Some binomial sums involving absolute values

We consider several families of binomial sum identities whose definition involves the absolute value function. In particular, we consider centered double sums of the form $$S_{\alpha,\beta}(n) := \sum_{k,\;\ell}\binom{2n}{n+k}\binom{2n}{n+\ell} |k^\alpha-\ell^\alpha|^\beta,$$ obtaining new results in the cases $\alpha = 1, 2$. We show that there is a close connection between these double sums in the case $\alpha=1$ and the single centered binomial sums considered by Tuenter.Comment: 15 pages, 19 reference

Online-Computation Approach to Optimal Control of Noise-Affected Nonlinear Systems with Continuous State and Control Spaces

A novel online-computation approach to optimal control of nonlinear, noise-affected systems with continuous state and control spaces is presented. In the proposed algorithm, system noise is explicitly incorporated into the control decision. This leads to superior results compared to state-of-the-art nonlinear controllers that neglect this influence. The solution of an optimal nonlinear controller for a corresponding deterministic system is employed to find a meaningful state space restriction. This restriction is obtained by means of approximate state prediction using the noisy system equation. Within this constrained state space, an optimal closed-loop solution for a finite decisionmaking horizon (prediction horizon) is determined within an adaptively restricted optimization space. Interleaving stochastic dynamic programming and value function approximation yields a solution to the considered optimal control problem. The enhanced performance of the proposed discrete-time controller is illustrated by means of a scalar example system. Nonlinear model predictive control is applied to address approximate treatment of infinite-horizon problems by the finite-horizon controller

Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents

Motivated by the statistical mechanics description of stationary 2D-turbulence, for a sinh-Poisson type equation with asymmetric nonlinearity, we construct a concentrating solution sequence in the form of a tower of singular Liouville bubbles, each of which has a different degeneracy exponent. The asymmetry parameter $\gamma\in(0,1]$ corresponds to the ratio between the intensity of the negatively rotating vortices and the intensity of the positively rotating vortices. Our solutions correspond to a superposition of highly concentrated vortex configurations of alternating orientation; they extend in a nontrivial way some known results for $\gamma=1$. Thus, by analyzing the case $\gamma\neq1$ we emphasize specific properties of the physically relevant parameter $\gamma$ in the vortex concentration phenomena

Meteor showers of comet C/1917 F1 Mellish

December Monocerotids and November Orionids are weak but established annual meteor showers active throughout November and December. Analysis of a high quality orbits subset of the SonotaCo video meteor database shows that the distribution of orbital elements, geocentric velocity and also the orbital evolution of the meteors and potential parent body may imply a common origin for these meteors coming from the parent comet C/1917 F1 Mellish. This is also confirmed by the physical properties and activity of these shower meteors. An assumed release of meteoroids at the perihelion of the comet in the past and the sky-plane radiant distribution reveal that the December Monocerotid stream might be younger than the November Orionids. A meteoroid transversal component of ejection velocity at the perihelion must be larger than 100 m/s. A few authors have also associated December Canis Minorids with the comet C/1917 F1 Mellish. However, we did not find any connection.Comment: 11 pages, 11 figures and 5 table

Do various imaging modalities provide potential early detection and diagnosis of medication-related osteonecrosis of the jaw? A review

Objective: Patients with medication-related osteonecrosis of the jaw (MRONJ) often visit their dentists at advanced stages and subsequently require treatments that greatly affect quality of life. Currently, no clear diagnostic criteria exist to assess MRONJ, and the definitive diagnosis solely relies on clinical bone exposure. This ambiguity leads to a diagnostic delay, complications, and unnecessary burden. This article aims to identify imaging modalities' usage and findings of MRONJ to provide possible approaches for early detection. Methods: Literature searches were conducted using PubMed, Web of Science, Scopus, and Cochrane Library to review all diagnostic imaging modalities for MRONJ. Results: Panoramic radiography offers a fundamental understanding of the lesions. Imaging findings were comparable between non-exposed and exposed MRONJ, showing osteolysis, osteosclerosis, and thickened lamina dura. Mandibular cortex index Class II could be a potential early MRONJ indicator. While three-dimensional modalities, CT and CBCT, were able to show more features unique to MRONJ such as a solid type periosteal reaction, buccal predominance of cortical perforation, and bone-within-bone appearance. MRI signal intensities of vital bones are hypointense on T1WI and hyperintense on T2WI and STIR when necrotic bone shows hypointensity on all T1WI, T2WI, and STIR. Functional imaging is the most sensitive method but is usually performed in metastasis detection rather than being a diagnostic tool for early MRONJ. Conclusion: Currently, MRONJ-specific imaging features cannot be firmly established. However, the current data are valuable as it may lead to a more efficient diagnostic procedure along with a more suitable selection of imaging modalities

Antisymmetrization of a Mean Field Calculation of the T-Matrix

The usual definition of the prior(post) interaction $V(V^\prime )$ between projectile and target (resp. ejectile and residual target) being contradictory with full antisymmetrization between nucleons, an explicit antisymmetrization projector ${\cal A}$ must be included in the definition of the transition operator, $T\equiv V^\prime{\cal A}+V^\prime{\cal A}GV.$ We derive the suitably antisymmetrized mean field equations leading to a non perturbative estimate of $T$. The theory is illustrated by a calculation of forward $\alpha$-$\alpha$ scattering, making use of self consistent symmetries.Comment: 30 pages, no figures, plain TeX, SPHT/93/14

Off-diagonal Wave Function Monte Carlo Studies of Hubbard Model I

We propose a Monte Carlo method, which is a hybrid method of the quantum Monte Carlo method and variational Monte Carlo theory, to study the Hubbard model. The theory is based on the off-diagonal and the Gutzwiller type correlation factors which are taken into account by a Monte Carlo algorithm. In the 4x4 system our method is able to reproduce the exact results obtained by the diagonalization. An application is given to investigate the half-filled band case of two-dimensional square lattice. The energy is favorably compared with quantum Monte Carlo data.Comment: 9 pages, 11 figure

Classification and nondegeneracy of SU(n+1)SU(n+1) Toda system with singular sources

We consider the following Toda system \Delta u_i + \D \sum_{j = 1}^n a_{ij}e^{u_j} = 4\pi\gamma_{i}\delta_{0} \text{in}\mathbb R^2, \int_{\mathbb R^2}e^{u_i} dx -1$,$\delta_0$is Dirac measure at 0, and the coefficients$a_{ij}$form the standard tri-diagonal Cartan matrix. In this paper, (i) we completely classify the solutions and obtain the quantization result:$$\sum_{j=1}^n a_{ij}\int_{\R^2}e^{u_j} dx = 4\pi (2+\gamma_i+\gamma_{n+1-i}), \;\;\forall\; 1\leq i \leq n.$$This generalizes the classification result by Jost and Wang for$\gamma_i=0$,$\forall \;1\leq i\leq n$. (ii) We prove that if$\gamma_i+\gamma_{i+1}+...+\gamma_j \notin \mathbb Z$for all$1\leq i\leq j\leq n$, then any solution$u_i$ is \textit{radially symmetric} w.r.t. 0. (iii) We prove that the linearized equation at any solution is \textit{non-degenerate}. These are fundamental results in order to understand the bubbling behavior of the Toda system.Comment: 28 page