Predicting the performance of a full-scale direct contact membrane distillation (DCMD) module based on experimental lab-scale results is rather difficult, since DCMD dependent many different process parameters. Hence, there need for methodology to perform system up-scaling results. In this study, we devise an approach scale up systems by using Buckingham's Pi theorem group parameters into eight...

The sum of external angles of a polygon is always constant, π 2 . There are several elementary proofs of this fact. In the similar way, there is an invariant in polyhedron that is π 4 . To see this, let us consider a regular tetrahedron as an example. Tetrahedron has four vertices. Three regular triangles gather at each vertex. Developing the tetrahedron around each vertex, there is an open ang...

The classical Shannon sampling theorem states that a signal f with Fourier transform F ∈ L(R) having its support contained in (−π, π) can be recovered from the sequence of samples (f(n))n∈Z via f(t) = ∑ n∈Z f(n) sin(π(t− n)) π(t− n) (t ∈ R). In this article we prove a generalization of this result under the assumption that F ∈ E (R) has support contained in (−π, π).

and Applied Analysis 3 2. Theorem of Existence, Uniqueness, and Asymptotics In this paper, we consider the following initial-boundary value problem for the damped Boussinesq equation: utt − auttxx − 2butxx 2b a ut −cuxxxx uxx − p2u β ( u2 ) xx , u 0, t u π, t 0, t > 0, uxx 0, t uxx π, t 0, t > 0, u x, 0 ε2φ x , ut x, 0 ε2ψ x , x ∈ 0, π , 2.1 where a, b, c, and p are positive constants, β ∈ R, a...

and Applied Analysis 3 In two main theorems in 20 , the authors proved the following results, which we now incorporate in the next theorem. Theorem A. Assume p ≥ 1 and φ is a holomorphic self-map of Π . Then the following statements true hold. a The operator Cφ : H Π → A∞ Π is bounded if and only if sup z∈Π Im z ( Imφ z )1/p < ∞. 1.8 b The operator Cφ : H Π → B∞ Π is bounded if and only if sup ...

19.1 Independent Sets (Continuation) In the previous lecture, we discussed the problem of sampling independent sets uniformly at random. Input: Graph G = (V,E) on |V | = n vertices. Goal: Sample an independent set of vertices of G uniformly at random. Definition 19.1 Call a Markov Chain with state space Ω equal to the independent sets of G and uniform stationary distribution γ-cautious if, in a...

The goal of this work is to study homomorphism problems (from a computational point of view) on two superclasses of graphs: 2-edge-coloured graphs and signed graphs. On the one hand, we consider the H-Colouring problem when H is a 2-edge-coloured graph, and we show that a dichotomy theorem would imply the dichotomy conjecture of Feder and Vardi. On the other hand, we prove a dichotomy theorem f...

The theory of equivalential deductive systems, as introduced by Prucnal and Wroński and further developed by Czelakowski, is abstracted to cover the case of logical systems formalized as π-institutions. More precisely, the notion of an N-equivalence system for a given π-institution is introduced. A characterization theorem for N-equivalence systems, previously proven for N-parameterized equival...

Let K/Q be a number field. Let π and π′ be cuspidal automorphic representations of GLd(AK) and GLd′(AK), and suppose that either both d and d′ are at most 2 or at least one of π and π′ is self-dual. We prove an effective log-free zero density estimate for the Rankin-Selberg L-function L(s, π ⊗ π′,K) when at least one of L(s, π,K) and L(s, π′,K) satisfies the generalized Ramanujan conjecture. Wi...