A quick proof of the 1, 2, 4, 8 theorem

AbstractGhenciu and Lewis introduced the notion of a strong Dunford–Pettis set and used this notion to study the presence or absence of isomorphic copies of c0 in Banach spaces. The authors asserted that they could obtain a fundamental result of J. Elton without resorting to Ramsey theory. While the stated theorems are correct, unfortunately there is a flaw in the proof of the first theorem in the paper which also affects subsequent corollaries and theorems. The difficulty is discussed, and Elton's results are employed to establish a Schauder basis proposition which leads to a quick proof of the theorem in question. Additional results where questions arise are discussed on an individual basis.

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A Quick Proof of Wagner's Equivalence Theorem

Journal of the London Mathematical Society ◽

10.1112/jlms/s2-3.4.661 ◽

1971 ◽

Vol s2-3 (4) ◽

pp. 661-664 ◽

Cited By ~ 7

Author(s):

H. Peyton Young

Keyword(s):

Equivalence Theorem ◽

Quick Proof

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Mixing solutions for the Muskat problem with variable speed

Journal of Evolution Equations ◽

10.1007/s00028-020-00655-1 ◽

2020 ◽

Author(s):

Florent Noisette ◽

László Székelyhidi

Keyword(s):

Weak Solutions ◽

Variable Speed ◽

Mixing Speed ◽

Link Type ◽

Muskat Problem ◽

Quick Proof ◽

Math Phys

AbstractWe provide a quick proof of the existence of mixing weak solutions for the Muskat problem with variable mixing speed. Our proof is considerably shorter and extends previous results in Castro et al. (Mixing solutions for the Muskat problem, 2016, arXiv:1605.04822) and Förster and Székelyhidi (Comm Math Phys 363(3):1051–1080, 2018).

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Inverse semigroup homomorphisms via partial group actions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700019778 ◽

2001 ◽

Vol 64 (1) ◽

pp. 157-168 ◽

Cited By ~ 6

Author(s):

Benjamin Steinberg

Keyword(s):

Inverse Semigroup ◽

Group Actions ◽

Inverse Semigroups ◽

Inverse Monoid ◽

Partial Group ◽

Inverse Monoids ◽

Maximal Group ◽

Special Cases ◽

Quick Proof ◽

Group Component

This papar constructs all homomorphisms of inverse semigroups which factor through an E-unitary inverse semigroup; the construction is in terms of a semilattice component and a group component. It is shown that such homomorphisms have a unique factorisation βα with α preserving the maximal group image, β idempotent separating, and the domain I of β E-unitary; moreover, the P-representation of I is explicitly constructed. This theory, in particular, applies whenever the domain or codomain of a homomorphism is E-unitary. Stronger results are obtained for the case of F-inverse monoids.Special cases of our results include the P-theorem and the factorisation theorem for homomorphisms from E-unitary inverse semigroups (via idempotent pure followed by idempotent separating). We also deduce a criterion of McAlister–Reilly for the existence of E-unitary covers over a group, as well as a generalisation to F-inverse covers, allowing a quick proof that every inverse monoid has an F-inverse cover.

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The Rank of the Trip Matrix of a Positive Knot Diagram

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216597000194 ◽

1997 ◽

Vol 06 (02) ◽

pp. 299-301 ◽

Cited By ~ 10

Author(s):

Louis Zulli

Keyword(s):

Jones Polynomial ◽

Quick Proof

In this note we show that the rank of the trip matrix of a positive knot diagram is exactly twice the genus of the associated positive knot. From this, we give a quick proof of the following result of Murasugi: The term of lowest degree in the Jones polynomial of a positive knot is 1 · tg, where g is the genus of the knot.

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KAWACHI'S INVARIANT FOR NORMAL SURFACE SINGULARITIES

International Journal of Mathematics ◽

10.1142/s0129167x98000270 ◽

1998 ◽

Vol 09 (05) ◽

pp. 623-640 ◽

Cited By ~ 1

Author(s):

VLADIMIR MAŞEK

Keyword(s):

Linear Systems ◽

A Priori ◽

Normal Surface ◽

Numerical Invariant ◽

Normal Surfaces ◽

Log Canonical ◽

Surface Singularities ◽

Canonical Surface ◽

Quick Proof ◽

Canonical Singularities

We study a useful numerical invariant of normal surface singularities, introduced recently by T. Kawachi. Using this invariant, we give a quick proof of the (well-known) fact that all log-canonical surface singularities are either elliptic Gorenstein or rational (without assuming a priori that they are ℚ-Gorenstein). In Sec. 2 we prove effective results (stated in terms of Kawachi's invariant) regarding global generation of adjoint linear systems on normal surfaces with boundary. Such results can be used in proving effective estimates for global generation on singular threefolds. The theorem of Ein–Lazarsfeld and Kawamata, which says that the minimal center of log-canonical singularities is always normal, explains why the results proved here are relevant in that situation.

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