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Engineering and Scientific Calculations
Published in David E. Clough, Steven C. Chapra, Introduction to Engineering and Scientific Computing with Python, 2023
David E. Clough, Steven C. Chapra
Another related function is the sign (also called signum) function, abbreviated sgn. It can be expressed in terms of the absolute value function by the formula sgn(x)=|x|x
A correlation-hole approach to the electric double layer with counter-ions only
Published in Molecular Physics, 2018
Ivan Palaia, Martin Trulsson, Ladislav Šamaj, Emmanuel Trizac
The normalisation constant is determined by the electroneutrality condition (8) through . There exists a simple way to obtain the explicit value of ; it will prove useful below and we thus present it in its simplest clothing. We first differentiate the -potential (15) with respect to z: where sgn denotes the standard signum (sign) function. The integral vanishes due to the anti-symmetric property of the function under integration in the rhs with respect to the interchange transformation . From (14), we get Consequently, we have from (17) that The density vanishes as , so that which is nothing but the contact-value theorem (9). We see that, within the PB theory, the normalisation (8) automatically ensures the contact-value theorem (9), and vice versa. Under the gauge , the contact-value relation (20) fixes in (14). It is easy to check that under this normalisation, the PB solution (10) satisfies Equations (14) and (15).
Finite-time consensus for controlled dynamical systems in network
Published in International Journal of Control, 2018
Naim Zoghlami, Rhouma Mlayeh, Lotfi Beji, Azgal Abichou
Throughout this paper, we use to denote the set of real number. is the n-dimensional real vector space, and ‖.‖ denotes the Euclidian norm. is the set of n × n matrices. diag{m1, m2,… , mn} denotes a n × n diagonal matrix. is the identity matrix. The symbol ⊗ is the Kronecker product of matrices. We use sgn(.) to denote the signum function. For a scalar x, note that ϕα(x) = sgn(x)|x|α. We use , x = (x1,… , xN)T to denote the vector in . Let φα(xi) = (ϕα(xi1), …, ϕα(xin))T with φα(x) = (φα(xi),… , φα(xN))T. For z = (z1,… , zn) vector in , δ(z) = (|z1|,… , |zn|)T and δγ(z) = (|z1|γ,… , |zn|γ)T for γ > 0. Let 1n = (1,… , 1)T. The exponent T is the transpose.
Disturbance rejection of T–S fuzzy systems: a membership function-dependent EID method
Published in International Journal of Systems Science, 2023
Shengnan Tian, Kang-Zhi Liu, Manli Zhang, Chengda Lu, Luefeng Chen, Min Wu, Jinhua She
Notations: denotes the n-dimensional real vector space. represents the transpose of U. U<0 on behalf of that U is a negative definite matrix. means . is defined as diagonal matrix. stands for the n-dimensional identity matrix. sgn denotes the signum function.