The Integral Mean of the Discrepancy of the Sequence (nα)

 For a real number x let be the fractional part of x and for any set M let c _M be the characteristic function of M. For and a positive integer N let

The Discrepancy of the Sequence (nα + (log n)β)

In this paper we give an upper bound for the discrepancy of the sequence (nα + (log n)β) where α = (α₁, ..., α _s ) R ^s , which satisfies that 1, α₁, ..., α _s are linearly independent over Z, is of finite type η or is of constant type.     

The Integral of Mean Curvatures on the Tube Hypersurfaces

The calculation on the tube hypersurfaces is very interesting and important,especially H.Weyl gave a formula of the volume of tubes which contains a seriesof curvature invariants of submanifolds. Let N be an oriented compact n-dimensional submanifold in the m-dimensional     

On the Integral Curvature of Curve-(III)

In this paper we prove the following theorem.It is a generalization of Tenchel's theorem on theintegral curvature of curve.Theorem.If 1 is the length of a curve C=AB and φ is the angle between the tangent vectors ofC at A,B,then the integral curvature of C     

The Relation Between the Petits Integral and the McShane Integral of Banach-valued Functions Defined on R~n

R.A.Gordon[1]generalizedthedefinitionoftheMcShaneintegralforrealvaluedfunctionstotheabstractfunctionsfromR1toBanachspaces.Inthispaper,wedefinetheMcShaneintegralforabstractfunctionsfromRntoBanachspacesandprovethatifeveryfunctionffromIoinRntoBanachspacesisMcShaneintegrablethenfisPettisintegrable,andiftheunitballB(X)inX*isrelativelyweaklycompact,fisPetitsintegrablethenfisMcShaneintegrableonIo.ThisgivesapositiveanswertoanopenproblemleftbyR.A.Gordonin[1].ThroughoutthispaperXwilldenoteare…     

On the (n − 2)-transversals of n convex subsets of the plane

In this paper we consider families of distinct ovals in the plane, with the property that certain subfamilies have stabbing lines (transversals). Our main result says that if any k member of the family can be stabbed by a line avoiding all the other ovals and k is large enough, then the family consists of at most k+1 ovals. For any n4 we show a family of n ovals, whose n–2 element subfamilies have, but the n–1 element subfamilies do not have, transversals.     

Integral mean square estimation for the error term related to n xλ2(n2)

Let λf(n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) ∈Sk(Γ).We establish that, for any ε 0,1/Xintegral from n=1 to x|sum λ~2f~((n~2)) from n≤x to - c_2x|2dx ?ε X154/101+ε,which improves previous results.     

On the Integral Extension of Lattices

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Zygmund Type Mean Lipschitz Spaces on the Unit Ball of ℂ n

On the unit ball of ? ⁿ , the space of those holomorphic functions satisfying mean Lipschitz condition $$ \int_0^1 \omega^*_p(t, f)^q \frac{dt}{t^{1+ q}} <\infty$$ is characterized by integral growth conditions of the tangential derivatives as well as the radial derivatives, where \(\omega^*_p(t, f)\) denotes the double difference L ^p modulus of continuity defined in terms of the unitary transformations of ? ⁿ .     

The Representation of Continuous Linear Operator by Integral

LetXoverthefieldKstandforacompletelocallyconvexspace,andweassumethatitstopologyisgeneratedbyafamilyPofcontinuousseminormsonX,Let(O,Z,p)beafinitepositivemeasurespace.Afunctionf:Q-Xissaidtobep-integrable(forthedefinition,see[1]),ifthereexistsasequence{ft}ofsimplefunctionssuchthat.,ThislimitexistsandisinX.Denote,p)asthespaceofcIassesofNintegrablefunctionswhereLfj=Lgjifandonlyiff=gP-a.e..Putq(f)=j.p(f)dp,foreachpeP.ThetopologyonL1(p,Z)consideredisthese,generatedbyalltheq.L.EggheLlIhaspro…     

The partition function p(n) in terms of the classical Möbius function

The Ramanujan Journal - In this paper, we investigate decompositions of the partition function p(n) from the additive theory of partitions considering the famous Möbius function $$mu (n)$$...     

The First Power Mean of the Inversion of L-Functions Weighted by Quadratic Gauss Sums

The main purpose of this paper is to use estimates for character sums and analytic methods to study the first power mean of the inversion of Dirichlet L-functions with the weight of general quadratic Gauss sums, and three asymptotic formulae are obtained.