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Short Proofs for the Determinant Identities
[article]

2013
*
arXiv
*
pre-print

This yields a solution to a basic open problem in propositional

arXiv:1112.6265v2
fatcat:ymrua6bsyfg4hj3udd4ujqjmr4
*proof*complexity, namely, whether there are polynomial-size NC^2-Frege*proofs**for**the**determinant**identities*and*the*hard matrix*identities*... We show that matrix*identities*like AB=I → BA=I (*for*matrices over*the*two element field) as well as basic properties of*the**determinant*have polynomial-size NC^2-Frege*proofs*, and quasipolynomial-size ... To understand our construction of*short*arithmetic*proofs**for**the**determinant**identities*, let us consider*the*following example. ...##
###
Short proofs for the determinant identities

2012
*
Proceedings of the 44th symposium on Theory of Computing - STOC '12
*

This yields a solution to a basic open problem in propositional

doi:10.1145/2213977.2213998
dblp:conf/stoc/HrubesT12
fatcat:e3opnkgbc5gapinaqe5slidhza
*proof*complexity, namely, whether there are polynomial-size NC 2 -Frege*proofs**for**the**determinant**identities*and*the*hard matrix*identities*... We show that matrix*identities*like AB = I → BA = I (*for*matrices over*the*two element field) as well as basic properties of*the**determinant*have polynomial-size NC 2 -Frege*proofs*, and quasipolynomial-size ... To understand our construction of*short*arithmetic*proofs**for**the**determinant**identities*, let us consider*the*following example. ...##
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Short Proofs for the Determinant Identities

2015
*
SIAM journal on computing (Print)
*

This yields a solution to a basic open problem in propositional

doi:10.1137/130917788
fatcat:q6qgppfiv5agtkcg2iuazpwmre
*proof*complexity, namely, whether there are polynomial-size NC 2 -Frege*proofs**for**the**determinant**identities*and*the*hard matrix*identities*... We show that matrix*identities*like AB = I → BA = I (*for*matrices over*the*two element field) as well as basic properties of*the**determinant*have polynomial-size NC 2 -Frege*proofs*, and quasipolynomial-size ... To understand our construction of*short*arithmetic*proofs**for**the**determinant**identities*, let us consider*the*following example. ...##
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A short proof of a symmetry identity for the $q$-Hahn distribution

2014
*
Electronic Communications in Probability
*

We give a

doi:10.1214/ecp.v19-3674
fatcat:hno72oacmbgstf27hrvakhz344
*short*and elementary*proof*of a (q, μ, ν)-deformed Binomial distribution*identity*arising in*the*study of*the*(q, μ, ν)-Boson process and*the*(q, μ, ν)-TASEP. ... This was used in turn to derive exact formulas*for*a large class of observables of both these processes. ... In*the*following, we give a new*proof*of this*identity*. * Université Paris-Diderot, France. ...##
###
Page 82 of The American Mathematical Monthly Vol. 54, Issue 2
[page]

1947
*
The American Mathematical Monthly
*

*The*Bazin-Reiss-Picquet theorem. Recall

*the*meaning of B[A(J{”)/B(J$”)], or B[A/B]

*for*

*short*, from section 2. THEOREM. ... 82 SOME

*IDENTITIES*IN

*THE*THEORY OF

*DETERMINANTS*[February, Furthermore, by (3.3) and (4.2) | A®||adjA| =| 4 |e» (6.6) = (Dayy + EO), Thus every value of dn,

*for*which |A (| vanishes is a zero of |.A | ...

##
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A short proof of an identity of Euler

1951
*
Proceedings of the American Mathematical Society
*

Now

doi:10.1090/s0002-9939-1951-0043808-6
fatcat:nrj5pfhejnepzjelka323qqctm
*the*partial product, P", is*the*first term of Fn (s = 0 in*the**determinants*gives Bkk = l/Fk. ...*The*same results hold*for*families of regular bilinear mappings (n = 2, H = K, Hi = Ki) and*for*families of groups of class 1 or 2. ...##
###
A Short Proof of an Identity of Euler

1951
*
Proceedings of the American Mathematical Society
*

Now

doi:10.2307/2032076
fatcat:kjfti63htzhxpgez6rglc2j23m
*the*partial product, P", is*the*first term of Fn (s = 0 in*the**determinants*gives Bkk = l/Fk. ...*The*same results hold*for*families of regular bilinear mappings (n = 2, H = K, Hi = Ki) and*for*families of groups of class 1 or 2. ...##
###
Proof of George Andrews's and David Robbins's q-TSPP conjecture

2011
*
Proceedings of the National Academy of Sciences of the United States of America
*

*The*conjecture that

*the*orbit-counting generating function

*for*totally symmetric plane partitions can be written as an explicit product formula, has been stated independently by George Andrews and David ... We present a

*proof*of this long-standing conjecture. ... D.Z. was supported in part by

*the*US National Science Foundation. ...

##
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Two Short Proofs of Kemp's Identity for Rooted Plane Trees

1984
*
European journal of combinatorics (Print)
*

Two

doi:10.1016/s0195-6698(84)80040-2
fatcat:utwmnu33ljdszmufb5tpxyp4hy
*short**proofs*of this result are given here, a 'bijective' one, and one making use of continued fract10n generating functions-both avoiding explicit expressions*for**the*numbers involved. ... has recently presented a nice*identity*relating different sets of rooted plane trees numerically. ...*The*close connection between*the*two (finite) continued fractions thus obtained again leads to a*short**proof*of Kemp's*identity*-without*determining**the*ordinary generating functions explicitly. ...##
###
A Short and Elementary Proof of the Two-Sidedness of the Matrix Inverse

2017
*
The College Mathematics Journal
*

This

doi:10.4169/college.math.j.48.5.366
fatcat:ynkw7ypklvgwfi5zg4f7x2qnye
*proof*underscores*the*importance of a basis and provides a*proof*of*the*invertible matrix theorem. ... An elementary*proof*of*the*two-sidedness of*the*matrix-inverse is given using only linear independence and*the*reduced row-echelon form of a matrix. ... Sandomierski [6] gives another elementary*proof*using an under-*determined*homogeneous linear system. ...##
###
A comment of the combinatorics of the vertex operator Γ_(t|X)
[article]

2017
*
arXiv
*
pre-print

We provide a combinatorial

arXiv:1701.02516v2
fatcat:dhnahbee5zfldfphrxvsdrvfea
*proof**for**the**identity*Γ_(t|X) s_α = σ[tX] s_α[x-1/t] due to Thibon et al. ... We include an overview of all*the*combinatorial ideas behind this beautiful*identity*, including a combinatorial description*for**the*expansion of s_(n,α) [X] in*the*Schur basis,*for*any integer value of ... While it is a well-known theorem that*the*previous*identity*holds*for*any partition (see*for*example*the*beautiful combinatorial*proof*of Gessel and Viennot using lattive paths),*the*Jacobi-Trudi*determinant*...##
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Existence of short proofs for nondivisibility of sparse polynomials under the extended Riemann hypothesis

1992
*
Papers from the international symposium on Symbolic and algebraic computation - ISSAC '92
*

*The*authors thank M. Singer

*for*

*the*number of interesting discussions. IEEE, 1990, pp. 840-846. [K 89] Karpinski, M., Boolean References ... Introduction We display an existence of

*the*

*short*(polynomial size)

*proofs*

*for*nondivisibility of two sparse multivariat,e polynomials under

*the*Extended Riemanu Hypothesis (ERH). ... In this case g divides ~if and only if j_/g ~const, arguing as in

*the*

*proof*of Lemma 2

*the*latter

*identity*is equivalent to

*the*equalities q(p) = . . . = q(p2tz+1 ). ...

##
###
A short proof of Grinshpon's theorem
[article]

2009
*
arXiv
*
pre-print

We give a very

arXiv:0904.0906v1
fatcat:qh3rarbufbgtjarjxna4mrlvgi
*short**proof*of*the*result. ... If*the*entries of A commute with each other, then there exists a commutative subring T of R such that A, B ∈ M n (T ).*Proof*. Let I denote*the**identity*of M n (R). ... Grinshpon's theorem is immediate from*the*following result by taking B = A −1 and using*the*fact that a square matrix over a commuta- tive ring is invertible iff its*determinant*is invertible. ...##
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A Short Proof of the Fact That the Matrix Trace Is the Expectation of the Numerical Values

2015
*
The American mathematical monthly
*

Using

doi:10.4169/amer.math.monthly.122.8.782
fatcat:ir4czrnuiffq7gw4v64oh25suu
*the*fact that*the*normalised matrix trace is*the*unique linear functional f on*the*algebra of n × n matrices which satisfies f (I) = 1 and f (AB) = f (BA)*for*all n × n matrices A and B, we derive ... a well-known formula expressing*the*normalised trace of a complex matrix A as*the*expectation of*the*numerical values of A; that is*the*function Ax, x , where x ranges*the*unit sphere of C n . ...*The*above formula is a particular version of a more general*identity**for*symmetric 2-tensors on Riemannian manifolds (consult e.g. [2] ; see also [1]*for**the**proof*1 ). ...##
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Some composition determinants

2006
*
Linear Algebra and its Applications
*

Appl. 23 (2001) 459-471] and [A polynomial generalization of

doi:10.1016/j.laa.2005.11.013
fatcat:rsyv4rog2jex5fwnlzjfixyapi
*the*power-compositions*determinant*, Linear Multilinear Algebra, in press], and they prove two conjectures of*the*second author [Advanced*determinant*... These results generalize previous*determinant*evaluations due to*the*first and fourth author [SIAM J. Matrix Anal. ... Since n−1 i=0 |C(i, k)| = n−1 i=0 i + k − 1 k − 1 = n + k − 1 k*the*only missing piece*for**the**proof*of*the*corollary is*the*verification of*the**identity*∈C(n,k) ! = β 1 +···+β k =n β 1 ! · · · β k ! ...
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