer. Math. Soc

1. INTRODUCTION. In this article we explore a variety of pleasing connections between analysis, number theory, and operator theory, while revisiting a number of beautiful inequalities originating with Hilbert, Hardy and others. We shall first discuss the aforementioned Hilbert inequality [14], [18] and then apply it to various multiple zeta values. In consequence we obtain the norm of the classical Hilbert matrix, in the process illustrating the interplay of numerical and symbolic computation with classical mathematics. 2. HILBERT'S (EASIER) INEQUALITY. The inequality in question is

In November and December 1915, Hilbert presented two communications to the Gottingen Academy of Sciences under the common title ‘The Foundations of Physics’. Versions of each eventually appeared in the Nachrichten of the Academy. Hilbert's first communication has received significant reconsideration in recent years, following the discovery of printer's proofs of this paper, dated 6 December 1915. The focus has been primarily on the ‘priority dispute’ over the Einstein field equations. Our contention, in contrast, is that the discovery of the December proofs makes it possible to see the thematic linkage between the material that Hilbert cut from the published version of the first communication and the content of the second, as published in 1917. The latter has been largely either disregarded or misinterpreted, and our aim is to show that (a) Hilbert's two communications should be regarded as part of a wider research program within the overarching framework of ‘the axiomatic method’ (as Hilbert expressly stated was the case), and (b) the second communication is a fine and coherent piece of work within this framework, whose principal aim is to address an apparent tension between general invariance and causality (in the precise sense of Cauchy determination), pinpointed in Theorem I of the first communication. This is not the same problem as that found in Einstein's ‘hole argument’—something that, we argue, never confused Hilbert.

In the context of Hilbert’s irreducibility theorem, it is an open question whether there exists a bound for the least hilbertian specialization in N that is polynomial in the degree d and the logarithmic height log(H) of the polynomial P (T,Y ) in question. A positive answer would be useful, notably for algorithmic applications. We obtain a polynomial bound in log(H) and dHi(P) where Hi(P) — the Hilbert index of P — is a pure group-theoretical invariant we define and which we show to be absolutely bounded for many classes of polynomials. We also discuss further questions related to effectiveness in Hilbert’s irreducibility theorem. 2000 MSC. Primary 12E25, 14G05 ; Secondary 11C08, 12E05

In this section we shall be interested in discussing proofs, generalizations and geometric formulations of the so-called Hilbert Irreducibility Theorem (HIT in the sequel).

There are many beautiful constructions of the Hilbert scheme of closed subschemes of projective schemes. In many cases where Hilbert schemes are involved, it suffices to know they exist. On the other hand, there are many situations when it is crucial to have an explicit description of the Hilbert schemes. In this article we give a simple construction of Hilbert schemes of points, under general circumstances, that provides such a description. In addition to being useful for computations the construction is short, elementary, and explicit. Our methods rely on simple algebraic constructions. It gives explicit expressions of members of an affine covering of the Hilbert schemes, involving few variables satisfying natural equations. In particular, it explains the connections with commuting schemes of matrices. We also give some examples showing how our method can be used.

International audience; We generalize to Hilbert spaces a theorem of Glaeser concerning minimal Lipschitz extensions of m-jets of one variable with range in R. The results contained in this paper can be seen as a small contribution to the general problem of the minimal Lipschitz extensions from m-jets for a Hilbert space to another Hilbert space.

In this work we shall give a characterization of the Hilbert algebras given by the order and we will prove a duality for finite Hilbert algebras by means of finite ordered sets endowed with a distinguished set of subsets. We will also study the case when the finite Hilbert algebras are join-semilattices or meet-semilattices relative to the natural order defined by the implication. Finally we will prove that Hilbert do not admit a natural duality.

A finite Hilbert transformation associated with a polynomial is the analogue of a Hilbert transformation associated with an entire function which is a generalization of the classical Hilbert transformation. The weighted Hilbert inequality, which has applications in analytic number theory, is closely related to the finite Hilbert transformation associated with a polynomial. In this note, we study a relation between the finite Hilbert transformation and the weighted Hilbert’s inequality. A question is posed about the finite Hilbert transformation, of which an affirmative answer implies the weighted Hilbert inequality.

A finite Hilbert transformation associated with a polynomial is the analogue of a Hilbert transformation associated with an entire function, which is a generalization of the classical Hilbert transformation (cf. [4]). Hilbert's inequality, which has applications in analytic number theory (see Vaaler [11]), is closely related to the finite Hilbert transformation associated with a polynomial. In this paper, an explicit construction is given for the finite Hilbert transformation.

A generalization of the Hilbert basis theorem in the geometric setting is proposed. It asserts that, for any well-describable (in a certain sense) family of polynomials, there exists a number C such that if P is an everywhere dense (in a certain sense) subfamily of this family, a is an arbitrary point, and the first C polynomials in any sequence from P vanish at the point a, then all polynomials from P vanish at a.

This paper presents an algorithm for computing numerical evidence for a conjecture whose validity is predicted by the requirement that the equivariant Tamagawa number conjectures for Tate motives as formulated by Burns and Flach are compatible with the functional equation of the Artin L-series. The algorithm includes methods for the computation of Fitting ideals and projective lattices over the integral group ring.

Hilbert modules over semigroups are essentially the same thing as uniformly bounded representations of semigroups on Hilbert spaces. We develop extension theory of Hilbert modules over semigroups and mainly apply them to study questions from operator theory and representations of nonamenable groups.

We develop an “arithmetic” of -type vectors in order to discuss the relationship between -type vectors and Hilbert functions. We then use this arithmetic to determine for which Hilbert functions it is true that every set of points with that Hilbert function has the Cayley-Bacharach Property. As a second application, we determine the degree of every point in a -configuration.

This work has focused on explicitly determining in what degree the maximal growth of the Hilbert function of a -algebra begins to persist (in the sense of Gotzmann). We have shown that this persistence index is solely dependent upon the a-invariant of and the coefficients of the Hilbert polynomial of . We have also established a relatively efficient algorithm for computing the persistence index. By giving a structured decomposition of the persistence index, we have been able to prove properties of Betti numbers of lex-segment ideals that extend the Bigatti-Hulett-Pardue result of 1993.

By introducing three parameters A , B and λ , we give a generalization of Hilbert’s inequality with a best constant factor and some particular results. Mathematics subject classification (2000): 26D15.

In this paper we study the problem of dilating unital completely positive (CP) semigroups (quantum dynamical semigroups) to weak Markov flows and then to semigroups of endomorphisms (E0-semigroups) using the language of Hilbert modules. This is a very effective, representation free approach to dilation. In this way we are able to identify the right algebra (maximal in some sense) for endomorphisms to act. We are led inevitably to the notion of tensor product systems of Hilbert modules and units for them, generalizing Arveson's notions for Hilbert spaces. In the course of our investigations we are not only able to give new natural and transparent proofs of well-known facts for semigroups on [Formula: see text], but also extend the results immediately to much more general setups. For instance, Arveson classifies E0-semigroups on [Formula: see text] up to cocycle conjugacy by product systems of Hilbert spaces.5 We find that conservative CP-semigroups on arbitrary unital C*-algebras are classified up to cocycle conjugacy by product systems of Hilbert modules. Looking at other generalizations, it turns out that the role played by E0-semigroups on [Formula: see text] in dilation theory for CP-semigroups on [Formula: see text] is now played by E0-semigroups on [Formula: see text], the full algebra of adjointable operators on a Hilbert module E. We have CP-semigroup versions of many results proved by Paschke27 for CP maps.

We give explicit constructions of semigroups and operators with particular properties. First we build a bounded C0-semigroup which is invertible and which is not similar to a semigroup of contractions. Afterwards we exhibit operators which admit bounded imaginary powers of angle ω > 0 on a Hilbert space but which do not admit a bounded functional calculus on the sector of angle ω. (This gives the limit of McIntosh's fundamental result.) Finally we build, in the 2-dimensional Hilbert space, an operator which is not the negative generator of a semigroup of contractions, although its imaginary powers are bounded by eπ|s|/2.

It is a well-known fact that if K is a field, then the Hilbert series of a quotient of the polynomial ring by a homogeneous ideal is of the form we call the polynomial q(t) the Hilbert numerator of the quotient algebra. We will generalize this concept to a class of non-finitely generated, graded, commuta-tive algebras, which are endowed with a surjective “co-filtration” of finitely generated algebras. Then, although the Hilbert series themselves can not be defined (since the sub-vector-spaces involved have infinite dimension), we get a sequence of Hilbert numerators qn (t), which we show converge to a power series in Z[[t]]. This power series we call the (generalized) Hilbert numerator of the non-finitely generated algebra. The question of determining when this power series is in fact a polynomial is the topic of the last part of this article. We show that quotients of the ring R' by homogeneous ideals that are generated by finitely many monomials have polynomial Hilbert numerator, as have quotients of R'...

The main goal of this paper is to generalize the criterion for complete intersections well known in commutative algebra to the case of an arbitrary variety of PI-(super)algebras. As a consequence, formulae are obtained for computing the Hilbert series of the even and odd parts of relatively free superalgebras and free special Jordan superalgebras.

We introduce a natural class of functions, the {\em pseudomultipliers}, associated with a general Hilbert function space, prove an extension theorem which justifies the definition, give numerous examples and establish the nature of the 1-pseudomultipliers of Hilbert spaces of analytic functions under mild hypotheses.

Let M \mathcal {M} be a real W ∗ W^{*} -algebra of J J -real bounded operators containing no central summand of type I 2 I_{2} in a complex Hilbert space H H with conjugation J J . Denote by P P the quantum logic of all J J -orthogonal projections in the von Neumann algebra N = M + i M {\mathcal {N}}={\mathcal {M}}+ i{\mathcal {M}} . Let μ : P → [ 0 , 1 ] \mu :P\rightarrow [0,1] be a probability measure. It is shown that N \mathcal {N} contains a finite central summand and there exists a normal finite trace τ \tau on N \mathcal {N} such that μ ( p ) = τ ( p ) \mu (p)=\tau (p) , ∀ p ∈ P \forall p\in P .

It is shown that in a Hilbert space the only infinite equidistant systems of points are those obtained by translation and scaling from (in the complex case: almost) orthonormal systems.

Recently, Rashidi-Kouchi et al. devoted their efforts to giving analogs of the Casazza–Christensen general perturbation theorem of Hilbert space frames for the case of g-frames in Hilbert C*-modules. We found, however, that in the proof they did not obtain the lower bound inequality of g-frames in Hilbert C*-modules. In this paper, we establish an equivalent definition of g-frames in Hilbert C*-modules, using it we present a complete proof for the stability result given by them.

A new proof of Durier's Theorem [3] concerning a characterization of Hilbert spaces in terms of the Fermat-Weber location problem will be presented.

The $T$-graph of a multigraded Hilbert scheme records the zero and one-dimensional orbits of the $T=(K^*)^n$ action on the Hilbert scheme induced from the $T$-action on $\mathbb{A}^n$. It has vertices the $T$-fixed points, and edges the onedimensional $T$-orbits. We give a combinatorial necessary condition for the existence of an edge between two vertices in this graph. For the Hilbert scheme of points in the plane, we give an explicit combinatorial description of the equations defining the scheme parameterizing all one-dimensional torus orbits whose closures contain two given monomial ideals. For this Hilbert scheme we show that the $T$-graph depends on the ground field, resolving a question of Altmann and Sturmfels., Oberwolfach Preprints;2011,34

Let $M$ be a finite module and let $I$ be an arbitrary ideal over a Noetherian local ring. We define the generalized Hilbert function of $I$ on $M$ using the 0th local cohomology functor. We show that our definition re-conciliates with that of Ciuperc$\breve{{\rm a}}$. By generalizing Singh's formula (which holds in the case of $\lambda(M/IM), Comment: arXiv admin note: text overlap with arXiv:1101.2281

Hilbert's programme is shown to have been inspired in part by what we can call Kronecker's programme in the foundations of an arithmetic theory of algebraic quantities.

In this paper we extend "up to powers" Hilbert’s Theorem 90 to arbitrary finite Galois extensions. In the case of algebraic number fields with class number equal to 1 1 , we completely determine the kernel and image of the norm map.

We discuss the Hilbert program for the axiomatization of physics in the context of what Hilbert and von Neumann came to call the analytical apparatus and its conditions of reality. We suggest that the idea of a physical logic is the basis for a physical mathematics and we use quantum mechanics as a paradigm case for axiomatics in the sense of Hilbert. Finite probability theory requires nite derivations in the measurement theory of QM and we give a polynomial formulation of local complementation for the metric induced on the topology of the Hilbert space. The conclusion hints at a constructivist physics.