Combinatorial proofs of some properties of tangent and Genocchi numbers

The tangent number $T_{2n+1}$ is equal to the number of increasing labelled complete binary trees with $2n+1$ vertices. This combinatorial interpretation immediately proves that $T_{2n+1}$ is divisible by $2^n$. However, a stronger divisibility property is known in the studies of Bernoulli and Genocchi numbers, namely, the divisibility of $(n+1)T_{2n+1}$ by $2^{2n}$. The traditional proofs of this fact need significant calculations. In the present paper, we provide a combinatorial proof of the latter divisibility by using the hook length formula for trees. Furthermore, our method is extended to $k$-ary trees, leading to a new generalization of the Genocchi numbers

Multiphase Flow Packed-Bed Reactor Modeling: Combining CFD and Cell Network Model

The Performance of a Multiphase Flow Packed-Bed Reactor Was Evaluated by a Mixing-Cell Network Model in Which the Cell-Scale Flow Field Information Was Provided by Multifluid CFD Modeling. Such a Sequentially Combined Modeling Approach Was Able to Show the Heterogeneity of Liquid Species Conversion in a Trickle-Bed Reactor that Has Been Confirmed by in Situ Magnetic Resonance Experiments. 1 a Conventional Plug Flow Model (PFM) and Axial Dispersion Model (ADM) Showed Either overpredicted or Underpredicted Conversion Compared with the Conversion Computed from the CFD-Cell Network Model. Preliminary Case Study Has Shown the Ability of the CFD-Cell Network Model in Assessing the Impact of Flow Distribution on the Trickle-Bed Reactor Performance, which is a Great Help to Trickle-Bed Reactor Design and Diagnosis. © 2005 American Chemical Society

Quantum theory of light diffraction

At present, the theory of light diffraction only has the simple wave-optical approach. In this paper, we study light diffraction with the approach of relativistic quantum theory. We find that the slit length, slit width, slit thickness and wave-length of light have affected to the diffraction intensity and form of diffraction pattern. However, the effect of slit thickness on the diffraction pattern can not be explained by wave-optical approach, and it can be explained in quantum theory. We compare the theoretical results with single and multiple slits experiment data, and find the theoretical results are accordance with the experiment data. Otherwise, we give some theory prediction. We think all the new prediction will be tested by the light diffraction experiment.Comment: 10 page

Modeling of Trickle-Bed Reactors with Exothermic Reactions using Cell Network Approach

One-Dimensional (1D) and Two-Dimensional (2D) Cell Network Models Were Developed to Simulate the Steady-State Behavior of Trickle-Bed Reactors Employed for the Highly Exothermic Hydrotreating of Benzene. the Multiphase Mass Transfer-Reaction Model and Novel Solution Method Are Discussed in This Report. the 1D Model Was Shown to Satisfactorily Simulate the Axial Temperature Field Observed Experimentally for Multiphase Flow with Exothermic Reactions. the 2D Reactor Modeling Provided Valuable Information About Local Hot Spot Behavior within the Multiphase Reactor, Identifying Situations in Which Hot Spots May Form. the Model Took into Consideration the Heterogeneous Nature of Liquid Distribution, Including Radial Liquid Maldistribution and Partial External Wetting. This Approach Was Proven to Be Stable and Efficient in Dealing with the Complex Interaction of Phase Vaporization and Temperature Rise. through Analysis and Discussion, This Report Established the Cell Network Model as a Valid Representation of the Flow Environment Produced in a Trickle Bed with Exothermic Reactions. © 2007 Elsevier Ltd. All Rights Reserved

Is the formal energy of the mid-point rule convergent?

AbstractWe obtain some formulae for calculation of the coefficients of four special types of terms in τ2k, k = 1, 2, … (1−1 corresponding to four type of (2k + 1)-vertex free unlabeled trees, k = 1, 2, …, respectively), for a fixed step size τ, in the tree-expansion of the formal energy of the mid-point rule. And, we give an estimate of the difference between the formal energy H and the standard Hamiltonian H in some domain Ω under the assumptions 1.(i)|H is smooth and bounded in Ω, and2.(ii)|the absolute values of the coefficients of the terms in τ2k are uniformly bounded by ησ2k for some constants η ≥ 1, σ > 0 and for any k ≥ 1