scholarly journals A New Development of the Classical Single Ladder Problem via Converting the Ladder to a Staircase

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 339
Author(s):  
Ralph Høibakk ◽  
Dag Lukkassen ◽  
Annette Meidell ◽  
Lars-Erik Persson
Keyword(s):  
Algebraic Equation ◽  
Rational Solutions ◽  
Main Result Theorem ◽  
New Development ◽  
Result Theorem

Our purpose is to shed some new light on problems arising from a study of the classical Single Ladder Problem (SLP). The basic idea is to convert the SLP to a corresponding Single Staircase Problem. The main result (Theorem 1) shows that this idea works fine and new results can be obtained by just calculating rational solutions of an algebraic equation. Some examples of such concrete calculations are given and examples of new results are also given. In particular, we derive a number of integer SLPs with congruent ladders, where a set of rectangular boxes with integer sides constitutes a staircase along a common ladder. Finally, the case with a regular staircase along a given ladder is investigated and illustrated with concrete examples.

scholarly journals A note on the paper "Optimality conditions for nonsmooth interval-valued and multiobjective semi-infinite programming"

2020 ◽  
Author(s):  
Nazih Abderrazzak Gadhi ◽  
Aissam Ichatouhane
Keyword(s):  
Optimality Conditions ◽  
Programming Problem ◽  
Optimality Condition ◽  
Short Proof ◽  
Necessary Optimality Condition ◽  
Main Result Theorem ◽  
Correct Formulation ◽  
Result Theorem ◽  
Infinite Programming ◽  
Interval Valued

A nonsmooth semi-infinite interval-valued vector programming problem is solved in the paper by Jennane et all. (RAIRO-Oper. Res. doi: 10.1051/ro/2020066, 2020). The necessary optimality condition obtained by the authors, as well as its proof, is false. Some counterexamples are given to refute some results on which the main result (Theorem 4.5) is based. For the convinience of the reader, we correct the faulty in those results, propose a correct formulation of Theorem 4.5 and give also a short proof.

Closure Theorem for Analytic Subgroups of Real Lie Groups

1976 ◽  
Vol 19 (4) ◽  
pp. 435-439 ◽  
Author(s):  
D. Ž. Djoković
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Closed Subgroup ◽  
Main Result Theorem ◽  
Closure Theorem ◽  
Result Theorem ◽  
Analytic Subgroup

Let G be a real Lie group, A a closed subgroup of G and B an analytic subgroup of G. Assume that B normalizes A and that AB is closed in G. Then our main result (Theorem 1) asserts that .This result generalizes Lemma 2 in the paper [4], G. Hochschild has pointed out to me that the proof of that lemma given in [4] is not complete but that it can be easily completed.

scholarly journals On radicals of finite near-rings

1984 ◽  
Vol 27 (3) ◽  
pp. 247-259 ◽  
Author(s):  
K. Kaarli
Keyword(s):  
Main Result Theorem ◽  
Result Theorem

In this paper the study of radicals of finite near-rings is initiated. The main result (Theorem 4.3) gives a description of hereditary radicals having hereditary semisimple classes too. Also it is shown that there exist non-hereditary radicals having hereditary semisimple classes.

Absorption Probabilities for Sums of Random Variables Defined on a Finite Markov Chain

1962 ◽  
Vol 58 (2) ◽  
pp. 286-298 ◽  
Author(s):  
H. D. Miller
Keyword(s):  
Markov Chain ◽  
Asymptotic Expression ◽  
Random Variables ◽  
Partial Sums ◽  
Finite Markov Chain ◽  
Initial State ◽  
Main Result Theorem ◽  
Sums Of Random Variables ◽  
Result Theorem ◽  
Image Position

SummaryThis paper is essentially a continuation of the previous one (5) and the notation established therein will be freely repeated. The sequence {ξr} of random variables is defined on a positively regular finite Markov chain {kr} as in (5) and the partial sums and are considered. Let ζn be the first positive ζr and let πjk(y), the ‘ruin’ function or absorption probability, be defined by The main result (Theorem 1) is an asymptotic expression for πjk(y) for large y in the case when , the expectation of ξ1 being computed under the unique stationary distribution for k0, the initial state of the chain, and unconditional on k1.

scholarly journals Independent inner functions in the classical domains

1987 ◽  
Vol 29 (2) ◽  
pp. 229-236
Author(s):  
Tomasz M. Wolniewicz
Keyword(s):  
Unit Ball ◽  
Hardy Spaces ◽  
Isometric Embedding ◽  
Symmetric Domain ◽  
Type I ◽  
Bounded Symmetric Domains ◽  
Inner Functions ◽  
Main Result Theorem ◽  
Complemented Subspace ◽  
Result Theorem

Let Bn denote the unit ball and Un the unit polydisc in Cn. In this paper we consider questions concerned with inner functions and embeddings of Hardy spaces over bounded symmetric domains. The main result (Theorem 2) states that for a classical symmetric domain D of type I and rank(D) = s, there exists an isometric embedding of Hl(Us) onto a complemented subspace of Hl(D). This should be compared with results of Wojtaszczyk [13] and Bourgain [3, 4] which state that H1(Bn) is isomorphic to Hl(U) while for n>m, Hl(Un) cannot be isomorphically embedded onto a complemented subspace of H1(Um). Since balls are the only bounded symmetric domains of rank 1 and they are of type I, Theorem 2 shows that if rank(D1) = 1, rank(D2) > 1 then H1(D1) is not isomorphic to H1(D2). It is natural to expect this to hold always when rank(D1 ≠ rank(D2) but unfortunately we were not able to prove this.

Choice principles, the bar rule and autonomously iterated comprehension schemes in analysis

1983 ◽  
Vol 48 (1) ◽  
pp. 63-70 ◽  
Author(s):  
S. Feferman ◽  
G. Jäger
Keyword(s):  
General Result ◽  
Recursive Relation ◽  
Transfinite Induction ◽  
Conservative Extension ◽  
Main Result Theorem ◽  
Functional Interpretation ◽  
Result Theorem ◽  
Primitive Recursive ◽  
Comprehension Axiom

In [10] Friedman showed that (-AC) is a conservative extension of (-CA)<ε0for-sentences wherei= min(n+ 2, 4), i.e.,i= 2, 3, 4 forn= 0, 1, 2 +m. Feferman [5], [7] and Tait [11], [12] reobtained this result forn= 0, 1 and even with (-DC) instead of (-AC). Feferman and Sieg established in [9] the conservativeness of (-DC) over (-CA)<ε0for-sentences (ias above) for alln. In each paper, different methods of proof have been used. In particular, Feferman and Sieg showed how to apply familiar proof-theoretical techniques by passing through languages with Skolem functionals.In this paper we study the same choice principles in the presence of theBar Rule(BR), which permits one to infer the scheme of transfinite induction on a primitive recursive relation ≺ when it has been proved that ≺ is wellfounded. The main result (Theorem 1 below) characterizes (-DC) + (BR) as a conservative extension of a system of the autonomously iterated-comprehension axiom for-sentences (idepending onnas above). Forn= 0 this has been proved by Feferman in the form that (-DC) + (BR) is a conservative extension of (-CA)<Γ0; this was first done in [8] by use of the Gödel functional interpretation for the stronger systemZω+μ+ (QF-AC) + (BR) and then more recently by the simpler methods of [9]. Jäger showed how the latter methods could also be used to obtain the general result of Theorem 1 below.

Corrigendum: Spectral properties of two parameter eigenvalue problems II

1990 ◽  
Vol 115 (1-2) ◽  
pp. 87-90 ◽  
Author(s):  
Paul Binding ◽  
Patrick J. Browne ◽  
R. H. Picard
Keyword(s):  
Spectral Properties ◽  
Eigenvalue Problems ◽  
Main Result Theorem ◽  
Result Theorem ◽  
Two Parameter

SynopsisThere are some mistakes in [1, Section 4], and since the main result, Theorem 4.4, is central to the theory and has already been applied in various contexts, we felt it advisable to give a complete statement and proof. The applications of vTheorem 4.4 made to date have fortunately been in situations where the results are correct. For convenience, we restate our notation.

The connection between the universal enveloping C*-algebra and the universal enveloping von Neumann algebra of a JW-algebra

1992 ◽  
Vol 112 (3) ◽  
pp. 575-579 ◽  
Author(s):  
Fatmah B. Jamjoom
Keyword(s):  
Von Neumann Algebra ◽  
Main Result Theorem ◽  
Von Neumann ◽  
C Algebra ◽  
Result Theorem ◽  
The Relationship

AbstractThis article aims to study the relationship between the universal enveloping C*-algebra C*(M) and the universal enveloping von Neumann algebra W*(M), when M is a JW-algebra. In our main result (Theorem 2·7) we show that C*(M) can be realized as the C*-subalgebra of W*(M) generated by M.

scholarly journals Effectiveness of transposed inverse sets in faber regions

1983 ◽  
Vol 6 (2) ◽  
pp. 285-295 ◽  
Author(s):  
J. A. Adepoju ◽  
M. Nassif
Keyword(s):  
Lower Bound ◽  
The Other ◽  
Main Result Theorem ◽  
Simple Set ◽  
Other Hand ◽  
Basic Set ◽  
Result Theorem ◽  
The One ◽  
The Given ◽  
Class Of Functions

The effectiveness properties, in Faber regions, of the transposed inverse of a given basic set of polynominals, are investigated in the present paper. A certain inevitable normalizing substitution, is first formulated, to be undergone by the given set to ensure the existence of the transposed inverse in the Faber region. The first main result of the present work (Theorem 2.1), on the one hand, provides a lower bound of the class of functions for which the normalized transposed inverse set is effective in the Faber region. On the other hand, the second main result (Theorem 5.2) asserts the fact that the normalized transposed inverse set of a simple set of polynomials, which is effective in a Faber region, should not necessarily be effective there.

Idempotent-Separating Extensions Of Inverse Semigroups

1969 ◽  
Vol 9 (1-2) ◽  
pp. 211-217 ◽  
Author(s):  
H. D'Alarcao
Keyword(s):  
Inverse Semigroup ◽  
Main Tool ◽  
Inverse Semigroups ◽  
Main Result Theorem ◽  
Result Theorem ◽  
Standard Terminology ◽  
Points Of View ◽  
Definition Of

Extensions of semigroups have been studied from two points of view; ideal extensions and Schreier extension. In this paper another type of extension is considered for the class of inverse semigroups. The main result (Theorem 2) is stated in the form of the classical treatment of Schreier extensions (see e.g.[7]). The motivation for the definition of idempotentseparating extension comes primarily from G. B. Preston's concept of a normal set of subsets of a semigroup [6]. The characterization of such extensions is applied to give another description of bisimple inverse ω-semigroups, which were first described by N. R. Reilly [8]. The main tool used in the proof of Theorem 2 is Preston's characterization of congruences on an inverse semigroup [5]. For the standard terminology used, the reader is referred to [1].

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