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Continuum topological derivative - A novel application tool for segmentation of CT and MRI images. [PDF]

Neuroimage Rep
Muthukrishnan V, Jaipurkar S, Damodaran N.   +2 more
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Relatively separated transcendental field extensions

Archiv der Mathematik, 1973
Mordeson, J. N., Vinograde, B.
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Recognizing Simple Subextensions of Purely Transcendental Field Extensions

Applicable Algebra in Engineering, Communication and Computing, 2000
Let \(k(X)/k\) be a finitely generating purely transcendental field extension, where \(X\) is algebraically independent over \(k\). It is known that all intermediate fields \(K\) of \(k(X)/k\) with \(\text{transdeg} (K/k)=1\) are simple extensions of \(k\).
Mueller-Quade, Jörn, Steinwandt, Rainer
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Kronecker Function Rings of Transcendental Field Extensions

Communications in Algebra, 2010
We consider the ring Kr(F/D), where D is a subring of a field F, that is the intersection of the trivial extensions to F(X) of the valuation rings of the Zariski–Riemann space consisting of all valuation rings of the extension F/D and investigate the ideal structure of Kr(F/D) in the case where D is an affine algebra over a subfield K of F and the ...
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Transcendental Field Extensions

2018
Looking at the domain of rational numbers \(\mathbb Q\), it was realized quite early that certain familiar “numbers,” such as \(\sqrt {2}\), are not rational and are hence irrational, as one began to say. There were several attempts to classify irrational numbers.
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ON THE EXTENSION OF THE RING TOPOLOGY OF A $ \sigma$-BOUNDED FIELD TO A SIMPLE TRANSCENDENTAL EXTENSION OF THE FIELD

Mathematics of the USSR-Sbornik, 1988
See the review in Zbl 0631.12020.
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Prolongations of valuations to simple transcendental extensions with given residue field and value group

Mathematika, 1991
Let \(K_ 0(x)\) be a simple transcendental extension of a field \(K_ 0\), \(v_ 0\) be a non-trivial valuation of \(K_ 0\) with value group \(G_ 0\) and residue field \(k_ 0\). Suppose we are given a finite extension k of \(k_ 0\) and an inclusion \(G_ 0\subseteq G_ 1\subseteq G\) of totally ordered abelian groups with \([G_ 1 : G_ 0]
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On the residual transcendental extension of a valuation on a fields K to K (X, σ)

Results in Mathematics, 2004
Let \(K\) be a field, \(\sigma\) an automorphism of \(K\) and \(v\) a \(\sigma\)-invariant valuation on \(K\). The main goal of the paper under review is to study all the extensions of \(v\) to the skew field \(K(X,\sigma)\) which consists of the left quotients of the skew polynomial ring \(K[X,\sigma]\) where \(Xa=\sigma(a)X\) for any \(a\in K\). When
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Symbolic manipulation of integrodifferential expressions and factorization of linear ordinary differential operators over transcendental extensions of a differential field

Proceedings of the 1997 international symposium on Symbolic and algebraic computation - ISSAC '97, 1997
We discuss the problem of manipulation of expressions involving indefinite functions, integration operators and inverses of linear ordinary differential operators (LODO). Using Loewy-Ore formal theory we obtain some subnormal forms for such expressions and algorithms for verification of identities involving such expressions.
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Using Groebner bases to determine the algebraic and transcendental nature of field extensions: Return of the killer tag variables

1993
Suppose I is a prime ideal in k[X1, ...,Xn] with a given finite generating set and k(q1,...,qm) is a finitely generated subfield of the field of fractions Z of k[X1, ..., Xn]/I and c is an element of Z. We present Groebner basis techniques to determine: if c is transcendental over k(q1,...,qm), a minimal polynomial for c if c is algebraic ...
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