On the exponent of e-regular primitive matrices
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Let Pnr be the set of n-by-n r-regular primitive (0, 1)-matrices. In this paper, an explicit formula is found in terms of n and r for the minimum exponent achieved by matrices in Pnr. Moreover, matrices achieving that exponent are given in this paper. Gregory and Shen conjectured that bnr = (n/r)2 + 1 is an upper bound for the exponent of matrices in Pnr. Matrices achieving the exponent bnr are presented for the case when n is not a multiple of r. In particular, it is shown that b2r+1,r is the maximum exponent attained by matrices in P2r+1,r. When n is a multiple of r, it is conjectured that the maximum exponent achieved by matrices in Pnr is strictly smaller than bnr. Matrices attaining the conjectured maximum exponent in that set are presented. It is shown that the conjecture is true when n = 2r.Keywords:
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In this note, we show that a Toy Conjecture made by (Boyle, Ishai, Pass, Wootters, 2017) is false, and propose a new one. Our attack does not falsify the full (non-toy) conjecture in that work, and it is our hope that this note will help further the analysis of that conjecture. Independently, (Boyle, Holmgren, Ma, Weiss, 2021) have obtained similar results.
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The first two authors have shown [KK99,KK00] that the sum the exponent (and thus the number) of maximal repetitions of exponent at least 2 (also called runs) is linear in the length of the word. The exponent 2 in the definition of a run may seem arbitrary. In this paper, we consider maximal repetitions of exponent strictly greater than 1.
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The main objective of this paper is to present an answer to Bressoud's conjecture for the case $j=0$, resulting in a complete solution to the conjecture. The case for $j=1$ has been recently resolved by Kim. Using the connection established in our previous paper between the ordinary partition function $B_0$ and the overpartition function $\overline{B}_1$, we found that the proof of Bressoud's conjecture for the case $j=0$ is equivalent to establishing an overpartition analogue of the conjecture for $j=1$. By generalizing Kim's method, we obtain the desired overpartition analogue of Bressoud's conjecture for $j=1$, which eventually enables us to confirm Bressoud's conjecture for the case $j=0$.
abc conjecture
Lonely runner conjecture
Elliott–Halberstam conjecture
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Dedicated to 60-th anniversary of E. Zelmanov Abstract. We show how to use generating exponent matrices to study the quivers of exponent matrices. We also describe the admissible quivers of 3 × 3 exponent matrices.
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Let Pnr be the set of n-by- nr -regular primitive (0,1)-matrices. In this paper, an explicit formula is found in terms of n and r for the minimum exponent achieved by matrices in Pnr. Moreover, matrices achieving that exponent are given in this paper. Gregory and Shen conjectured that bnr = n 2 + 1 is an upper bound for the exponent of matrices in Pnr .M atrices achieving the exponent bnr are presented for the case when n is not a multiple of r. In particular, it is shown that b2r+1,r is the maximum exponent attained by matrices in P2r+1,r .W henn is a multiple of r ,i t is conjectured that the maximum exponent achieved by matrices in Pnr is strictly smaller than bnr. Matrices attaining the conjectured maximum exponent in that set are presented. It is shown that the conjecture is true whenn =2 r.
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Abstract This paper deals with seismic activity represented by a hazard curve through a single parameter – exponent k as given in EN 1998-1, as well as with its implications on importance factors. We have used the SHARE project dataset for calculation of exponent k for the wider European area and limited number of separate national studies for comparison of results since comparison to the SHARE results on the same dataset resulted with values of exponent k smaller by 1–1.5. The results indicate that recommended value of exponent k of 3 is rather an exception than expected value in seismically active regions, and that with the exclusion of Vrancea zone, for majority of Europe exponent k is well below assumed in EN 1998-1, which consequently indicate that importance factors for these locations should be larger than recommended in EN 1998-1.
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At the end of 1960's, Lawrence Zalcman posed a conjecture that the coefficients of univalent functions $f(z) = z + \sum\limits_2^\infty a_n z^n$ on the unit disk satisfy the sharp inequality $|a_n^2 - a_{2n-1}| \le (n-1)^2$, with equality only for the Koebe function. This remarkable conjecture implies the Bieberbach conjecture, investigated by many mathematicians, and still remains a very difficult open problem for all n > 3; it was proved only in certain special cases. We provide a proof of Zalcman's conjecture based on results concerning the plurisubharmonic functionals and metrics on the universal Teichmüller space. As a corollary, this implies a new proof of the Bieberbach conjecture. Our method gives also other new sharp estimates for large coefficients.
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In this paper, we generalize the Cosmetic Surgery Conjecture to an $n$-cusped hyperbolic $3$-manifold and prove it under the assumption of another well-known conjecture in number theory, so called the Zilber-Pink Conjecture. For $n=1$ and $2$, we show them without the assumption.
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The exponent of a group is a basic conception of group theory,which indicates the property character of orders of elements of a group.This paper makes primary research into the exponent of group,and has the conclusion as following.Let the orders of elements of group G are finite,and sup|a|a∈G=sup|a|a∈C(G),then,the exponent of G expG=sup|a|a∈C(G).Moreover,the structure of the group is represented with exponent further.
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Let Pnr be the set of n-by-n r-regular primitive (0, 1)-matrices. In this paper, an explicit formula is found in terms of n and r for the minimum exponent achieved by matrices in Pnr. Moreover, matrices achieving that exponent are given in this paper. Gregory and Shen conjectured that bnr = (n/r)2 + 1 is an upper bound for the exponent of matrices in Pnr. Matrices achieving the exponent bnr are presented for the case when n is not a multiple of r. In particular, it is shown that b2r+1,r is the maximum exponent attained by matrices in P2r+1,r. When n is a multiple of r, it is conjectured that the maximum exponent achieved by matrices in Pnr is strictly smaller than bnr. Matrices attaining the conjectured maximum exponent in that set are presented. It is shown that the conjecture is true when n = 2r.
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