An optimal-control based integrated model of supply chain

Problems of supply chain scheduling are challenged by high complexity, combination of continuous and discrete processes, integrated production and transportation operations as well as dynamics and resulting requirements for adaptability and stability analysis. A possibility to address the above-named issues opens modern control theory and optimal program control in particular. Based on a combination of fundamental results of modern optimal program control theory and operations research, an original approach to supply chain scheduling is developed in order to answer the challenges of complexity, dynamics, uncertainty, and adaptivity. Supply chain schedule generation is represented as an optimal program control problem in combination with mathematical programming and interpreted as a dynamic process of operations control within an adaptive framework. The calculation procedure is based on applying Pontryagin’s maximum principle and the resulting essential reduction of problem dimensionality that is under solution at each instant of time. With the developed model, important categories of supply chain analysis such as stability and adaptability can be taken into consideration. Besides, the dimensionality of operations research-based problems can be relieved with the help of distributing model elements between an operations research (static aspects) and a control (dynamic aspects) model. In addition, operations control and flow control models are integrated and applicable for both discrete and continuous processes.supply chain, model of supply chain scheduling, optimal program control theory, Pontryagin’s maximum principle, operations research model,

The NLO jet vertex for Mueller-Navelet and forward jets in the small-cone approximation

We calculate in the next-to-leading order the impact factor (vertex) for the production of a forward high-$p_T$ jet, in the approximation of small aperture of the jet cone in the pseudorapidity-azimuthal angle plane. The final expression for the vertex turns out to be simple and easy to implement in numerical calculations.Comment: 4 pages, 4 figures; presented at the XX International Workshop on Deep-Inelastic Scattering and Related Subjects, 26-30 March 2012, University of Bon

Pitt's inequalities and uncertainty principle for generalized Fourier transform

We study the two-parameter family of unitary operators $$\mathcal{F}_{k,a}=\exp\Bigl(\frac{i\pi}{2a}\,(2\langle k\rangle+{d}+a-2 )\Bigr) \exp\Bigl(\frac{i\pi}{2a}\,\Delta_{k,a}\Bigr),$$ which are called $(k,a)$-generalized Fourier transforms and defined by the $a$-deformed Dunkl harmonic oscillator $\Delta_{k,a}=|x|^{2-a}\Delta_{k}-|x|^{a}$, $a>0$, where $\Delta_{k}$ is the Dunkl Laplacian. Particular cases of such operators are the Fourier and Dunkl transforms. The restriction of $\mathcal{F}_{k,a}$ to radial functions is given by the $a$-deformed Hankel transform $H_{\lambda,a}$. We obtain necessary and sufficient conditions for the weighted $(L^{p},L^{q})$ Pitt inequalities to hold for the $a$-deformed Hankel transform. Moreover, we prove two-sided Boas--Sagher type estimates for the general monotone functions. We also prove sharp Pitt's inequality for $\mathcal{F}_{k,a}$ transform in $L^{2}(\mathbb{R}^{d})$ with the corresponding weights. Finally, we establish the logarithmic uncertainty principle for $\mathcal{F}_{k,a}$.Comment: 16 page

The γ∗γ∗\gamma^* \gamma^* total cross section in next-to-leading order BFKL and LEP2 data

We study the total cross section for the collision of two highly-virtual photons at large energies, taking into account the BFKL resummation of energy logarithms with full next-to-leading accuracy. A necessary ingredient of the calculation, the next-to-leading order impact factor for the photon to photon transition, has been calculated by Balitsky and Chirilli using an approach based on the operator expansion in Wilson lines. We extracted the result for the photon impact factor in the original BFKL calculation scheme comparing the expression for the photon-photon total cross section obtained in BFKL with the one recently derived by Chirilli and Kovchegov in the Wilson-line operator expansion scheme. We perform a detailed numerical analysis, combining different, but equivalent in next-to-leading accuracy, representations of the cross section with various optimization methods of the perturbative series. We compare our results with previous determinations in the literature and with the LEP2 experimental data. We find that the account of Balitsky and Chirilli expression for the photon impact factor reduces the BFKL contribution to the cross section to very small values, making it impossible to describe LEP2 data as the sum of BFKL and leading-order QED quark box contributions.Comment: 20 pages, 8 figures; two sentences and some references added, a few typos removed; version to be published on JHE