scholarly journals Integral representations for local dilogarithm and trilogarithm functions

2021 ◽  
Vol 5 (1) ◽  
pp. 337-352
Author(s):  
Masato Kobayashi ◽  
Keyword(s):  
Lower Bound ◽  
Unit Interval ◽  
Integral Representations ◽  
Euler Sums ◽  
Dilogarithm Function

We show new integral representations for dilogarithm and trilogarithm functions on the unit interval. As a consequence, we also prove (1) new integral representations for Apéry, Catalan constants and Legendre \(\chi\) functions of order 2, 3, (2) a lower bound for the dilogarithm function on the unit interval, (3) new Euler sums.

scholarly journals A nonlinear Lazarev–Lieb theorem: L2-orthogonality via motion planning

2020 ◽  
pp. 1-17
Author(s):  
Florian Frick ◽  
Matt Superdock
Keyword(s):  
Lower Bound ◽  
Motion Planning ◽  
Unit Circle ◽  
Smooth Function ◽  
Unit Interval ◽  
Inner Product ◽  
Lower Bounding ◽  
Integrable Functions ◽  
Equivariant Topology ◽  
Norm Bound

Lazarev and Lieb showed that finitely many integrable functions from the unit interval to [Formula: see text] can be simultaneously annihilated in the [Formula: see text] inner product by a smooth function to the unit circle. Here, we answer a question of Lazarev and Lieb proving a generalization of their result by lower bounding the equivariant topology of the space of smooth circle-valued functions with a certain [Formula: see text]-norm bound. Our proof uses a variety of motion planning algorithms that instead of contractibility yield a lower bound for the [Formula: see text]-coindex of a space.

Lower and upper bounds for the linear arrangement problem on interval graphs

2018 ◽  
Vol 52 (4-5) ◽  
pp. 1123-1145
Author(s):  
Alain Quilliot ◽  
Djamal Rebaine ◽  
Hélène Toussaint
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Feasible Solution ◽  
Interval Graph ◽  
Upper Bounds ◽  
Unit Interval ◽  
Lower And Upper Bounds ◽  
Linear Arrangement ◽  
Arrangement Problem ◽  
Linear Arrangement Problem

We deal here with theLinear Arrangement Problem(LAP) onintervalgraphs, any interval graph being given here together with its representation as theintersectiongraph of some collection of intervals, and so with relatedprecedenceandinclusionrelations. We first propose a lower boundLB, which happens to be tight in the case ofunit intervalgraphs. Next, we introduce the restriction PCLAP of LAP which is obtained by requiring any feasible solution of LAP to be consistent with theprecedencerelation, and prove that PCLAP can be solved in polynomial time. Finally, we show both theoretically and experimentally that PCLAP solutions are a good approximation for LAP onintervalgraphs.

New integral representations of the polylogarithm function

2006 ◽  
Vol 463 (2080) ◽  
pp. 897-905 ◽  
Author(s):  
Djurdje Cvijović
Keyword(s):  
Bernoulli Polynomials ◽  
Integral Representations ◽  
The Other ◽  
Important Special Case ◽  
Polylogarithm Function ◽  
Physical Problems ◽  
The Riemann Zeta Function ◽  
Dilogarithm Function ◽  
Riemann Zeta ◽  
Special Case

Maximon has recently given an excellent summary of the properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function Li s ( z ). The polylogarithm function appears in several fields of mathematics and in many physical problems. We, by making use of elementary arguments, deduce several new integral representations of the polylogarithm Li s ( z ) for any complex z for which | z |<1. Two are valid for all complex s , whenever Re  s >1. The other two involve the Bernoulli polynomials and are valid in the important special case where the parameter s is a positive integer. Our earlier established results on the integral representations for the Riemann zeta function ζ (2 n +1), n ∈ N , follow directly as corollaries of these representations.

scholarly journals Euler Sums and Contour Integral Representations

1998 ◽  
Vol 7 (1) ◽  
pp. 15-35 ◽  
Author(s):  
Philippe Flajolet ◽  
Bruno Salvy
Keyword(s):  
Integral Representations ◽  
Contour Integral ◽  
Euler Sums

scholarly journals Characterizations of Positive Operator-Monotone Functions and Monotone Riemannian Metrics via Borel Measures

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1162
Author(s):  
Pattrawut Chansangiam ◽  
Sorin V. Sabau
Keyword(s):  
Integral Representation ◽  
Positive Operator ◽  
Riemannian Metrics ◽  
Unit Interval ◽  
Integral Representations ◽  
Monotone Functions ◽  
Borel Measures ◽  
Operator Monotone ◽  
Operator Monotone Functions ◽  
Harmonic Means

We show that there is a one-to-one correspondence between positive operator-monotone functions on the positive reals, monotone Riemannian metrics, and finite positive Borel measures on the unit interval. This correspondence appears as an integral representation of weighted harmonic means with respect to that measure on the unit interval. We also investigate the normalized/symmetric conditions for operator-monotone functions. These conditions turn out to characterize monotone metrics and Morozowa–Chentsov functions as well. Concrete integral representations of such functions related to well-known monotone metrics are also provided. Moreover, we use this integral representation to decompose positive operator-monotone functions. Such decomposition gives rise to a decomposition of the associated monotone metric.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

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