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### Short Proofs for the Determinant Identities [article]

Pavel Hrubes, Iddo Tzameret
2013 arXiv   pre-print
This yields a solution to a basic open problem in propositional 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

Pavel Hrubes, Iddo Tzameret
2012 Proceedings of the 44th symposium on Theory of Computing - STOC '12
This yields a solution to a basic open problem in propositional 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

Pavel Hrubeš, Iddo Tzameret
2015 SIAM journal on computing (Print)
This yields a solution to a basic open problem in propositional 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.  ...

### A short proof of a symmetry identity for the \$q\$-Hahn distribution

Guillaume Barraquand
2014 Electronic Communications in Probability
We give a 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 |  ...

### A short proof of an identity of Euler

Daniel Shanks
1951 Proceedings of the American Mathematical Society
Now 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

Daniel Shanks
1951 Proceedings of the American Mathematical Society
Now 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

C. Koutschan, M. Kauers, D. Zeilberger
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.  ...

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

Volker Strehl
1984 European journal of combinatorics (Print)
Two 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

Pietro Paparella
2017 The College Mathematics Journal
This 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]

Mercedes Helena Rosas
2017 arXiv   pre-print
We provide a combinatorial 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  ...

### Existence of short proofs for nondivisibility of sparse polynomials under the extended Riemann hypothesis

Dima Yu. Grigoriev, Marek Karpinski, Andrew M. Odlyzko
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]

Dinesh Khurana
2009 arXiv   pre-print
We give a very 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.  ...

### A Short Proof of the Fact That the Matrix Trace Is the Expectation of the Numerical Values

Tomasz Kania
2015 The American mathematical monthly
Using 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 ).  ...

### Some composition determinants

J.M. Brunat, C. Krattenthaler, A. Lascoux, A. Montes
2006 Linear Algebra and its Applications
Appl. 23 (2001) 459-471] and [A polynomial generalization of 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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