A generalization of the quotient integral formula is presented and some of its properties are investigated. Also the relations between two function spaces related to the spacial homogeneous spaces are derived by using general quotient integral formula. Finally our results are supported by some examples.
Let H be a compact subgroup of a locally compact group G. In this paper we define a convolution on M (G/H), the space of all complex bounded Radon measures on the homogeneous space G/H. Then we prove that the measure space M (G/H, * ) is a non-unital Banach algebra that possesses an approximate identity. Finally, it is shown that the Banach algebra M (G/H, * ) is not involutive and also L 1 (G/H, * ) is a two-sided ideal of it.
Abstract. Let G be a graph with eigenvalues λ 1 (G) ≥ · · · ≥ λn (G). In this paper we find all simple graphs G such that G has at most twelve vertices and G has exactly two non-negative eigenvalues.In other words we find all graphs G on n vertices such that n ≤ 12 and λ 1 (G) ≥ 0, λ 2 (G) ≥ 0 and λ 3 (G) < 0. We obtain that there are exactly 1575 connected graphs G on n ≤ 12 vertices with λ 1 (G) > 0, λ 2 (G) > 0 and λ 3 (G) < 0. We find that among these 1575 graphs there are just two integral graphs.
Objective: Although Levodopa-carbidopa intestinal gel (LCIG) treatment has shown to be efficacious in motor and some non-motor symptoms (NMS), not all the patients with advanced Parkinsons disease (PD) are ideal candidates. To improve their selection analysis knowledge of prognostic factors is of great importance. We aimed to develop a novel machine learning model to predict the clinical outcomes of patients with advanced PD at 2 years under the LCIG therapy. Methods This was a longitudinal 24-month, observational study of 59 patients with advanced PD of a Greek multicenter registry under LCIG treatment from September 2019 to September 2021. Motor status was assessed with the Unified Parkinsons Disease Rating Scale (UPDRS) part III (off) and IV. NMS were assessed by the NMS Questionnaire (NMSQ) and the Geriatric Depression Scale (GDS), the quality of life by PDQ-39 and severity by Hoehn &Yahr (HY). Multivariate linear regression, ARIMA, SARIMA, and Long Short-Term Memory- recurrent neural network (LSTM-RNN) models were used. Results Dyskinesia duration and quality of life were significantly improved with LCIG (19% and 10% greater improvement for men than women, respectively). Multivariate linear regression models showed that UPDRS-III was decreased by 1.5 and 4.39 units per one unit of increase of the PDQ-39, UPDRS-IV indexes, respectively. Among all the time series models, the LSTM-RNN model predicts these clinical characteristics with highest accuracy (mean square error =0.0069) Conclusions Τhe LSTM-RNN model predicts with highest accuracy sex dependent clinical outcomes of patients with advanced PD after two years of LCIG therapy.
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