Rn In Terms Research Articles

A Fourier Integral Formula for Logarithmic Energy

AbstractA formula which expresses logarithmic energy of Borel measures on $$\mathbb {R}^n$$ R n in terms of the Fourier transforms of the measures is established and some applications are given. In addition, using similar techniques a (known) formula for Riesz energy is reinvented.

Telescoping continued fractions for the error term in Stirling’s formula

In this paper, we introduce telescoping continued fractions to find lower bounds for the error term rn in Stirling’s approximation n!=2πnn+1/2e−nern. This improves lower bounds given earlier by Cesàro (1922), Robbins (1955), Nanjundiah (1959), Maria (1965) and Popov (2017). The expression is in terms of a continued fraction, together with an algorithm to find successive terms of this continued fraction. The technique we introduce allows us to experimentally obtain upper and lower bounds for a sequence of convergents of a continued fraction in terms of a difference of two continued fractions.

On Persson’s formula: an étale groupoid approach

Persson’s formula expresses the infimum of the essential spectrum of a suitable self-adjoint Schrödinger operator in ℝn in terms of the lower spectral points of a family of restrictions of the operator to complements of relatively compact subsets. It has been extended to other situations. We present an approach based on C*-algebras associated to étale groupoids. In such a setting there are intrinsic versions, referring to self-adjoint elements of the groupoid C*-algebras. When representations in Hilbert spaces are considered, the results not always involve only the essential spectrum. The range of applications is quite distinct from the traditional one. We indicate examples related to symbolic dynamics and band dominated operators on discrete metric spaces. The treatment needs only a small amout of symmetry. Even when group actions are involved, restrictions to non-invariant subsets are needed and have to be treated carefully.

Faà di Bruno's formula and inversion of power series

Faà di Bruno's formula gives an expression for the derivatives of the composition of two real-valued functions. In this paper we prove a multivariate and synthesised version of Faà di Bruno's formula in higher dimensions, providing a combinatorial expression for the derivatives of chain compositions F(1)∘…∘F(m) of functions F(l):RN→RN in terms of sums over labelled trees. We give several applications of this formula, including a new involution formula for the inversion of multivariate power series. We use this framework to outline a combinatorial approach to studying the invertibility of polynomial mappings, giving a purely combinatorial restatement of the Jacobian conjecture. Our methods extend naturally to the non-commutative case, where we prove a free version of Faà di Bruno's formula for multivariate power series in free indeterminates, and use this formula as a tool for obtaining a new inversion formula for free power series.

The Lebesgue differentiation theorem revisited

We prove a general version of the Lebesgue differentiation theorem where the averages are taken on a family of sets that may not shrink nicely to any point. These families of sets involve the unit ball and its dilated by negative integers of an expansive linear map. We also give a characterization of the Lebesgue measurable functions on Rn in terms of approximate continuity associated to an expansive linear map.

On the generalized circle problem for a random lattice in large dimension

In this note we study the error term Rn,L(x) in the generalized circle problem for a ball of volume x and a random lattice L of large dimension n. Our main result is the following functional central limit theorem: Fix an arbitrary function f:Z+→R+ satisfying limn→∞⁡f(n)=∞ and f(n)=Oε(eεn) for every ε>0. Then, the random functiont↦12f(n)Rn,L(tf(n)) on the interval [0,1] converges in distribution to one- dimensional Brownian motion as n→∞. The proof goes via convergence of moments, and for the computations we develop a new version of Rogers' mean value formula from [18]. For the individual kth moment of the variable (2f(n))−1/2Rn,L(f(n)) we prove convergence to the corresponding Gaussian moment more generally for functions f satisfying f(n)=O(ecn) for any fixed c∈(0,ck), where ck is a constant depending on k whose optimal value we determine.

A note on the Blaschke-Petkantschin formula, Riesz distributions, and Drury’s identity

Abstract The Blaschke-Petkantschin formula is a variant of the polar decomposition of the k-fold Lebesgue measure on ℝ n in terms of the corresponding measures on k-dimensional linear subspaces of ℝ n . We suggest a new elementary proof of this famous formula and discuss its connection with Riesz distributions associated with fractional powers of the Cayley-Laplace operator on matrix spaces. Another application of our proof is the celebrated Drury identity that plays a key role in the study of mapping properties of the Radon-John k-plane transforms. Our proof gives precise meaning to the constants in Drury’s identity and to the class of admissible functions.

Unboundedness properties of smoothness Morrey spaces of regular distributions on domains

We study unboundedness of smoothness Morrey spaces on bounded domains Ω ⊂ Rn in terms of growth envelopes. It turns out that in this situation the growth envelope function is finite—in contrast to the results obtained by Haroske et al. (2016) for corresponding spaces defined on Rn. A similar effect was already observed by Haroske et al. (2017), where classical Morrey spaces Mu,p(Ω) were investigated. We deal with all cases where the concept is reasonable and also include the tricky limiting cases. Our results can be reformulated in terms of optimal embeddings into the scale of Lorentz spaces Lp,q(Ω).

Criterion of the continuation of harmonic functions in the ball of n­dimensional space and representation of the generalized orders of the entire harmonic functions in ℝn in terms of approximation error

A growth of harmonic functions in the whole space ℝn is examined. We found the estimate for a uniform norm of spherical harmonics in terms of the best approximation of harmonic function in the ball by harmonic polynomials. An approximation error of harmonic function in the ball is estimated by the maximum modulus of an entire harmonic function in space, as well as the maximum modulus of an entire harmonic function in space in terms of the maximum modulus of some entire function of one complex variable or the maximal term of its power series. These results allowed us to obtain the necessary and sufficient conditions under which a harmonic function in the ball of an n-dimensional space, n≥3, can be continued to the entire harmonic one. This result is formulated in terms of the best approximation of the given function by harmonic polynomials. In order to characterize growth of an entire harmonic function, we used the generalized and the lower generalized orders. Formulae for the generalized and the lower generalized orders of an entire harmonic function in space are expressed in terms of the approximation error by harmonic polynomials of the function that continues. We also investigated the growth of functions of slow increase. The obtained results are analogues to classical results, which are known for the entire functions of one complex variable.The conducted research is important due to the fact that the harmonic functions occupy a special place not only in many mathematical studies, but also when applying mathematical analysis to physics and mechanics, where these functions are often employed to describe various stationary processes

Disentangling the f(R)-duality

Motivated by UV realisations of Starobinsky-like inflation models, we study generic exponential plateau-like potentials to understand whether an exact f(R)-formulation may still be obtained when the asymptotic shift-symmetry of the potential is broken for larger field values. Potentials which break the shift symmetry with rising exponentials at large field values only allow for corresponding f(R)-descriptions with a leading order term Rn with 1<n<2, regardless of whether the duality is exact or approximate. The R2-term survives as part of a series expansion of the function f(R) and thus cannot maintain a plateau for all field values. We further find a lean and instructive way to obtain a function f(R) describing m2ϕ2-inflation which breaks the shift symmetry with a monomial, and corresponds to effectively logarithmic corrections to an R+R2 model. These examples emphasise that higher order terms in f(R)-theory may not be neglected if they are present at all. Additionally, we relate the function f(R) corresponding to chaotic inflation to a more general Jordan frame set-up. In addition, we consider f(R)-duals of two given UV examples, both from supergravity and string theory. Finally, we outline the CMB phenomenology of these models which show effects of power suppression at low-ℓ.

Upper and lower bounds for the iterates of order-preserving homogeneous maps on cones

We define upper bound and lower bounds for order-preserving homogeneous of degree one maps on a proper closed cone in Rn in terms of the cone spectral radius. We also define weak upper bounds for these maps. For a proper closed cone C⊂Rn, we prove that any order-preserving homogeneous of degree one map f:intC→intC has a lower bound. If C is polyhedral, we prove that the map f has a weak upper bound. We give examples of weak upper bounds for certain order-preserving homogeneous of degree one maps defined on the interior of R+n.

Cauchy's Continuum

Cauchy's sum theorem of 1821 has been the subject of rival interpretations ever since Robinson proposed a novel reading in the 1960s. Some claim that Cauchy modified the hypothesis of his theorem in 1853 by introducing uniform convergence, whose traditional formulation requires a pair of independent variables. Meanwhile, Cauchy's hypothesis is formulated in terms of a single variable x, rather than a pair of variables, and requires the error term rn = rn(x) to go to zero at all values of x, including the infinitesimal value generated by explicitly specified by Cauchy. If one wishes to understand Cauchy's modification/clarification of the hypothesis of the sum theorem in 1853, one has to jettison the automatic translation-to-limits.

On asymptotic linearity of L-estimates

Abstract A theorem on asymptotic linearity of L-estimates is proved under general set of regularity conditions, allowing the sampled distribution to be non-integrable. The main result is the improvement in the order of the remainder term in the formula for asymptotic linearity of L-statistic. It is shown that in the case of the integral coefficients this term R n = O P(1/n) and the case of functional coefficients is also covered.

Hazard rates and subexponential distributions

A distribution function F on the nonnegative halfline is called subexponential if limx??(1?F*n(x))/(1?F(x)) = n for all n&gt;_ 2. We obtain new sufficient conditions for subexponential distributions and related classes of distribution functions. Our results are formulated in terms of the hazard rate. We also analyze the rate of convergence in the definition and discuss the asymptotic behaviour of the remainder term Rn(x) = 1?F*n(x)?n(1?F(x)). We use the results in studying subordinated distributions and we conclude the paper with some multivariate extensions of our results.

Symmetry at Infinity

Since the graphs of y = x2 and y at the points (0, 0) and (1, 1) and are symmetric about the line y = x, the question seems like a reasonable one. However, rather than simply answer the question for y = x2 and y = x-, I replied by asking the same question concerning any pair of functions of the form y = xn and y = /, where n is a positive integer, since they possess the same points of intersection and symmetry. This paper is devoted to answering that question. Thus for each positive integer n, define fn,(x) = xn and f(x) = r = x1/n. For n = 1, fi (x) = x = f'1(x), so that no area is bounded by f, (x) and f-'1(x). However, for n > 1, fn (x) and fn-'(x) bound a region Rn of finite area between (0, 0) and (1, 1). We proceed now to derive a general formula for the coordinates (x,, yn) of the centroid Cn of Rn in terms of n (n > 1). Since any invertible function and its inverse are symmetric about the line y = x, we have x, = y,. (See Figure 1.) Thus a single function of n provides the coordinates (x,, Yn) of Cn. At first glance, many students conjecture that the region Rn is also symmetric about the line y = 1 x. (See Figure 2.) Based on the assumption of this additional symmetry, the students conclude that Rn must have centroid Cn (1, 1) at the intersection of y = x and y = 1 x. However, if f(x) > g(x) for a < x < b and R is a plane lamina of constant density p whose area is bounded by f(x) and g(x) over the interval [a, b], then the mass and first moments of R are

An Error Expansion for some Gauss–Turán Quadratures and L1-Estimates of the Remainder Term

Our aim in this paper is to obtain error expansions in the Gauss–Turan quadrature formula ∫−11f(t)w(t) dt=∑ν=1n∑i=02sAi,νf(i)(τν)+Rn,s(f), in the case when f is an analytic function in some region of the complex plane containing the interval [−1,1] in its interior. Using a representation of the remainder term Rn,s(f) in the form of contour integral over confocal ellipses, we obtain Rn,1(f) for the four Chebyshev weights and Rn,2(f) for the Chebyshev weight of the first kind. Also, we get a few new L1-estimates of the remainder term, which are stronger than the previous ones. Some numerical results, illustrations and comparisons are also given.

THE RATE OF CONVERGENCE FOR SUBEXPONENTIAL DISTRIBUTIONS AND DENSITIES

A distribution function F on the nonnegative real line is called subexponential if limx↑∞(1-F*n(x)/(1 - F(x)) = n for all n ≥ 2, where F*n denotes the n‐fold Stieltjes convolution of F with itself. In this paper, we consider the rate of convergence in the above definition and in its density analogue. Among others we discuss the asymptotic behavior of the remainder term Rn (x) defined by Rn(x) = 1 - F*n(x) - n(1 - F(x)) and of its density analogue rn (x) = -(Rn (x))'. Our results complement and complete those obtained by several authors. In an earlier paper, we obtained results of the form n(x) = O(1)f(x)R(x), where f is the density of F and R(x) = ∫0x(1-F(y))dy. In this paper, among others we obtain asymptotic expressions of the form Rn(x)=2nR2(x) + O(1)(-f'(x))R2(x) where f' is the derivative of f.

Compactness properties for perturbed semigroups and application to transport equation

AbstractUsing the comparsion results for positive compact operators by Aliprantis and Burkinshow, Mokhtar Kharroubi investigated cimpactness properties of positive semigroups on Banach latttices. The aim of this paper is to study these properties in general Banach spaces (without positivity). Our results generalize a part fo those obtained by Mokhtar-Kharroubi to general Banach spaces context. More specifically, we derive conditions which ensure the compactness of the remainder term Rn(t) for some inteter n. The improvement here is that it can applied directly to the neutron transport equation for a wide class of collision operators.

Jung's Theorem for Alexandrov Spaces of Curvature Bounded Above

The classical Jung theorem gives an optimal upper estimate for the radius of a bounded subset of Rn in terms of its diameter and the dimension. In this note we present an analogue of this result for metric spaces of curvature bounded above in the sense of Alexandrov.

Traces of Sobolev Functions on Fractal Type Sets and Characterization of Extension Domains

We describe traces of Sobolev functionsu∈W1, p(Rn), 1<p⩽∞, on certain subsets of Rnin terms of Sobolev spaces on metric spaces [7]. Our results apply to smooth submanifolds, fractal subsets, as well as to open subsets of Rn. In particular if 0⊂Rnis a John domain, then we characterize thoseW1, p(Ω) functions which can be extended toW1, p(Rn). IfΩis uniform, then this result implies Jones' extension theorem [14]. In the case of traces on fractal subsets our results are related to those of Jonsson and Wallin [16].