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Register a new userSolution of the Diophantine Equation $ x_{1}x_{2}x_{3}cdots x_{m-1}=z^n $
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This work determine the entire family of positive integer solutions of the diophantine equation. The solution is described in terms of $frac{(m-1)(m+n-2)}{2} $ or $frac{(m-1)(m+n-1)}{2}$ positive parameters depending on $n$ even or odd. We find the solution of a diophantine system of equations by using the solution of the diophantine equation. We generalized all the results of the paper [5].
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Suppose that $n$ is a positive integer. In this paper, we show that the exponential Diophantine equation $$(n-1)^{x}+(n+2)^{y}=n^{z}, ngeq 2, xyz eq 0$$ has only the positive integer solutions $(n,x,y,z)=(3,2,1,2), (3,1,2,3)$. The main tools on the proofs are Bakers theory and Bilu-Hanrot-Voutiers result on primitive divisors of Lucas numbers.
We explore the utility of CI as an alternative high-fidelity gas mass tracer for Galactic molecular clouds. We evaluate the X$_{rm CI}$-factor for the 609 $mu$m carbon line, the analog of the CO X-factor, which is the ratio of the H$_2$ column density to the integrated $^{12}$CO(1-0) line intensity. We use 3D-PDR to post-process hydrodynamic simulations of turbulent, star-forming clouds. We compare the emission of CI and CO for model clouds irradiated by 1 and 10 times the average background and demonstrate that CI is a comparable or superior tracer of the molecular gas distribution for column densities up to $6 times 10^{23}$ cm$^{-2}$. Our results hold for both reduced and full chemical networks. For our fiducial Galactic cloud we derive an average $X_{rm CO}$ of $3.0times 10^{20}$ cm$^{-2}$K$^{-1}$km$^{-1}$s and $X_{rm CI}$ of $1.1times 10^{21}$ cm$^{-2}$K$^{-1}$km$^{-1}$s.
We have studied, using double ratio of QCD (spectral) sum rules, the ratio between the masses of $T_{cc}$ and X(3872) assuming that they are respectively described by the $D-{D}^*$ and $D-bar{D}^*$ molecular currents. We found (within our approximation) that the masses of these two states are almost degenerate. Since the pion exchange interaction between these mesons is exactly the same, we conclude that if the observed X(3872) meson is a $Dbar{D}^*+c.c.$ molecule, then the $DD^*$ molecule should also exist with approximately the same mass. An extension of the analysis to the $b$-quark case leads to the same conclusion. We also study the SU(3) breakings for the $T^s_{QQ}/T_{QQ}$ mass ratios. Motivated by the recent Belle observation of two $Z_b$ states, we revise our determination of $X_b$ by combining results from exponential and FESR sum rules.
We show that in the Standard Model the parametrically leading (by a factor 1/alpha_s) contribution to the inclusive CP asymmetry in B->X_{s,d}+gamma decays arises from a long-distance effect in the interference of the electromagnetic dipole amplitude with the amplitude for an up-quark penguin transition accompanied by soft gluon emission. This contribution is governed by a single hadronic parameter Lambda_{17}^u related to a matrix elements of a non-local operator. In view of current experimental data, a future precision measurement of the flavor-averaged CP asymmetry in B->X_s+gamma will signal the presence of new physics only if a value below -2% is found. A cleaner probe of new physics is offered by the difference of the CP asymmetries in charged versus neutral B-meson decays.
Lately, the LHCb Collaboration reported the discovery of two new states in the $B^+rightarrow D^+D^- K^+$ decay, i.e., $X_0(2866)$ and $X_1(2904)$. In the present work, we study whether these states can be understood as $D^*bar{K}^*$ molecules from the perspective of their two-body strong decays into $D^-K^+$ via triangle diagrams and three-body decays into $D^*bar{K}pi$. The coupling of the two states to $D^*bar{K}^*$ are determined from the Weinberg compositeness condition, while the other relevant couplings are well known. The obtained strong decay width for the $X_0(2866)$, in marginal agreement with the experimental value within the uncertainty of the model, hints at a large $D^*bar{K}^*$ component in its wave function. On the other hand, the strong decay width for the $X_1(2904)$, much smaller than its experimental counterpart, effectively rules out its assignment as a $D^*bar{K}^*$ molecule.
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