and Applied Analysis 3 Theorem C. Let Aj z /≡ 0 j 0, 1 be entire functions with σ Aj < 1, and let a, b be complex constants such that ab / 0 and arga/ arg b or a cb 0 < c < 1 . If ψ z /≡ 0 is an entire function with finite order, then every solution f /≡ 0 of 1.2 satisfies λ f − ψ λ f ′ − ψ λ f ′′ − ψ ∞. Furthermore, let d0 z , d1 z ,and d2 z be polynomials that are not all equal to zero, and l...

||N ||2 √ |T | ≤ 4σ √ n √ |T | producing the contradiction. If σ ≤ ε 4 /4 √ d, (1) is satisfied. Furthermore, if σ > cε/d, then, with high probability, a CONSTANT FRACTION of the columns j violate the condition ||N·,j ||1 ≤ ε required by previous algorithms to hold for EVERY column. Lemma 2. Suppose k = 1, n ≤ c0d, ||C·,j ||1 = 1 for all j and N has i.i.d. entries drawn from N (0, σ), where, σ ...

∆Σ modulators [1] are widely used in AD (Analog-toDigital) and DA (Digital-to-Analog) converters, in which high performance can be obtained with coarse quantizers. A fundamental issue in designing ∆Σ modulators is noise shaping in the frequency domain [1]. A usual solution to this is to insert accumulator(s) in the feedback loop to attenuate the gain of the noise transfer function (NTF) in low ...

and Applied Analysis 3 where τ t ≤ t, σ t ≤ t, τ ′ t τ0 > 0, 0 ≤ p t ≤ p0 < ∞, and the authors obtained some oscillation criteria for 1.7 . However, there are few results regarding the oscillatory problem of 1.1 when τ t ≥ t and σ t ≥ t. Our aim in this paper is to establish some oscillation criteria for 1.1 under the case when τ t ≥ t and σ t ≥ t. The paper is organized as follows. In Section ...

We give a simple proof of the Cafiero theorem based on a matrix method approach in the form of Lemma 2.4 in the σ-additive context. Based on a version of Drewnowski lemma for an SCP-ring we obtain an extension of Cafiero theorem for exhaustive finitely additive set functions defined on an SCP-ring. As consequences, the well-known Nikodým and Brooks-Jewett convergence theorems are obtained. AMS ...

and Applied Analysis 3 Proposition 1.2. Function σ t is μ-measurable. Proof. Pick a dense set {ri}i 1 in 0,∞ and set Bk { t ∈ T : M ( t, 1 2 rk ) 1 2 M t, rk } , qk t rkχBk t k ∈ N . 1.7 It is easy to see that for all k ∈ N, σ t ≥ qk t μ-a.e on T . Hence, supk≥1qk t ≤ σ t . For μ-a.e t ∈ T , arbitrarily choose ε ∈ 0, σ t . Then, there exists rk ∈ σ t − ε, σ t such that M t, 1/2 rk 1/2 M t, rk ,...

We construct a σ-ideal of subsets of the Cantor space which is productive but does not have the Weak Fubini Property. In the construction we use a combinatorial lemma which is of its own interest.

and Applied Analysis 3 For the rest of the paper we need the following assumption: C3 0 < ∑m−2 i 1 αiφ1 ηi < 1. Lemma 2.2 see 1 . Assuming that (C2) and (C3) hold. Let y ∈ C ρ 0 , σ 1 . Then boundary value problem xΔ∇ t a t xΔ t b t x t y t 0, t ∈ 0, 1 T , x ( ρ 0 ) 0, x σ 1 m−2 ∑ i 1 αix ( ηi ) 2.3 is equivalent to integral equation

To solve variational indefinite problems, one uses classically the Banach–Nečas–Babuška theory. Here, we study an alternate theory to solve those problems: T-coercivity. Moreover, we prove that one can use this theory to solve the approximate problems, which provides an alternative to the celebrated Fortin lemma. We apply this theory to solve the indefinite problem div σ∇u = f set in H1 0 , wit...

In the paper A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees [1], Proposition 2.7 is not true. To avoid this error and correct Proposition 2.7, the definition of the property R1,א1 is changed. In [1], all proofs of lemmas and theorems but Lemma 6.9 are valid about this definition without changing the proofs. We give a new statement an...