Displaying similar documents to “A new estimate for the sum M ( x ) = ∑ n ≤ x μ ( n ) ”

On linear normal lattices configurations

Mordechay B. Levin, Meir Smorodinsky (2005)

Journal de Théorie des Nombres de Bordeaux

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In this paper we extend Champernowne’s construction of normal numbers in base b to the ℤ d case and obtain an explicit construction of the generic point of the ℤ d shift transformation of the set { 0 , 1 , . . . , b - 1 } ℤ d . We prove that the intersection of the considered lattice configuration with an arbitrary line is a normal sequence in base b .

The ratio and generating function of cogrowth coefficients of finitely generated groups

Ryszard Szwarc (1998)

Studia Mathematica

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Let G be a group generated by r elements g 1 , … , g r . Among the reduced words in g 1 , … , g r of length n some, say γ n , represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of γ 2 n has a limit, called the cogrowth exponent with respect to the generators g 1 , … , g r . We show by analytic methods that the numbers γ n vary regularly, i.e. the ratio γ 2 n + 2 / γ 2 n is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function...

Polydisc slicing in ℂ n

Krzysztof Oleszkiewicz, Aleksander Pełczyński (2000)

Studia Mathematica

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Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in ℂ n of codimension 1, v o l 2 n - 2 ( D n - 1 ) ≤ v o l 2 n - 2 ( H ∩ D n ) ≤ 2 v o l 2 n - 2 ( D n - 1 ) . The lower bound is attained if and only if H is orthogonal to the versor e j of the jth coordinate axis for some j = 1,...,n; the upper bound is attained if and only if H is orthogonal to a vector e j + σ e k for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify ℂ n with ℝ 2 n ; by v o l k ( · ) we denote the usual k-dimensional volume in ℝ 2 n . The result is a complex counterpart of Ball’s [B1]...

The d X ( t ) = X b ( X ) d t + X σ ( X ) d W equation and financial mathematics. II

Josef Štěpán, Petr Dostál (2003)

Kybernetika

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This paper continues the research started in [J. Štěpán and P. Dostál: The d X ( t ) = X b ( X ) d t + X σ ( X ) d W equation and financial mathematics I. Kybernetika 39 (2003)]. Considering a stock price X ( t ) born by the above semilinear SDE with σ ( x , t ) = σ ˜ ( x ( t ) ) , we suggest two methods how to compute the price of a general option g ( X ( T ) ) . The first, a more universal one, is based on a Monte Carlo procedure while the second one provides explicit formulas. We in this case need an information on the two dimensional distributions of ℒ ( Y ( s ) , τ ( s ) ) for s ≥ 0 , where Y ...