A CLASS OF �-CONSERVATIVE MATRICES

By using the concept of ℐ -convergence defined by Kostyrko et al. in 2001, the ℐ -limit superior of real sequences was introduced and the inequality 
 ℐ − lim ⁡ sup ⁡ ( A x ) ≤ ℐ − lim ⁡ sup ⁡ ( x ) for all x ∈ l ∞ was studied by Demirci in 2001. In this paper, we have characterized 
a class of ℐ -conservative matrices by studying some new inequalities related to the ℐ -limit superior.


Introduction
Let ∞ and c be the Banach spaces of bounded and convergent sequence x = (x k ) with the usual supremum norm.Let σ be a one-to-one mapping of N, the set of positive integers, into itself and T : ∞ → ∞ a linear operator defined by Tx = (Tx k ) = (x σ(k) ).An element φ ∈ ∞ , the conjugate space of ∞ , is called an invariant mean or a σ-mean if and only if (i) φ(x) ≥ 0 when the sequence x = (x k ) has x k ≥ 0 for all k, (ii) φ(e) = 1 where e = (1,1,1,...), and (iii) φ(Tx) = φ(x) for all x ∈ ∞ .Let M be the set of all σ-means on ∞ .A sublinear functional P on ∞ is said to generate σ-means if φ ∈ ∞ and φ ≤ P ⇒ φ is a σ-mean, and to dominate σ-means if φ ≤ P for all φ ∈ M, where φ ≤ P means that φ(x) ≤ P(x) for all x ∈ ∞ .
It is shown [8] that the sublinear functional both generates and dominates σ-means, where σ at n. Thus, a σ-mean extends the limit functional onto c in the sense that φ(x) = limx for all x ∈ c [9].Consequently, c ⊂ V σ .By (iii), it is clear that (Tx − x) ∈ Z for x ∈ ∞ , where Z is the set of all σ-convergent sequences with σ-limit zero.
For x ∈ ∞ , we write It is known that V (x) = W(x) on ∞ [8].
Let A = (a nk ) be an infinite matrix of real numbers and x = (x k ) a real sequence such that Ax = (A n (x)) = ( k a nk x k ) exists for each n.Then, the sequence Ax = (A n (x)) is called an A-transform of x.For two sequence spaces E and F, we say that the matrix A maps E into F if Ax exits and belongs to F for each x ∈ E. By (E,F), we denote the set of all matrices which map E into F.
Let E be a subset of N. Natural density δ of E is defined by where the vertical bars indicate the number of elements in the enclosed set.The number sequence x = (x k ) is said to be statistically convergent to the number l if for every ε, δ{k : |x k − l| ≥ ε} = 0 [4].In this case, we write st − limx = l.
A matrix A ∈ (c,c) reg is said to be regular and it is known [1, page 21] that A is regular if and only if A < ∞, lim n a nk = 0 for each k, and lim n k a nk = 1.For a given nonnegative regular matrix A, the number ) is said to be A-statistical convergent to a number s if for every ε > 0, the set {k : |x k − s| ≥ ε} has A-density zero [5].In this case, we write st A − limx = s.By st A , we denote the set of all A-statistically convergent sequences.Let Ꮾ = (Ꮾ i ) = (b nk (i)) be a sequence of infinite matrices.Then, a bounded sequence x is said to be Ꮾ summable to the value l if The matrix Ꮾ is regular [11] if and only if Ꮾ < ∞, lim n b nk (i) = 0 for all k, uniformly in i, and lim n k b nk (i) = 1 uniformly in i, where Ꮾ = sup n,i k |b nk (i)|.For a given nonnegative regular matrix sequence Ꮾ, Kolk [6] introduced the Ꮾ-density of a subset of N as follows.
The number is said to be Ꮾ-density of E if it exists.In the cases Ꮾ = (A) and Ꮾ = (C,1), the Cesàro matrix, the Ꮾ-density reduces to the A-density and natural density, respectively.A sequence x = (x k ) is said to be Ꮾ-statistically convergent [6] to a number s if for every ε > 0, the set {k : |x k − s| ≥ ε} has Ꮾ-density zero.The set of all Ꮾ-statistically convergent sequences is denoted by st Ꮾ . Let Let Ᏽ be a nontrivial ideal in N. A sequence x = (x k ) is said to be Ᏽ-convergent to a number l if for every ε > 0, {k : |x k − l| > ε} ∈ Ᏽ [7].In this case, we write Ᏽ − limx = l.It is clear that a Ᏽ-convergent sequence need not be bounded.Let F Ᏽ (b) be the set of all Ᏽ-convergent and bounded sequences.
Note that in the cases the Ᏽ-convergence is reduced to the statistically convergence, Astatistically convergence, and Ꮾ-statistically convergence, respectively.
An admissible ideal Ᏽ in N is said to satisfy the additive property if for every countable system {Y 1 ,Y 2 ,...} of mutually disjoint sets in Ᏽ, there exist sets Z j ⊆ N ( j = 1,2,...) such that the symmetric differences Y j ∆Z j ( j = 1,2,...) are finite and j Z j ∈ Ᏽ [7].
Demirci [3] has introduced the concepts Ᏽ-limit superior and inferior.For a real number sequence x, let B x and A x denote the sets {b ∈ R : {k : x k > b} / ∈ Ᏽ} and {a ∈ R : {k : x k < a} / ∈ Ᏽ}, respectively, and also let Ᏽ be admissible.Then, For all x ∈ ∞ , the inequality has been studied in [3].
In this paper, we have characterized a class of matrices A ∈ (c,F Ᏽ (b)) by studying some new inequalities related to the Ᏽ-limit superior and limit inferior.

The main results
Firstly, we will begin with the following lemma.
For the necessity of the other conditions it is enough to consider the sequences (e k ) and e, respectively, where (e k ) is the sequence whose kth place is 1 and the others are all zero.Conversely, suppose that the conditions (2.1)-( 2.3) hold.Let x ∈ c and limx = l.Then, for any given ε > 0, there exists a k 0 ∈ N such that |x k − l| ≤ ε whenever k ≥ k 0 .Now, we can write ( By an easy calculation, one can see that So, by applying Ᏽ − lim n in (2.4), we get that This completes the proof.
In what follows, a matrix A ∈ (c,F Ᏽ (b)) is said to be Ᏽ-conservative.In the case A is Ᏽ-conservative, the number is said to be Ᏽ-characteristic of A.
To the proof of our main results, we need two lemmas which can be proved by the same technique used in [2, Lemmas 2.3-2.4 ], respectively.Lemma 2.2.Let A be Ᏽ-conservative and λ > 0.Then, C. C ¸akan and H. C ¸os ¸kun 3447 if and only if (2.9) Then there exists a y ∈ ∞ such that y ≤ 1 and (2.10) Theorem 2.4.Let A be Ᏽ-conservative.Then, for some constant λ ≥ |K Ᏽ | and for all x ∈ ∞ , if and only if which yields (2.12).Conversely, let (2.12) hold and x ∈ ∞ .Then, for any ε > 0, there exits a

.14)
Since A is Ᏽ-conservative and by Lemma 2.2, we obtain which yields (2.11), since ε is arbitrary.
When K Ᏽ > 0 and λ = K Ᏽ , we can conclude from Theorem 2.4 the following result.
Theorem 2.5.Let A be Ᏽ-conservative.Then, for all x ∈ ∞ , if and only if (2.17) In the cases Ᏽ = Ᏽ δᏮ and Ᏽ = Ᏽ δA , we respectively have the following results from Theorem 2.4.
if and only if Then, for some constant λ ≥ |K A | and for all x ∈ ∞ ,
Conversely, suppose that (2.12) and (2.23) hold.Then, for any x ∈ ∞ , we have (2.24).Hence, since ( We also see from the assumption that (2.11) holds.Thus, by taking infimum over z ∈ Z in (2.11), we observe that (2.26) On the other hand, since (2.27) Since W(x) = V (x) for all x ∈ ∞ , we conclude that (2.22) holds and the proof is completed.
When K Ᏽ > 0 and λ = K Ᏽ , we have the following result.
The following results can be derived from Theorem 2.7 for the special cases if and only if (2.19) holds and

31) if and only if (2.21) holds and
Further, for Theorem 2.10.Let A and λ be as in Theorem 2.4.Then, for all x ∈ ∞ , if and only if (2.12) holds and for every Proof.If (2.33) holds, since γ(x) ≤ L(x) and γ(−x) ≤ −l(x), (2.12) follows from Theorem 2.4.To show the necessity of (2.34), for any E ∈ Ᏽ, let us define a matrix D = (d nk ) by d nk = a nk − t k , k ∈ E; otherwise, it equals zero for all n.Then, clearly, D satisfies the conditions of Lemma 2.2, and therefore there exists a y ∈ ∞ such that y ≤ 1 and (2.35) Now, for the same E, we choose the sequence y as Then, since Ᏽ − lim y = γ(y) = γ(−y) = 0, (2.33) implies that which yields (2.34).Conversely, suppose that the conditions of the theorem hold and x ∈ ∞ .Let E 1 = {k : and ε is arbitrary.
When K Ᏽ > 0 and λ = K Ᏽ , we have the following result.
We can choose Ᏽ = Ᏽ δᏮ and Ᏽ = Ᏽ δA in Theorem 2.10 to obtain the following results.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning .23) Proof.Let (2.22) hold.Then, since V (x) ≤ L(x) and V (−x) ≤ −l(x) for all x ∈ ∞ , (2.12) follows from Theorem 2.4.Define a matrix C = (c nk ) by c nk = (b nk − b n,σ(k) ) for all n, k, where b nk is defined as in Theorem 2.4.Then, we have the hypothesis of Lemma 2.3.Now, choose the sequence y such that y k = 0 for k / ∈ σ(N).Then, (y k − y σ(k) ) ∈ Z and also, by the same argument used in [10, Theorem 23], one can easily see that k b nk y k − y σ(k) = k c nk y σ(k) .

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation