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We show that the structure of a block outside the critical hyperplanes of category O over a symmetrizable Kac-Moody algebra depends only on the corresponding integral Weyl group and its action on the parameters of the Verma modules.
      
We also prove the uniqueness of Verma embeddings outside the critical hyperplanes.
      
Using the matrix approach we prove that the sequence of sampling functions is always complete in the cases of critical sampling and oversampling.
      
An important example is the Moser-Trudinger inequality where limiting Sobolev behavior for critical exponents provides significant understanding of geometric analysis for conformal deformation on a Riemannian manifold [5, 6].
      
On Generating Tight Gabor Frames at Critical Density
      
Finally, we show that microlocally away from a critical set the continuity estimate can be mproved:
      
Riesz property plays an important role in any wavelet-based compression algorithm and is critical for the stability of any wavelet-based numerical algorithms.
      
We establish spectral estimates at a critical energy level for h-pseudo-differential operators.
      
We consider tight Gabor frames (h,a=1,b=1) at critical density with h of the form Z-1(Zg/|Zg|).
      
At last, we discuss the series expressions of these functions and give a Box-counting dimension estimation of "critical" fractal interpolation functions by using our smoothness results.
      
Bootstrap critical point for circular mean direction and its applications
      
On critical exponents for semilinear heat equations with nonlinear boundary conditions
      
The coefficient k in the equation is found to be a critical parameter.
      
there exists a C∞ rank-1 map f: I2→R1 such that f(A) has nonempty interior for some subset A ?I2 of critical points with finite Hausdorff dimension.
      
Since there is at most one limit cycle that may be created from critical point O by Hopf bifurcation, the number of limit cycles depends on the different situations of separatrix cycle to be formed around O.
      
When β is less than some critical value the boundary periodic solution (xs(t), 0, zs(t)) is locally stable, and when β is larger than the critical value there are periodic oscillations in substrate, prey and predator.
      
The critical micelle concentration (cmc) and surface tension of the novel zwitterionic Gemini surfactant in aqueous solution at 15°C are 7.2×10-5 mol/L and 34.5 mN/m, respectively.
      
Then, from the curves, the critical micelle concentration (CMC) and the thermodynamic standard formation functions (ΔH?m, ΔG?m and ΔS?m) were obtained through thermodynamic theories.
      
Active sites on the surface of ITO are critical to the formation of the crystal nucleus and a uniform and compact CdS nano-film.
      
Based on electromagnetic theory, a computing model of the Talbot interferometer is established with the temporal and the spatial coherence taken into account and with the turning angle of the reflectors as a critical parameter.
      
 

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