Sentence examples for exists a representation from inspiring English sources

We assume that there exists a representation x=sum_{k}lambda_{k}g^{ k)}(r,s).

For frequently used kernels, such as the Gaussian (radial basis function) kernel, there exists a representation in which classical RR is applied to a model with infinitely many predictors, nevertheless yielding finite predictions.

For a large class of LM-measures, as mentioned earlier, there exists a "robust representation" type theorem essentially a representation, via convex duality, as a function of conditional expectation.

Define a Hilbert-type operator (T:l_{p,Phi }rightarrow l_{p,Psi^{1-p}}) as follows: For any (a={a_{m}} _{|m|=1}^{infty}in l_{p,Phi}), there exists a unique representation (c=Tain l_{p,Psi^{1-p}}).

Define a half-discrete Hardy-Hilbert-type operator (T_{1} L_{p,Phi_{delta}}(mathbf{R}_)rightarrow l_{p,Psi ^{1-p}}) as follows: For any (fin L_{p,Phi_{delta}}(mathbf{R}_)), the exists a unique representation (T_{1}f=cin l_{p,Psi^{1-p}}).

Define a half-discrete Hilbert's operator as follows: T ˜ : l q, Ψ → L q, Φ 1 − q ( b, c ), for a ∈ l q, Ψ, there exists a unified representation T ˜ a ∈ L q, Φ 1 − q ( b, c ) satisfying ( T ˜ a ) ( x ) = ∑ n = n 0 ∞ a n ( u ( x ) + v ( n ) ) λ, x ∈ ( b, c ).

Define a Hardy-Hilbert-type operator (T: l_{p,varphi }rightarrow l_{p,psi ^{1-p}}) as follows: For ({a_{m}geq 0}), (a={a_{m}}_{m=1}^{infty }in l_{p,varphi }), there exists a unique representation (Ta=hin l_{p,psi ^{1-p}}).

Define a half-discrete Hardy-Hilbert-type operator (T_{2}:l_{q,widetilde{Psi}}rightarrow L_{q,Phi_{delta }^{1-q}}(mathbf{R}_)) as follows: For any (a={a_{n}}_{n=1}^{infty}in l_{q,widetilde{Psi}}), there exists a unique representation (T_{2}a=hin L_{q,Phi _{delta}^{1-q}}(mathbf{R}_)).

Define a half-discrete Hardy-Hilbert-type operator (T_{2}:l_{q,widehat{Psi }}rightarrow L_{q,Phi _{delta }^{1-q}}(mathbf{R}_)) as follows: For any (a={a_{n}}_{n=1}^{infty }in l_{q,widehat{Psi }}), there exists a unique representation (T_{2}a=hin L_{q,Phi _{delta }^{1-q}}(mathbf{R}_)).

Define a half-discrete Hardy-Hilbert-type operator (T_{2}:l_{q,Psi}rightarrow L_{q,Phi_{delta}^{1-q}}(mathbf {R}_)) as follows: For any (a={a_{n}}_{n=1}^{infty}in l_{q,Psi}), there exists a unique representation (T_{2}a=hin L_{q,Phi_{delta}^{1-q}}(mathbf {R}_)).

Define a Hardy-Mulholland-type operator (T:l_{p,Phi_{lambda}} to l_{p,Psi_{lambda}^{1 - p}}) as follows: For any (a = { a_{m}}_{m = 2}^{infty} in l_{p,Phi_{lambda}}), there exists a unique representation (Ta = c in l_{p,Psi_{lambda}^{1 - p}}).