###### Controlled G-Frames and Their G-Multipliers in Hilbert spaces

.P. Antoine, L. Jacques, M. Morvidone, Stereographic wavelet frames on the sphere, Applied Comput. Harmon. Anal. (19) (2005) 223-252. [8] P.G. Casazza, G. Kutyniok, Frames of Subspaces, Wavelets, Frames and Operator Theory, Amer. Math. Soc. 345 (2004) 87-113. [9] O. Christensen, Y.C. Eldar, Oblique dual frames and shift-invariant spaces, Appl. Comput. Harmon. Anal. 17 (2004) 48-68. [10] Ph. Depalle, R. Kronland-Martinet, B. Torresani, Time-frequency multipliers for sound synthesis. In Proceedings of the Wavelet XII conference

###### Analysis of models for viscoelastic wave propagation

{array}{} \displaystyle \frac{\phi(x)}{\min\{1,x\}\psi_\star(x)} \end{array}$ 1 + r 0 β σ This proves the continuity properties for u and σ as well as the estimates (32a) and (32c) . Note that zero initial values hold whenever the output function is continuous (see Corollary 18 ). To obtain the bounds for ∥ u ( t )∥ Ω we can use a simple shifting argument, since if ( f , α , β ) ↦ u̇ (i.e., we use the operators multiplied by s and with values in L 2 (Ω)), then ( f (−1) , α (−1) , β (−1) ) ↦ u . In general (see Proposition 10 ) 1 ψ ⋆ ( t

###### On the stabilizing effect of chemotaxis on bacterial aggregation patterns

eigenvalue, and U ∊ L 2 (Ω L ) is the associated eigenfunction. The right hand side of ( 22 ) defines a closed, densely defined operator in L 2 (Ω L ), namely L U = D ( ϕ ) ( U z z + U y y ) + ( 2 D ( ϕ ) z + c ) U z + ( D ( ϕ ) z z + g ′ ( ϕ ) ) U , L : D ( L ) = H 2 ( Ω L ) ⊂ L 2 ( Ω L ) → L 2 ( Ω L ) , $$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {{\cal L}U} \hfill & = \hfill & {D(\phi )({U_{zz}} + {U_{yy}}) + (2D{{(\phi )}_z} + c){U_z} + (D{{(\phi )}_{zz}} + {g^\prime }(\phi ))U,} \hfill \\ {{\cal L}} \hfill & : \hfill & {{\cal D

###### Noether’s theorems for the relative motion systems on time scales

. M. Torres, The second Euler-Lagrange equation of variational calculus on time scales. Eur. J. Control, 2011,1: 9-18 Bartosiewicz Z. Martins N. Torres D. F. M. The second Euler-Lagrange equation of variational calculus on time scales Eur. J. Control 2011 1 9 18 [9] M. Bohner, G. S. H. Guseinov, Double integral calculus of variations on time scales, Comput. Math. Appl, 2007, 54:45-57 10.1016/j.camwa.2006.10.032 Bohner M. Guseinov G. S. H. Double integral calculus of variations on time scales Comput. Math. Appl 2007 54 45 57 [10] R. A. C. Ferreira, D. F. M

###### Travelling waves and conservation laws of a (2+1)-dimensional coupling system with Korteweg-de Vries equation

) are constructed in Section 3 by employing the multiplier approach [ 26 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 ]. Finally concluding remarks are presented in Section 4 . 2 Travelling wave solutions of (2a) In this section we use Lie symmetry analysis together with the ( G ′/ G )–expansion method to obtain travelling wave solutions of (2a) . 2.1 Lie point symmetries and symmetry reductions of (2a) Lie symmetry analysis was introduced by Marius Sophus Lie (1842-1899), a Norwegian mathematician, in the later half of the nineteenth century. He

###### Modeling and Simulation of Equivalent Circuits in Description of Biological Systems - A Fractional Calculus Approach

) are also compared for both integer and fractional order model responses. Methodology Complex electrical circuit in the Laplace space and its application to distributed elements In order to apply the general theory to the skin type system, we propose an equivalent electrical circuit for each component of the skin. Fig. 1 shows the schematic diagram to describe the proposed model of the skin, where R a and R b are the contact resistances of the electrodes, D e represents the equivalent circuit of the dermis ( R 1 , C 1 , R 2 , C 2 , R 3 , C 3 ), G

###### Singular value decomposition analysis of back projection operator of maximum likelihood expectation maximization PET image reconstruction

possible fix points of the algorithm can be obtained from the next equations (ratio is in a Hadamard sense) as the update process multiplies (also in Hadamard sense) the current estimate by 1 when the following is true: A T y r | A | = A T y r A T ∗ 1 ⋮ 1 L o r − s p a c e = 1 ⋮ 1 v o x e l s p a c e $$\begin{array}{} \displaystyle \frac{{\boldsymbol A}^{\boldsymbol T} y_r}{|{\boldsymbol A}|}=\frac{{\boldsymbol A}^{\boldsymbol T}y_r}{{\boldsymbol A}^{\boldsymbol T}*\begin{pmatrix} {1}\\{\vdots}\\1\end{pmatrix}_{Lor-space}}=\begin{pmatrix} {1}\\{\vdots}\\1\end

###### Multigrid method for the solution of EHL line contact with bio-based oils as lubricants

equations can be written as L h ( u h ) = f h , $$\begin{array}{} \displaystyle L^h(u^h)=f^h, \end{array} $$ where L is a non linear operator , u is the exact solution , f is a right hand side function and h is the mesh size of the uniform finer grid . Let v be an approximation to the exact solution u . Then e h = u h − v h ( e r r o r ) r h = f h − L ( v h ) ( r e s i d u a l ) . $$\begin{array}{c} \displaystyle e^h=u^h-v^h\quad {(error)}\\\displaystyle r^h=f^h-L(v^h)\quad {(residual)}. \end{array} $$ For simplicity, we shall

###### Designing optimal trajectories for a skimmer ship to clean, recover and prevent the oil spilled on the sea from reaching the coast

(\tau, x) {\text d} x {\text d} \tau, \end{array}$$ (4) where coef( x ) = λ 1 (1 − (dist( x )/max x ∈Ω dist( x ))) λ 2 + λ 3 is a weight function, with dist( x ) being the distance between x and the nearest point to the coast (i.e., in the boundary ∂ Ω c ). λ 1 , λ 2 and λ 3 are real parameters used to control the behaviour of coef(.). Function coef(.) gives more weight to the value of c for points near the coast than for points far from the coast. These formulations take into consideration the evolution in time and space of the pollution

###### New Complex and Hyperbolic Forms for Ablowitz–Kaup–Newell–Segur Wave Equation with Fourth Order

following result: U 3 x , y , t = − 1 18 γ γ t 3 + x + y − 3 i s e c h γ t 3 + x + y − 3 t a n h γ t 3 + x + y . $$\begin{array}{} \displaystyle {{\rm{U}}_3}\left( {{\rm{x}},{\rm{y}},{\rm{t}}} \right) = - \frac{1}{{18}}{\rm{\gamma }}\left( {\frac{{{\rm{\gamma t}}}}{3} + {\rm{x}} + {\rm{y}} - 3{\rm{isech}}\left[ {\frac{{{\rm{\gamma t}}}}{3} + {\rm{x}} + {\rm{y}}} \right] - 3{\rm{tanh}}\left[ {\frac{{{\rm{\gamma t}}}}{3} + {\rm{x}} + {\rm{y}}} \right]} \right).{\rm{\;}} \end{array}$$ (16) Case 4 G etting the following items into Eq. (10) with V = U ′; A