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Basic Ideas
Published in Steven G. Krantz, Complex Variables, 2019
It is not true that every non-constant polynomial with real coefficients has a real root. For instance, p(x) = x2 + 1 has no real roots. The Fundamental Theorem of Algebra states that every polynomial with complex coefficients has a complex root (see the treatment in §§1.2.4, 5.1.4, 9.3.3 below). The complex field ℂ is the smallest field that contains ℝ and has this so-called algebraic closure property.
The higher-order type 2 Daehee polynomials associated with p-adic integral on ℤ p
Published in Applied Mathematics in Science and Engineering, 2022
Waseem Ahmad Khan, Jihad Younis, Ugur Duran, Azhar Iqbal
Let in conjunction with and be the completion of the algebraic closure of , cf. [2–10], where p be a prime number and the normalized p-adic absolute value is provided by . For (g being a continuous map), the p-adic bosonic integral of g is given as follows: It is observed from (2) that where and , cf. [2–10].
t-closure Operators
Published in International Journal of General Systems, 2019
Mehmet Akif İnce, Funda Karaçal
Let real unit interval and . Consider the t-norms and on L defined as follows: and It can easily be obtained that Since and are algebraic closure operators, for all . Thence, is obtained. Note that and are not continuous.
Series concatenation of 2D convolutional codes by means of input-state-output representations
Published in International Journal of Control, 2018
Joan-Josep Climent, Diego Napp, Raquel Pinto, Rita Simões
Let be a finite field and let denote the algebraic closure of . Denote by the ring of polynomials in two indeterminates with coefficients in , by the field of fractions of and by the ring of formal powers series in two indeterminates with coefficients in .