A Logarithmic Inequality

Kalachev, Gleb; Sadov, Sergey

doi:10.1134/s0001434618010224

Cited by 2 publications

(17 citation statements)

References 3 publications

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“…For example, the set {1, 2, 3} can be block-partitioned as (123), (12)(3), (13)(2), (1) (23), and (1)(2)(3). A partition with k = 1, 2, 3 blocks can be ordered in k!…”

Section: Min-sums I: the Problem For An Individual Graphmentioning

confidence: 99%

“…It can be shown that min 1<z≤r−1 f (z, r) > 1 + ln ln(r − ln r). The calculation is straightforward but not too short; details can be found in [27, Solution of Problem 67B] (an even more precise estiimate for min z f (z, r) is given in [23]). The inequality (26) follows.…”

Section: Casementioning

confidence: 99%

“…One can find the asymptotics (r → ∞) of the minimum of the function f (z, r) (again, we refer to [27] or [23] for details) and obtain The asymptotic formula remains true if z is allowed to assume only integer values. Putting k = z and minimizing over k, we find that in the extremal case m Γ k ∧ = e ln(n + 1) + O((ln n) −1 ), which is equivalent to (27).…”

Section: Casementioning

confidence: 99%

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Three steps away from Shapiro's problem: lower bounds for graphic sums with functions `max' or `min' in denominators

Sadov

2021

Preprint

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Taking Shapiro's cyclic sums n i=1 x i /(x i+1 + x i+2 ) (assuming index addition mod n) as a starting point, we introduce a broader class of cyclic sums, called generalized Shapiro-Diananda sums, where the denominators are p-th order power means of the sets {x i+j 1 , . . . , x i+j k } with fixed distinct integers j 1 , . . . , j k and 1 ≤ i ≤ n.Generalizing further, we replace the set of arguments of the power mean in the i-th denominator by an arbitrary nonempty subset of {1, . . . , n} interpreted as the set of out-neighbors of the node number i in a directed graph with n nodes. We call such sums graphic power sums since their structure is controlled by directed graphs.The inquiry, as in the well-researched case of Shapiro's sums, concerns the greatest lower bound of the given "sum" as a function of positive variables x 1 , . . . , x n . We show that the cases of p = +∞ (max-sums) and p = −∞ (min-sums) are tractable.For the max-sum associated with a given graph the g.l.b. is always an integer; for a strongly connected graph it equals to graph's girth.For the similar min-sum, we could not relate the g.l.b. to a known combinatorial invariant; we only give some estimates and describe a method for finding the g.l.b., which has factorial complexity in n.A satisfactory analytical treatment is available for the secondary minimization -when the g.l.b.'s of min-sums for individual graphs are mininized over the class of strongly connected graphs with n nodes. The result (depending only on n) is found to be asymptotic to e ln n.

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“…For example, the set {1, 2, 3} can be block-partitioned as (123), (12)(3), (13)(2), (1) (23), and (1)(2)(3). A partition with k = 1, 2, 3 blocks can be ordered in k!…”

Section: Min-sums I: the Problem For An Individual Graphmentioning

confidence: 99%

Section: Casementioning

confidence: 99%

Section: Casementioning

confidence: 99%

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Three steps away from Shapiro's problem: lower bounds for graphic sums with functions `max' or `min' in denominators

Sadov

2021

Preprint

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“…Introduce the minimization problem: to determine f Γ,L (w, t) = min x x, w , where x ≥ 0 and p L (x) = t L for all L ∈ L. (11) It is a particular case of problem (5) and, obviously, f Γ,L (w, t) = f (w, t, A), where A is the arc-circuit incidence matrix for the pair (Γ, L). (The case L = ∅ corresponds to the unconstrained minimum; then f Γ,∅ (w, −) = 0.…”

Section: Arc Sums Cyclic Constraintsmentioning

confidence: 99%

“…The undetermined component x a of a candidate vector x will be called the value of the arc a, to distinguish it from the known weight w a . In formula (11) we wrote 'min' instead of 'inf' since the system of constraints is compact, so there is the unique minimizer by 16, 17.…”

Section: Arc Sums Cyclic Constraintsmentioning

confidence: 99%

Minimization of the sum under product constraints

Sadov

2020

Preprint

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We systematically explore a class of constrained optimization problems with linear objective function and constraints that are linear combinations of logarithms of the optimization variables. Such problems can be viewed as a generalization of the inequality between the arithmetic and geometric means.The existence and uniqueness of the minimizer is proved under natural assumptions in the general case.We study in detail special subclasses where the set of constraints is described in combinatorial terms (oriented graphs, rooted trees). In particular, given a directed, strongly connected graph, we seek to minimize the total of all arc values under cyclic product constraints.We obtain some estimates and asymptotics for the minimum in problems with given (large) number of variables.Also in this context we revisit an asymptotical result known as J. Shallit's minimization problem.The material is presented in the form of a problem book. Along with problems that constitute main theoretical threads, there are many exercises, some mini-paradoxes, and a touch of numerical methods.

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A Logarithmic Inequality

Cited by 2 publications

References 3 publications

Three steps away from Shapiro's problem: lower bounds for graphic sums with functions `max' or `min' in denominators

Three steps away from Shapiro's problem: lower bounds for graphic sums with functions `max' or `min' in denominators

Minimization of the sum under product constraints

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