Let L/K be a field extension. A K-vector space W can be extended to an L-vector space L⊗KW , and W embeds into L⊗KW by w 7→ 1⊗w. Under this embedding, when W 6= 0 a K-basis {ei} of W turns into an L-basis {1⊗ ei} of L⊗KW . Passing from W to L⊗KW is called ascent. In the other direction, if we are given an L-vector space V 6= 0, we may ask how to describe the K-subspaces W ⊂ V such that a K-basi...

The purpose of the present article is to define and study a new class of descent algebras, called twisted descent algebras. These algebras are associated to the Barratt-Joyal theory of twisted bialgebras in the same way than classical descent algebras are associated to classical bialgebras. The theory of twisted descent algebras is a refinement of the theory of descent algebras of twisted algeb...

2 (First order) Descent methods, rates 2 2.1 Gradient descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 What can we achieve? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Second order methods: Newton’s method . . . . . . . . . . . . . . . . . 7 2.4 Multistep first order methods . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4.1 Heavy ball method . ...

We consider in this work the enumeration of involutions by descent sets, and based on that, by descent numbers. Formulas of the number of involutions with a prescribed descent set, with a prescribed descent number and Frobenius-type formulas are given.

1Department of Urology, University General Hospital of Heraklion, University of Crete Medical School, Heraklion, Crete, Greece; 2First Pediatric Surgery Clinic, Aghia Sophia Children’s Hospital, Athens, Greece; 3Department of Urology, Tottori University School of Medicine, Yonago, Japan; 4Laboratory of Molecular Urology and Genetics of Human Reproduction, Department of Urology, Ioannina Univers...

We introduce a method for establishing that a reduction strategy is normalising and minimal, or dually, that it is perpetual and maximal, in the setting of abstract rewriting. While being complete, the method allows to reduce these global properties to the verification of local diagrams. We show its usefulness both by giving uniform proofs of some known results and by establishing new ones.

Introduction In classical C̆ech theory, we “compute” (or better: filter) the cohomology of a sheaf when given an open covering. Namely, if X is a topological space, U = {Ui} is an indexed open covering, and F is an abelian sheaf on X, then we get a C̆ech to derived functor spectral sequence E 2 = H (U,H(F ))⇒ H(X,F ), where H(F ) is the presheaf whose value on an open U is H(U,F |U ) (and we use ...

Jugular wave recognition requires looking for descents rather than rises. The dominant normal descent is systolic and should be called the X prime (X') descent. The X' is a rapid collapse caused by the descent of the base of the right ventricle and can be easily recognized by noting that the collapse falls onto the second heart sound or is simultaneous with the radial pulse. The V wave and Y de...

Following the setting of the Dai-Liao (DL) parameter in conjugate gradient (CG) methods‎, ‎we introduce two new parameters based on the modified secant equation proposed by Li et al‎. ‎(Comput‎. ‎Optim‎. ‎Appl‎. ‎202:523-539‎, ‎2007) with two approaches‎, ‎which use an extended new conjugacy condition‎. ‎The first is based on a modified descent three-term search direction‎, ‎as the descent Hest...