Conformally Invariant Path Integral Formulation of the Wess-Zumino-Witten →\to Liouville Reduction

The path integral description of the Wess-Zumino-Witten $\to$ Liouville reduction is formulated in a manner that exhibits the conformal invariance explicitly at each stage of the reduction process. The description requires a conformally invariant generalization of the phase space path integral methods of Batalin, Fradkin, and Vilkovisky for systems with first class constraints. The conformal anomaly is incorporated in a natural way and a generalization of the Fradkin-Vilkovisky theorem regarding gauge independence is proved. This generalised formalism should apply to all conformally invariant reductions in all dimensions. A previous problem concerning the gauge dependence of the centre of the Virasoro algebra of the reduced theory is solved.Comment: Plain TeX file; 28 Page

Duality in Liouville Theory as a Reduced Symmetry

The origin of the rather mysterious duality symmetry found in quantum Liouville theory is investigated by considering the Liouville theory as the reduction of a WZW-like theory in which the form of the potential for the Cartan field is not fixed a priori. It is shown that in the classical theory conformal invariance places no condition on the form of the potential, but the conformal invariance of the classical reduction requires that it be an exponential. In contrast, the quantum theory requires that, even before reduction, the potential be a sum of two exponentials. The duality of these two exponentials is the fore-runner of the Liouville duality. An interpretation for the reflection symmetry found in quantum Liouville theory is also obtained along similar lines.Comment: Plain TeX file; 9 page

Path Integral Formulation of the Conformal Wess-Zumino-Witten to Liouville Reduction

The quantum Wess-Zumino-Witten $\to$ Liouville reduction is formulated using the phase space path integral method of Batalin, Fradkin, and Vilkovisky, adapted to theories on compact two dimensional manifolds. The importance of the zero modes of the Lagrange multipliers in producing the Liouville potential and the WZW anomaly, and in proving gauge invariance, is emphasised. A previous problem concerning the gauge dependence of the Virasoro centre is solved.Comment: Plain TeX file, 15 page

The Two-exponential Liouville Theory and the Uniqueness of the Three-point Function

It is shown that in the two-exponential version of Liouville theory the coefficients of the three-point functions of vertex operators can be determined uniquely using the translational invariance of the path integral measure and the self-consistency of the two-point functions. The result agrees with that obtained using conformal bootstrap methods. Reflection symmetry and a previously conjectured relationship between the dimensional parameters of the theory and the overall scale are derived.Comment: Plain TeX File; 15 Page

Duality in Quantum Liouville Theory

The quantisation of the two-dimensional Liouville field theory is investigated using the path integral, on the sphere, in the large radius limit. The general form of the $N$-point functions of vertex operators is found and the three-point function is derived explicitly. In previous work it was inferred that the three-point function should possess a two-dimensional lattice of poles in the parameter space (as opposed to a one-dimensional lattice one would expect from the standard Liouville potential). Here we argue that the two-dimensionality of the lattice has its origin in the duality of the quantum mechanical Liouville states and we incorporate this duality into the path integral by using a two-exponential potential. Contrary to what one might expect, this does not violate conformal invariance; and has the great advantage of producing the two-dimensional lattice in a natural way.Comment: Plain TeX File; 36 page

Critical dynamics of nonconserved NN-vector model with anisotropic nonequilibrium perturbations

We study dynamic field theories for nonconserving $N$-vector models that are subject to spatial-anisotropic bias perturbations. We first investigate the conditions under which these field theories can have a single length scale. When N=2 or $N \ge 4$, it turns out that there are no such field theories, and, hence, the corresponding models are pushed by the bias into the Ising class. We further construct nontrivial field theories for N=3 case with certain bias perturbations and analyze the renormalization-group flow equations. We find that the three-component systems can exhibit rich critical behavior belonging to two different universality classes.Comment: Included RG analysis and discussion on new universality classe

ENHANCEMENT OF DIFFERENT IMAGES USING ADAPTIVE METHOD IN SVD

It has been determined within the zero noise, you'll manage to obtain exact renovation inside the original image after when using the full projection. Growing the quantity of projections signi_cantly cuts lower across the RMSE.However, once we boost the block size, the slope inside the RMSE versus M decreases, and thus bigger blocksizesrequire more projections to attain the identical RMSE.SVD might be a matrix factorization. The singular values are similar to a weighting factor inside the component images. We'll demonstrate three techniques of image demising through Singular Value Decomposition (SVD). Inside the _rest method, we'll use SVD to represent only one noisy image like a straight line combination of image components, that's cut lower at various terms. We'll then compare each image approximation and determine the electiveness of truncating every single term. The second technique stretches the concept of imagedemising via SVD, but uses block wise analysis to conduct demising. We'll show blockwiseSVD demising could be the least elective at eliminating noise as compared to other techniques. Finally, we will discuss image demising with block wise Principal Component Analysis (PCA) calculated through SVD.In contrast for your _rest two techniques; this can be frequently a great technique in reducing the appearance RMSE

Lower urinary tract symptoms evaluation with uroflowmetry in patients with benign prostatic hyperplasia

Background: Uroflowmetry, a non-invasive urodynamic technique, is commonly employed in evaluating patients with potential lower urinary tract dysfunction. Accurate assessment of the severity of lower urinary tract symptoms (LUTS) can be achieved through the utilization of various validated questionnaires, such as the International Prostate Symptom Score (IPSS). The objective of this study was to investigate the correlation between uroflowmetry parameters and the severity of symptoms. Methods: Fifty patients with LUTS caused by benign prostatic hyperplasia were evaluated by using uroflowmetry, IPSS, prostate volume estimation from May 2022 to December 2023. The correlations between these parameters were quantified by means of Spearman correlation co-efficients. Results: Significant statistical correlations were identified between the IPSS and uroflowmetry outcomes, including peak flow rate, average flow rate, and post-void residual urine. However, no correlation was observed between the IPSS and measurements of prostate volume. Conclusions: A positive correlation was observed between the measured peak flow rate through uroflowmetry and the severity of lower urinary tract symptoms

The first example of direct oxidation of sulfides to sulfones by an osmate molecular oxygen system

Osmate-exchanged Mg-Al layered double hydroxides catalysed the delivery of two oxygen atoms simultaneously via a 3 + 1 cycloaddition to sulfide to form sulfone directly for the first time, reminiscent of 3 + 2 cycloaddition in asymmetric dihydroxylation reactions

Path Integral Formulation of the Conformal Wess-Zumino-Witten to Toda Reductions

The phase space path integral Wess-Zumino-Witten $\to$ Toda reductions are formulated in a manifestly conformally invariant way. For this purpose, the method of Batalin, Fradkin, and Vilkovisky, adapted to conformal field theories, with chiral constraints, on compact two dimensional Riemannian manifolds, is used. It is shown that the zero modes of the Lagrange multipliers produce the Toda potential and the gradients produce the WZW anomaly. This anomaly is crucial for proving the Fradkin-Vilkovisky theorem concerning gauge invariance.Comment: Plain TeX file, 27 page