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A direct proof of the five element basis theorem [article]

Boban Velickovic, Giorgio Venturi
<span title="2016-03-01">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We present a direct proof of the consistency of the existence of a five element basis for the uncountable linear orders.  ...  Our argument is based on the approach of notion of saturation of Aronszajn trees considered by Koenig, Larson, Moore and Velickovic and simplifies the original proof of Moore.  ...  The purpose of this note is to present a direct proof of CAT, and therefore the existence of a five element linear basis, assuming the conjunction of BPFA and ϕ.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1012.0596v2">arXiv:1012.0596v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rdaqhgmbl5bjrfwzhyiyrokzsy">fatcat:rdaqhgmbl5bjrfwzhyiyrokzsy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200924081306/https://arxiv.org/pdf/1012.0596v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/eb/f0/ebf04d9de0495f069de988a5d1c297e8b8537c8d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1012.0596v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Page 1275 of Mathematical Reviews Vol. 45, Issue 5 [page]

<span title="">1973</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
A five page proof yields a characterization of M,,M,,M_ as the modules M eT satisfying J(M/(x+y)M)=0, J(O:a+y)y=0.  ...  If p has one of the forms 2° + 1, 3-2*+1, 5-2*+1 (k>0) then Ly, has a direct summand isomorphic to (FT)°. The proof requires 13 lemmas dealing with a minimal counter-example to the theorem. H. K.  ... 
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Page 1027 of Mathematical Reviews Vol. , Issue 82c [page]

<span title="">1982</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The paper gives a basis for the laws of each semigroup of order less than five and ends by producing an example of an n-element semigroup a basis for whose laws requires at least n variables.  ...  The following theorem is proven: A torsion free abelian group G is separable if every element of G is contained in a completely decomposable direct summand of G.  ... 
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Page 423 of American Journal of Mathematics Vol. 54, Issue 2 [page]

<span title="">1932</span> <i title="Johns Hopkins University Press"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_american-journal-of-mathematics" style="color: black;">American Journal of Mathematics </a> </i> &nbsp;
If we select the latter five elements as a new basis, calling them respectively @’,, e's, e’3, e's, and e’;, we obtain a new multiplication table which is identical with the original one with e; («= 1,2,3,4,5  ...  Thus the conditions of theorem 4. 1 are satisfied. The proof is similar when z is in Bu.  ... 
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Page 6450 of Mathematical Reviews Vol. , Issue 2002I [page]

<span title="">2002</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Note also that the theorem itself is not stated in Kazhdan and Lusztig’s paper. In the paper under review the author gives a direct proof of the theorem by combinatorial methods.  ...  But after a decade has passed, the direct proof is still not readily available. Further, there have been new developments in the last decade.  ... 
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André-Quillen homology in degree five

Yves Pitteloud
<span title="">1994</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/74pjbehpfraq5ogj3loetdwp7m" style="color: black;">Journal of Pure and Applied Algebra</a> </i> &nbsp;
We have then the following result, the proof of which will occupy the next two sections of this paper. Proof. This is a direct consequence of the following lemma. Cl Proof.  ...  In the general case, one has to use a theorem of Lazard (cf. [7, Theorem 1.2]), saying that a flat module is isomorphic to a direct limit of free modules.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0022-4049(94)90033-7">doi:10.1016/0022-4049(94)90033-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/p4knfafh5jbqberhfyhl4lbtje">fatcat:p4knfafh5jbqberhfyhl4lbtje</a> </span>
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On representations of anisotropic invariants

Liu I-Shih
<span title="">1982</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wu724rsyybdghixb2famiiw7jy" style="color: black;">International Journal of Engineering Science</a> </i> &nbsp;
The results enable us to obtain representations of anisotropic functions using the much well-known tables for representations of isotropic functions.  ...  Transverse isotropy. orthotropy and crystal classes of triclinic. monoclinic and rhombic systems are considered.  ...  of the elements in (7.4), one is tempted to replace ulu2 and u22r, by a single element uluz-u2ul and hence obtain a smaller basis of 18 instead of 19 elements.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0020-7225(82)90092-1">doi:10.1016/0020-7225(82)90092-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gg363zamqrdhnf3hlvmdpl72qa">fatcat:gg363zamqrdhnf3hlvmdpl72qa</a> </span>
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Bell's theorem without inequalities and only two distant observers [article]

P.K.Aravind
<span title="2002-07-07">2002</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
A proof of Bell's theorem without inequalities and involving only two observers is given by suitably extending a proof of the Bell-Kochen-Specker theorem due to Mermin.  ...  This proof is generalized to obtain an inequality-free proof of Bell's theorem for a set of n Bell states (with n odd) shared between two distant observers.  ...  I would like to thank David Mermin and Adán Cabello for several valuable comments on an earlier draft of this paper and Vlad Babau for some discussions during the early stages of this work.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/quant-ph/0104133v7">arXiv:quant-ph/0104133v7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/scuxwve4abhs5bjuikiilvb3ri">fatcat:scuxwve4abhs5bjuikiilvb3ri</a> </span>
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HOMOLOGICAL DECISION PROBLEMS FOR FINITELY GENERATED GROUPS WITH SOLVABLE WORD PROBLEM

W. A. BOGLEY, J. HARLANDER
<span title="">2002</span> <i title="World Scientific Pub Co Pte Lt"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/467zefb3vfbmdmbauxdzukpug4" style="color: black;">International journal of algebra and computation</a> </i> &nbsp;
Then the elements f (a) a , aA, form a basis for a free subgroup of G * H. Proof: Consider the free group F (A) with basis A and the homomorphism φ : F (A) → G * H given by φ(a) = f (a) a .  ...  Similarly, the right sides determine a basis for a free subgroup of rank five in F (α, β).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1142/s0218196702000973">doi:10.1142/s0218196702000973</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/c5oc22eqlve3hofelfnznd4y7e">fatcat:c5oc22eqlve3hofelfnznd4y7e</a> </span>
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More on Groups in Which Each Element Commutes with its Endomorphic Images

J. J. Malone
<span title="">1977</span> <i title="JSTOR"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/a64sw4kcsveajhudnssdwrwvze" style="color: black;">Proceedings of the American Mathematical Society</a> </i> &nbsp;
Faudree has given examples of nonabelian groups which have the property cited in the title. His groups are /»-groups such that (i)  ...  The question of when £(G) is a ring is treated in Maxson's paper [8] which has already been cited and by B. C. McQuarrie and J. J. Malone in [9] .  ...  This theorem and the result from near ring theory that iEiG), + ) is abelian if and only if multiplication is left and right distributive over addition tells us that Faudree's groups, for odd p, are nonabelian  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2307/2041892">doi:10.2307/2041892</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kqf54mn3ozh2lanob3vcygrwoa">fatcat:kqf54mn3ozh2lanob3vcygrwoa</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170825154546/http://www.ams.org/journals/proc/1977-065-02/S0002-9939-1977-0447351-X/S0002-9939-1977-0447351-X.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/6f/e0/6fe08341f7d424517b9d4bd35e9013ed6856eee5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2307/2041892"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> jstor.org </button> </a>

More on groups in which each element commutes with its endomorphic images

J. J. Malone
<span title="1977-02-01">1977</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/a64sw4kcsveajhudnssdwrwvze" style="color: black;">Proceedings of the American Mathematical Society</a> </i> &nbsp;
Faudree has given examples of nonabelian groups which have the property cited in the title. His groups are /»-groups such that (i)  ...  The question of when £(G) is a ring is treated in Maxson's paper [8] which has already been cited and by B. C. McQuarrie and J. J. Malone in [9] .  ...  This theorem and the result from near ring theory that iEiG), + ) is abelian if and only if multiplication is left and right distributive over addition tells us that Faudree's groups, for odd p, are nonabelian  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9939-1977-0447351-x">doi:10.1090/s0002-9939-1977-0447351-x</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bxyb34o64jgvthdhotwik23zza">fatcat:bxyb34o64jgvthdhotwik23zza</a> </span>
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Construction of a 45,694-dimensional simple module of Lyons' sporadic group over GF(2)

Holger W Gollan, Gerhard O Michler
<span title="">1997</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wsx3rzhpingfvewcn5nwhfkq3e" style="color: black;">Linear Algebra and its Applications</a> </i> &nbsp;
The construction has been performed on a parallel computer of type IBM SP1 with i0 nodes.  ...  Furthermore, an explicit 45,694-dimensional irreducible matrix representation of Ly over the field GF(2) with two elements is given.  ...  Our special thanks are due to the referee for several helpful suggestions improving the presentation of the paper and pointing out a gap in an argument in an earlier version.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0024-3795(96)00002-x">doi:10.1016/s0024-3795(96)00002-x</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ru3qtgz2s5bbxolktfdtbxz2cy">fatcat:ru3qtgz2s5bbxolktfdtbxz2cy</a> </span>
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Wang algebra and matroids

R. Duffin, T. Morley
<span title="">1978</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/sqa5g6kqfjdezfulhmsorb4fqy" style="color: black;">IEEE Transactions on Circuits and Systems</a> </i> &nbsp;
Moreover tke Wang shortcut ~method can still be used to find the joint resistance of an electrical network. Manuscri t  ...  Wang showed that these rules give a shortcut method for finding tbe joint resistance (or driving point reshance) of an electrical network.  ...  Let T be a tree and let x be an edge that is not in T, then Proof One direction follows from Theorems 4 and 5.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tcs.1978.1084531">doi:10.1109/tcs.1978.1084531</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/d5ivp77owvf7pf7mtdxv2svjsy">fatcat:d5ivp77owvf7pf7mtdxv2svjsy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170610233825/http://homepages.math.uic.edu:80/~kauffman/DuffinLong.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/51/37/5137d87fbceb312a0d71a258d33ec16fa386e86b.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tcs.1978.1084531"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Page 723 of School Science and Mathematics Vol. 24, Issue 7 [page]

<span title="">1924</span> <i title="Blackwell Publishing Ltd."> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_school-science-and-mathematics" style="color: black;">School Science and Mathematics </a> </i> &nbsp;
The geometry of the first half-year should include theorems from both plane and solid geometry and the basis of organiza- tion should be the method of proof.  ...  In the 9th year the pupil has used only direct proofs and he has omitted some of the more difficult theorems which can be proved directly. He has proved no theorems of solid geometry.  ... 
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An equational basis in four variables for the three-element tournament

G. Grätzer, A. Kisielewicz, B. Wolk
<span title="">1992</span> <i title="Institute of Mathematics, Polish Academy of Sciences"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qja4hqeadzam7ph6bhaks7ersu" style="color: black;">Colloquium Mathematicum</a> </i> &nbsp;
It would be interesting to find a direct equational theoretic proof of the equivalence. The existence of such a proof is known. 3 . 3 The Theorem. Our main result is the following:Theorem.  ...  Thus, in view of the Characterization Theorem of Z, quoted in §2, this proves that K ⊆ Z; since by the Lemma, K ⊇ Z, this completes the proof of the Theorem. Corollary 1. Let A be an algebra.  ... 
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