Ideal Theorem Research Articles

On ${\text{L}}^2 $ Sufficient Conditions and the Gradient Projection Method for Optimal Control Problems

${\text{L}}^2 $-local convergence and active constraint identification theorems are proved for gradient projection iterates in sets of ${\text{L}}^\infty $ functions on $[0,1]$ with range in a polyhedron $U \subset \mathbb{R}^m $. These theorems extend earlier results for $U = [0,\infty ) \subset \mathbb{R}^1 $ and are based on an infinite-dimensional variant of the Karush–Kuhn–Tucker second-order sufficient conditions in polyhedral subsets of $\mathbb{R}^n $. The new sufficient conditions and convergence results proved here are directly applicable to continuous-time optimal control problems with smooth nonconvex nonquadratic objective functions and Hamiltonians that are quadratic in the control input vector u. In particular, these theorems apply to nonconvex nonquadratic regulator problems with control-linear state equations and vector-valued inputs $u(t)$ satisfying unqualified affine inequality constraints at almost all t in $[0,1]$.

Numerical characterization study of Micrococcaceae associated with lamb spoilage

A computer-assisted characterization of 296 Micrococcaceae isolates obtained from aerobically chill-stored lamb carcasses was carried out using a probability matrix and Bayesian identification theorems, complemented with cluster analysis. Preliminary identification was done with an original probability matrix comprising 37 previously described taxa and 32 tests. Although its statistical quality was adequate, the percentage of identification of field strains to species level was only 70% (96.6% identified with genera). To achieve an improved characterization, cluster analysis was subsequently performed on this group and an additional 26% could be associated with defined species, with five more taxa defined. The combined use of both approaches was judged positive as new identifications and better discrimination could be achieved. The majority of our isolates belonged to the Staphylococcus species group. Many species and groups of staphylococci increased as the spoilage progressed.

A note on the exponential map of a real or p-adic Lie group

where the right-hand side is an absolutely convergent series, involving higher-order brackets of X and Y, and (1.2) is valid for all small X and Y. For an exposition of this and any other facts Soncer3ing real Lie groups see [l]. The reason that (1.1) holds is that if X and Y commute so do tX and tY for real t so that by (1.2), exp’ tX mexp t Y = expt(X + Y) for small t, since all the other terms in the formula are zero. But then, by the identity theorem for real analytic functions, this holds for ali t. Taking t = 1 gives the result. Now, since the log inverts exp locally at the origin, and X and Y are small in (1.2), it follows that an absolutely convergent series of these brackets is also small. Hence

A Lower Bound for the Class Number of a Real Quadratic Field of ERD-Type

AbstractIn this paper, we use the Lagrange neighbour and our equivalence theorem for primitive ideals to obtain lower bounds which are sharper than those given in the literature for class numbers of real quadratic fields in general, but applied to greatest advantage when d is of ERD type.

On the Gradient Projection Method for Optimal Control Problems with Nonnegative $\mathcal{L}^2 $ Inputs

Local convergence and active constraint identification theorems are proved for gradient-projection iterates in the cone of nonnegative ${\cal L}^2$ functions on $[0,1]$. The theorems are based on recently established infinite-dimensional extensions of the Kuhn-Tucker sufficient conditions and are directly applicable to a large class of continuous-time optimal control problems with smooth nonconvex nonquadratic objective fractions and Hamiltonians that are quadratic in the control input $u$.

The prüfer rings that are endomorphism rings of artinian modules

In this paper we characterize the (commutative) Priifer rings that can be realized as endomorphism rings of artinian modules over arbitrary associative rings with identity (Theorem 4.7). This characterization is obtained by determining the structure of ∑-pure-injective modules over Prufer rings (Theorems 3.4 and 3.5)

Fuzzy inference based on families of alpha -level sets

A fuzzy-inference method in which fuzzy sets are defined by the families of their alpha -level sets, based on the resolution identity theorem, is proposed. It has the following advantages over conventional methods: (1) it studies the characteristics of fuzzy inference, in particular the input-output relations of fuzzy inference; (2) it provides fast inference operations and requires less memory capacity; (3) it easily interfaces with two-valued logic; and (4) it effectively matches with systems that include fuzzy-set operations based on the extension principle. Fuzzy sets defined by the families of their alpha -level sets are compared with those defined by membership functions in terms of processing time and required memory capacity in fuzzy logic operations. The fuzzy inference method is then derived, and important propositions of fuzzy-inference operations are proved. Some examples of inference by the proposed method are presented, and fuzzy-inference characteristics and computational efficiency for alpha -level-set-based fuzzy inference are considered. >

A generalization of theorems of Eidelheit and Carleman concerning approximation and interpolation

We prove necessary and sufficient conditions for linear operators to approximate and interpolate unbounded continuous functions on certain subsets U ⊆ (−∞, ∞). The main application of our general theory is to simultaneous asymptotic approximation and interpolation by function series. Special cases of our results are a sharpened version of a theorem of Eidelheit for the solubility of infinite systems of linear equations and a generalization of a theorem of Carleman concerning the asymptotic approximation and interpolation of continuous functions by entire functions on the real axis. Moreover we can apply our general theorems to a moment problem of Pólya and to asymptotic approximation and interpolation by Dirichlet series. Our general approach to such problems is based on the use of certain complete approximation systems and on an essential identity theorem of functional analysis concerning approximations in normed linear spaces with certain additional restrictions by seminorms.

Compatibility Conditions and the Manifold of Solutions of some Nonlinear Cauchy-Riemann System in Several Complex Variables

The system of differential equations Is studied. On the assumption that the system admits certain class of solutions, necessary conditions of compatibility are deduced, and by introducing some changes of variables the system is reduced to a single differential equation, then conversely by solving it, the formula giving all local solutions of the system is infered. The validity of the factorization theorem and the identity theorem for solutions in the class is also treated.

On Bayer's deformation and the associativity formula

A usual technique in computational commutative algebra is to reduce the computation of invariants of ideals I⊆ k[ X] where k is a field, to the computation of the corresponding invariant of the monomial ideal M( I) which is associated to I(w.r.t. some term ordering) by means of Gröbner bases, via Buchberger's algorithm. An early instance of this technique is Macaulay's theorem: if I is homogeneous then: dim k ( I n )=dim k ( M( I) n ) for all n. In this paper we give a general version of Macaulay's theorem for ideals in polynomial rings over a noetherian ring R and any additive function λ. As a consequence, the computation of λ( I) for any ideal I can be reduced to the computation of λ( M( I)), for the associated monomial ideal. The result above is obtained by a study of the main properties of Bayer's deformation.

A Proof of van Douwen's Right Ideal Theorem

A Proof of van Douwen's Right Ideal Theorem

An identity theorem for multi-relator groups

In this paper, the Identity Theorem of R. C. Lyndon and the Freiheitssatz of W. Magnus are extended to a large class of multi-relator groups. Included are the two-relator groups introduced by I. L. Anshel in her thesis, where the Freiheitssatz was proved for those groups. The Identity Theorem provides cohomology computations and a classification of finite subgroups. The methods are geometric; technical tools include the original theorems of Magnus and Lyndon, as well as an amalgamation technique due to J. H. C. Whitehead.

Integral extensions of noncommutative rings

In this paper we study integral extensions of noncommutative rings. To begin, we prove that finite subnormalizing extensions are integral. This is done by proving a generalization of the Pare-Schelter result that a matrix ring is integral over the coefficient ring. Our methods are similar to those of Lorenz and Passman, who showed that finite normalizing extensions are integral. As corollaries we note that the (twisted) smash product over the restricted enveloping algebra of a finite dimensional restricted Lie algebra is integral over the coefficient ring and then prove a Going Up theorem for prime ideals in these ring extensions.

A proof of van Douwen’s right ideal theorem

In 1979 Eric K. van Douwen announced a powerful theorem about the Stone-Čech compactification of a discrete semigroup which he called The Right Ideal Theorem. Its proof, however, was lost with his untimely death. In this paper we present a proof of the theorem and a derivation of some of its corollaries.

An identity theorem for logarithmic potentials

An identity theorem for logarithmic potentials

A nil implies nilpotent theorem for left ideals

A nil implies nilpotent theorem for left ideals

Weighted Subspaces of Hardy Spaces

A function f in Hp on the unit disc U of the complex plane has the uniform growthWe consider in this paper a subspace of Hp with better uniform growthFor the previous results on see [5, 6, 7]. We start with proving an inequality on Hp which is related to the Hardy-Stein identity (Theorem 2.1) in Section 2. This is applied in the subsequent section to prove some space imbedding theorems related to (Theorems 3.1 and 3.5). These theorems have some known theorems as their corollaries. Finally we prove some coefficient relations on in the last section.The authors wish to thank Professor Patrick Ahern for the helpful conversations during his visit to Korea. Actually he suggested to the first author the possibility of Theorem 2.1 some years ago.

On convergence and quasiconformality of complex planar spline interpolants

There appear to be two main approaches for developing complex splines. One of these, which has been in use for quite some time, consists in defining splines on the boundary of a given region which are then extended into the interior by Cauchy's integral formula (see e.g. [1]). The other approach, which is of a more recent origin, is motivated in spirit by the theory of finite elements (see e.g. [10], p. 320) and is contained in [8] and [9]. Observing that the foregoing extension into the interior is not easy to execute numerically, certain continuous piecewise non-holomorphic functions, called complex planar splines have been studied in [8] and [9]. The choice of non-holomorphic functions is justified, since if we take the pieces to be holomorphic functions like polynomials, then by the well known identity theorem ([5], p. 132, theorem 60) the continuity of such a piecewise function implies that all the pieces represent just one holomorphic function. Thus, we shall consider polynomials in z and its conjugate z¯ of the formwhich are generally non-holomorphic functions. The numberwill be called the degree of q. For simplicity we also write q(z) for q(z, z¯).

Commutators of homogeneous multiplier operators

Denote, where ω is a homogeneous function of degree ι}, R -α m is the n-fold composition of the Taylor series operator, m=(m 1 ,, m n )∈Z n , Z is the set of non-negative integers, α∈(R k ) n . Define where α=(α 1 ,…, α n ), α 1 , f∈(?)(R K ), (α)=(α 1 )…(α n ), α=sum from i=1 to n α 1 , dα=dα 1 …dα n . Our main results are as follows:1. Several identity theorems between the class of operators (α, f) and the class of multilinear singular integral operators are proved.2. As an application, some boundedness conclusions of the following operators are given:

An isomorphism theorem for standard ideals in lattices

Pro@ T is a standard and therefore T is s tandard in H v T . If S ' = ( H A S ) v T is s tandard in H v T then by the Second Isomorphism Theo rem S ' /T is s tandard in ( H v T ) / T which is the first assertion of the theorem. By the Second Isomorphism Theo rem it follows that ( ( H v T ) / T ) / ( S ' / T ) ~ ( H v T ) / S ' . From H v T ~ _ S ' ~ _ T we get H v T = H v S ' and consequently (H v T ) /S '= ( H v S')/S'. By the First I somorphism Theo rem the f ight-hand side is isomorphic to HI(HAS ' ) . Using the fact that T is s tandard we have H A S ' = H A ((HA S) v T)) = ( H A S) v ( H A T) = H A S, i.e. H I ( H A S') = H I ( H A S). Again f rom the First I somorphism Theo rem we conclude H I ( H A S ) ~ ( H v S ) / S , that is, (H v S')/S' ~ (H v S)/S. Thus ( (H v T) /T) / (S ' /T ) ~(H v S)/S, i.e. (*) holds. W e remark that in [1, p. 90 and pp. 252-253] one finds a proof of a common generalization of both isomorphism theorems for groups. This result and the preceding theorem give rise to pose the following