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This study deals with two statistical downscaling (SD) techniques for their potential to improve the skill of forecasts seasonal rainfall over core monsoonal belt regions over South Asia. It constitute most comprehensive to date intercomparison of SD methods. This work uses Indian Summer Monsoon (ISMR) forecasts from global primary models during 1996 to 2010. Two popular downscaling methods were tested by using unequal weight to all studied models which is termed as Best Linear Unbiased Estimator (BLUE) or by equal weight known as Ensemble Mean (EM). The statistical downscaling methods Bias Correction and Spatial Downscaling (BCSD) and Bias Correction and Constructed Analogue (BCCA) from Model Output statistics (MOS) approach were used. The analysis of direct models forecasts reveals that the bias is high with respect to season across South Asia, hence the necessity of SD. Though the skill of SD methods varies strongly depending on regions and seasons considered, both BCCA and BCSD also reveal higher performance than any primary models. The BCCA SD models are most capable to forecasts ISMR even at local scale than any others. [ABSTRACT FROM AUTHOR]

Bias correction (BC) is a crucial step for satellite radiance data assimilation (DA). In this study, the traditional airmass BC scheme in the National Centers for Environmental Prediction (NCEP) Gridpoint Statistical Interpolation (GSI) is investigated for Cross-track Infrared Sounder (CrIS) DA. The ability of the airmass predictors to model CrIS biases is diagnosed. Correlations between CrIS observation-minus-background (O − B) samples and the two lapse rate–related airmass predictors employed by GSI are found to be very weak, indicating that the bias correction contributed by the airmass BC scheme is small. A modified BC scheme, which directly calculates the moving average of O − B departures from data of the previous 2 weeks with respect to scan position and latitudinal band, is proposed and tested. The impact of the modified BC scheme on CrIS radiance DA is compared with the variational airmass BC scheme. Results from 1-month analysis/forecast experiments show that the modified BC scheme removes nearly all scan-dependent and latitude-dependent biases, while residual biases are still found in some channels when the airmass BC scheme is applied. Smaller predicted root-mean-square errors of temperature and specific humidity and higher equivalent threat scores are obtained by the DA experiment using the modified BC scheme. If O − B samples are replaced by observation-minus-analysis (O − A) samples for bias estimates in the modified BC scheme, the forecast impacts are reduced but remain positive. A convective precipitation case that occurred on 21 August 2016 is investigated. Using the modified BC scheme, the atmospheric temperature structure and the geopotential height structures near trough/ridge areas are better resolved, resulting in better precipitation forecasts. [ABSTRACT FROM AUTHOR]

We prove the following two basis theorems for ∑12-sets of reals: (1) Every nonthin ∑12-set has a perfect A12-subset if and only if it has a nonthin A12-subset, and this is equivalent to the statement that there is a nonconstructible real. (2) Every uncountable ∑12-set has an uncountable A12-subset if and only if either every real is constructible or ωL2 is countable. We also apply the method that proves (2) to show that if there is a nonconstructible real, then there is a perfect Π12-set with no nonempty Π12-thin subset, strengthening a result of Harrington [4]. [ABSTRACT FROM AUTHOR]

In this article, local linear estimators are adapted for the unknown infinitesimal coefficients associated with continuous‐time asset return models with jumps, which can correct the bias automatically due to their simple bias representation. The integrated diffusion models with jumps, especially infinite activity jumps, are mainly investigated. In addition, under mild conditions, the weak consistency and asymptotic normality are provided through the conditional Lindeberg theorem as the time span T→∞ and the sample interval Δn→0. Furthermore, our method presents advantages in bias correction through simulation whether jumps belong to the finite activity case or infinite activity case. Finally, the estimators are illustrated empirically through the returns of stock index under 5‐minute high sampling frequency for real application. [ABSTRACT FROM AUTHOR]

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In this paper we work with multivariate time series that follow a Factor Model. In particular, we consider the setting where factors are dominated by highly persistent AutoRegressive (AR) processes and samples that are rather small. Therefore, the factors' AR models are estimated using small sample bias correction techniques. A Monte Carlo study reveals that bias-correcting the AR coefficients of the factors allows to obtain better results in terms of prediction interval coverage. As expected, the simulation shows that bias-correction is more successful for smaller samples. We present the results assuming the AR order and number of factors are known as well as unknown. We also study the advantages of this technique for a set of Industrial Production Indexes of several European countries. [ABSTRACT FROM AUTHOR]

Polyzetas, indexed by words, satisfy shuffle and quasi-shuffle identities. In this respect, one can explore the multiplicative and algorithmic (locally finite) properties of their generating series. In this paper, we construct pairs of bases in duality on which polyzetas are established in order to compute local coordinates in the infinite dimensional Lie groups where their non-commutative generating series live. We also propose new algorithms leading to the ideal of polynomial relations, homogeneous in weight, among polyzetas (the graded kernel) and their explicit representation (as data structures) in terms of irreducible elements. [ABSTRACT FROM AUTHOR]

We present two algorithms for computing hypergeometric solutions of second order linear differential operators with rational function coefficients. Our first algorithm searches for solutions of the form (1) exp ⁡ ( ∫ r d x ) ⋅ 2 F 1 ( a 1 , a 2 ; b 1 ; f ) where r , f ∈ Q ( x ) ‾ , and a 1 , a 2 , b 1 ∈ Q . It uses modular reduction and Hensel lifting. Our second algorithm tries to find solutions in the form (2) exp ⁡ ( ∫ r d x ) ⋅ ( r 0 ⋅ 2 F 1 ( a 1 , a 2 ; b 1 ; f ) + r 1 ⋅ 2 F 1 ′ ( a 1 , a 2 ; b 1 ; f ) ) where r 0 , r 1 ∈ Q ( x ) ‾ , as follows: It tries to transform the input equation to another equation with solutions of type (1) , and then uses the first algorithm. [ABSTRACT FROM AUTHOR]

We present here a new approach for computing Groebner bases for bilateral modules over an effective ring. Our method is based on Weispfenning notion of restricted Groebner bases and related multiplication. [ABSTRACT FROM AUTHOR]

The finiteness property is an important arithmetical property of beta-expansions. We exhibit classes of Pisot numbers β having the negative finiteness property, that is the set of finite (− β)-expansions is equal to $${\mathbb{Z}[\beta^{-1}]}$$ . For a class of numbers including the Tribonacci number, we compute the maximal length of the fractional parts arising in the addition and subtraction of (− β)-integers. We also give conditions excluding the negative finiteness property. [ABSTRACT FROM AUTHOR]

As a follow-up to our previous work [10] , we characterize the measure-preservation of 1-Lipschitz functions on F q [ [ T ] ] in terms of the three well-known bases: Carlitz polynomials, digit derivatives, and digit shifts. [ABSTRACT FROM AUTHOR]

This paper highlights an unexpected connection between expansions of real numbers to noninteger bases (so-called β-expansions ) and the infinite derivatives of a class of self-affine functions. Precisely, we extend Okamoto's function (itself a generalization of the well-known functions of Perkins and Katsuura) to a two-parameter family { F N , a : N ∈ N , 1 / ( N + 1 ) < a < 1 } . We first show that for each x , F N , a ′ ( x ) is either 0, ±∞, or undefined. We then extend Okamoto's theorem by proving that for each N , depending on the value of a relative to a pair of thresholds, the set { x : F N , a ′ ( x ) = 0 } is either empty, uncountable but Lebesgue null, or of full Lebesgue measure. We compute its Hausdorff dimension in the second case. The second result is a characterization of the set D ∞ ( a ) : = { x : F N , a ′ ( x ) = ± ∞ } , which enables us to closely relate this set to the set of points which have a unique expansion in the (typically noninteger) base β = 1 / a . Recent advances in the theory of β -expansions are then used to determine the cardinality and Hausdorff dimension of D ∞ ( a ) , which depend qualitatively on the value of a relative to a second pair of thresholds. [ABSTRACT FROM AUTHOR]

Decompositions of the score of a forecast represent useful tools for assessing its performance. We consider local score decompositions permitting detailed forecast assessments across a spectrum of conditions of interest. We derive corrections to the bias of the decomposition components in the framework of point forecasts of quantile-type functionals, and illustrate their performance by simulation. Related bias corrections have thus far only been known for squared error criteria. [ABSTRACT FROM AUTHOR]

Let f 1 , … , f s be formal power series (respectively polynomials) in the variable x . We study the semigroup of orders of the formal series in the algebra K 〚 f 1 , … , f s 〛 ⊆ K 〚 x 〛 (respectively the semigroup of degrees of polynomials in K [ f 1 , … , f s ] ⊆ K [ x ] ). We give procedures to compute these semigroups and several applications. We prove in particular that the space curve parametrized by f 1 , … , f s has a flat deformation into a monomial curve. [ABSTRACT FROM AUTHOR]

We shall present a new characterization of greedy bases and 1-greedy bases in terms of certain functionals defined using distances to one dimensional subspaces generated by the basis. We also introduce a new property that unifies the notions of unconditionality and democracy and allows us to recover a better dependence on the constants. [ABSTRACT FROM AUTHOR]

Riordan bases in a power series ring of several variables are introduced as a special type of Schauder bases. As a part of the method of generating differentials, interplay of Riordan bases generalizes Lagrange inversion formulas. The set of Riordan bases has a group law with a specified set of variables as the unit element. Riordan arrays appear in a representation of the group. [ABSTRACT FROM AUTHOR]

Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the quality of approximation of almost time- and almost band-limited functions by its expansion in two classical orthogonal polynomials bases: the Hermite basis and the ultraspherical polynomials bases (which include Legendre and Chebyshev bases as particular cases). As a corollary, this allows us to obtain the quality of approximation in the L 2 -Sobolev space by these orthogonal polynomials bases. Also, we obtain the rate of convergence of the Legendre series expansion of the prolate spheroidal wave functions. [ABSTRACT FROM AUTHOR]

For a given totally positive space of continuous functions, we analyze the construction of totally positive bases of the space of antiderivatives. If the functions of the totally positive space have continuous derivatives, normalization properties can be used to describe totally positive bases of the space of derivatives and relate them with properties of the Greville abscissae. [ABSTRACT FROM AUTHOR]

We briefly review Ramanujan's theories of elliptic functions to alternative bases, describe their analogues for levels 5 and 7, and develop new theories for levels 14 and 15. This gives rise to a rich interplay between theta functions, eta-products and Eisenstein series. Transformation formulas of degrees five and seven for hypergeometric functions are obtained, and the paper ends with some series for 1/π similar to ones found by Ramanujan. [ABSTRACT FROM AUTHOR]

In a region where ground-based gauge data are scarce, satellite rainfall estimates (SREs) are a viable option for proper space–time rainfall characterization. However, their accuracy and performances vary from region to region, and must be assessed. In this study, five high resolution satellite products (3B42V7, CMORPH, TAMSAT, SM2R-CCI, and CFSR) are compared and analyzed using the available rain gauge data in one of the most topographically and climatologically complex basin of Africa, the Upper Blue Nile basin (UBN). The basin rainfall is investigated systematically, and it is found that, at some locations, the difference in mean annual rainfall estimates between these SREs could be as much as about 2700 mm. Considering three goodness-of-fit indexes, correlation, bias and root mean square error (RMSE) between the SREs and ground-based gauge rainfall, CMORPH, TAMSAT and SM2R-CCI outperform the other two. Furthermore, a confusion matrix is used to investigate the detection ability of satellite rainfall products for different rainfall intensities. TAMSAT has the highest (91%) detection skill for dry days, followed by CFSR (77%). On the contrary, SM2R-CCI has the highest accuracy index for medium rainfall ranges (10–20 mm). The empirical cumulative distribution ( ecdf ) mapping technique is used to correct the intensities distribution givenby the SREs. This method provides a means to improve the rainfall estimation of all SREs, and the highest improvement is obtained for CMORPH (bias reduction from − 72% to − 1%). [ABSTRACT FROM AUTHOR]

The goal of our article is a study of related mathematical and physical objects: orthogonal pairs in sl( n) and mutually unbiased bases in ℂ . An orthogonal pair in a simple Lie algebra is a pair of Cartan subalgebras that are orthogonal with respect to the Killing form. The description of orthogonal pairs in a given Lie algebra is an important step in the classification of orthogonal decompositions, i.e., decompositions of the Lie algebra into a direct sum of Cartan subalgebras pairwise orthogonal with respect to the Killing form. One of the important notions of quantum mechanics, quantum information theory, and quantum teleportation is the notion of mutually unbiased bases in the Hilbert space ℂ . Two orthonormal bases { e}, { f} are mutually unbiased if and only if $$ {\left|\left\langle {e}_i\left|{f}_j\right.\right\rangle \right|}^2=\frac{1}{n} $$ for any i, j = 1 ,..., n. The notions of mutually unbiased bases in ℂ and orthogonal pairs in sl( n) are closely related. The problem of classification of orthogonal pairs in sl( n) and the closely related problem of classification of mutually unbiased bases in ℂ are still open even for the case n = 6 . In this article, we give a sketch of our proof that there is a complex four-dimensional family of orthogonal pairs in sl(6) . This proof requires a lot of algebraic geometry and representation theory. Further, we give an application of the result on the algebraic geometric family to the study of mutually unbiased bases. We show the existence of a real four-dimensional family of mutually unbiased bases in ℂ , thus solving a long-standing problem. Bibliography: 24 titles. [ABSTRACT FROM AUTHOR]

The chlorophyll-a (chl-a) concentration is an Essential Climate Variable, and the study of its variability at global scale requires a succession of satellite ocean colour missions to cover a period suitable for climate research. In the context of a multi-mission data record, inter-mission differences can introduce artefacts affecting trend evaluations, and the impact of the bias between the Sea-viewing Wide Field-of-view Sensor (SeaWiFS) and Moderate Resolution Imaging Spectroradiometer (MODIS) chl-a products is shown to be significant in a substantial part of the ocean. The assessment of trends can also be directly impacted by a drift in the chl-a time series resulting from sensor functions. These issues are addressed by a sensitivity analysis that compares slopes of linear regression obtained for varying levels of intermission bias and drift with respect to a 15-year reference series built with SeaWiFS and MODIS data. The relationship, constructed for a representative set of ocean provinces, between bias and the level of significance associated with the comparison of slopes shows that a bias on the order of ±5-6% generally induces a slope that is significantly different from the reference case, while a threshold on bias values not exceeding 2% largely alleviates this effect. Moreover, the study suggests that a drift larger than 2% per decade on the chl-a series can result in misleading conclusions from a trend analysis. All results have a clear regional dependence that needs to be taken into account in bias-correction and merging efforts. Low chl-a regions, such as the oligotrophic subtropical gyres, appear particularly sensitive to perturbations and require still higher levels of consistency and stability. [ABSTRACT FROM AUTHOR]

In this paper, twisted shift-invariant spaces in L 2 ( R 2 n ) are introduced. Characterizations of orthonormal system and Bessel sequence of twisted translates in L 2 ( R 2 n ) are obtained in terms of Weyl transform. These results appear to be totally different from the classical case of characterizations of orthonormal systems and Bessel sequences of translates in L 2 ( R n ) . Further, characterizations of frames, Riesz basis in twisted shift-invariant spaces in L 2 ( R 2 n ) are also obtained. [ABSTRACT FROM AUTHOR]

In this note we continue the study initiated in Albiac and Wojtaszczyk (2006) of greedy-like bases in the “isometric case”, i.e., in the case that the constants that arise in the context of greedy bases in their different forms are 1 . Here we settle the problem to find a satisfactory characterization of 1 -quasi-greedy bases in Banach spaces. We show that a semi-normalized basis in a Banach space is quasi-greedy with quasi-greedy constant 1 if and only if it is unconditional with suppression-unconditional constant 1 . [ABSTRACT FROM AUTHOR]

We study mutually unbiased bases in $${\mathbb {C}}^d\otimes {\mathbb {C}}^{2^{l}d'}$$ . A systematic way of constructing mutually unbiased maximally entangled bases (MUMEBs) in $${\mathbb {C}}^d\otimes {\mathbb {C}}^{2^{l}d'} (l\in {\mathbb {Z}}^{+})$$ from MUMEBs in $${\mathbb {C}}^d \otimes {\mathbb {C}}^{d'}(d'=kd, k\in {\mathbb {Z}}^+)$$ and a general approach to construct mutually unbiased unextendible maximally entangled bases (MUUMEBs) in $${\mathbb {C}}^d\otimes {\mathbb {C}}^{2^ld'} (l \in {\mathbb {Z}}^{+})$$ from MUUMEBs in $${\mathbb {C}}^d \otimes {\mathbb {C}}^{d'}(d'=kd+r, 0

We obtain closed-form conditions for the oscillation of the Green function of a boundary value problem for an ordinary differential equation of the fourth order supplemented with fixing conditions of a general form at the endpoints and conditions of elastic bases at several intermediate points of the interval on which the equation is defined. We also obtain necessary and sufficient conditions under which a Green function is an oscillation kernel. [ABSTRACT FROM AUTHOR]

Define ^ m to be the smallest strong pseudoprime to the first m prime bases. The exact value of is known for 1 < m < 8. Z. Zhang has found a 19-decimal-digit number Q11 = 3825 12305 6546413051 which is a strong pseudoprime to the first 11 prime bases and he conjectured that ^9 = ^>10 = ^>11 = Q11. We tabulate all the strong pseudoprimes n < Q11 to the first eight prime bases, and prove Zhang's conjecture.efine ^ m to be the smallest strong pseudoprime to the first m prime bases. The exact value of is known for 1 < m < 8. Z. Zhang has found a 19-decimal-digit number Q11 = 3825 12305 6546413051 which is a strong pseudoprime to the first 11 prime bases and he conjectured that ^9 = ^>10 = ^>11 = Q11. We tabulate all the strong pseudoprimes n < Q11 to the first eight prime bases, and prove Zhang's conjecture. [ABSTRACT FROM AUTHOR]

Abstract: We developed a novel algorithm which can automatically find the optimum angle for surface plasmon resonance (SPR) measurements based on fixed angle mode using MATLAB R2013a. The optimum angle is decided by the coefficient of determination of linear regression fitting on an SPR curve obtained from a miniaturized SPR system. The SPR biosensor based on the novel algorithm achieved a dynamic range of 0.00696RIU of and LOD of 3.1956×10−6. [Copyright &y& Elsevier]

As a local shape descriptor, normal-controlled coordinates (NCC) are well defined on the boundary vertices of open meshes, and proven being always parallel with the corresponding vertex normals, which means no tangential drift appears in various editing operations. These two properties make NCC outperform the previous differential coordinates in many mesh processing jobs. We develop an implicit editing algorithm based on NCC, which is efficient by subtly using vertex normals. In addition, by exploring the relationship between NCC and the normals of triangular faces, we present a linear method based on NCC to reconstruct a mesh from predefined face normals. It provides another convenient way that users can operate face normals to edit models. Experiments show that NCC perform better than the previous differential coordinates in both effectiveness and efficiency. [ABSTRACT FROM AUTHOR]

In this paper we show that the sets of F-jumping coefficients of ideals form discrete sets in certain graded F-finite rings. We do so by giving a criterion based on linear bounds for the growth of the Castelnuovo-Mumford regularity of certain ideals. We further show that these linear bounds exist for one-dimensional rings and for ideals of (most) two-dimensional domains. We conclude by applying our technique to prove that all sets of F-jumping coefficients of all ideals in the determinantal ring given as the quotient by 2 ×2 minors in a 2 × 3 matrix of indeterminates form discrete sets. [ABSTRACT FROM AUTHOR]

ABSTRACT Our objective was to develop a non-invasive magnetic resonance (MR) method to predict the structural properties of a healing anterior cruciate ligament (ACL) using volume and T2* relaxation time. We also compared our T2*-based structural property prediction model to a previous model utilizing signal intensity, an acquisition-dependent variable. Surgical ACL transection followed by no treatment (i.e., natural healing) or bio-enhanced ACL repair was performed in a porcine model. After 52 weeks of healing, high-resolution MR images of the ACL tissue were collected. From these images, ligament volumes and T2* maps were established. The structural properties of the ligaments were determined via tensile testing. Using the T2* histogram profile, each ligament voxel was binned based on its T2* value into four discrete tissue sub-volumes defined by specific T2* intervals. The linear combination of the ligament sub-volumes binned by T2* value significantly predicted maximum load, yield load, and linear stiffness ( R2 = 0.92, 0.82, 0.88; p < 0.001) and were similar to the previous signal intensity based method. In conclusion, the T2* technique offers a highly predictive methodology that is a first step towards the development of a method that can be used to assess ligament healing across scanners, studies, and institutions. © 2013 Orthopaedic Research Society. Published by Wiley Periodicals, Inc. J Orthop Res 32:492-499, 2014. [ABSTRACT FROM AUTHOR]

Abstract: Quantitative computed tomography (QCT) based nonlinear homogenized finite element (hFE) models of the human femur do not take bone׳s microstructure into account due to the low resolution of the QCT images. Models based on high-resolution peripheral quantitative computed tomography (HR-pQCT) are able to include trabecular orientation and allow the modeling of a cortical shell. Such a model showed improvements compared to QCT-based models when studying human vertebral bodies. The goal of this study was to compare the femoral strength prediction ability of subject specific nonlinear homogenized FE (hFE) models based on HR-pQCT and QCT images. Thirty-six pairs of femurs were scanned with QCT as well as HR-pQCT, and tested in one-legged stance (STANCE) and side-ways fall (SIDE) configurations up to failure. Non-linear hFE models were generated from HR-pQCT images (smooth meshes) and compared to recently published QCT based models (voxel meshes) as well as experiments with respect to ultimate force. HR-pQCT-based hFE models improved ultimate force (R 2=0.87 vs 0.80, p=0.02) predictions only in STANCE configuration but not in SIDE (R 2=0.86 vs 0.84, p=0.6). Damage locations were similar for both types of models. In conclusion, it was shown for the first time on a large femur dataset that a more accurate representation of trabecular orientation and cortex only improve FE predictions in STANCE configuration, where the main trabecular orientation is aligned with the load direction. In the clinically more relevant SIDE configuration, the improvements were not significant. [Copyright &y& Elsevier]

Abstract: We generalize several Cauchy-like inclusion regions for the zeros of a polynomial expressed in a basis defined by a three-term recurrence relation. Our results are obtained by applying linear algebra techniques to the comrade matrix of a polynomial. We pay special attention to the Newton and Chebyshev bases. [Copyright &y& Elsevier]

Abstract: This paper introduces a framework that allows to mitigate the impact of class imbalance on most scalar performance measures when used to evaluate the behavior of classifiers. Formally, a correction function is defined with the aim of highlighting those classification results that present moderately higher prediction rates on the minority class. Besides, this function punishes those scenarios that are biased towards the majority class, but also those that are strongly biased to favor the minority class. This strategy assumes a typical imbalance task, in which the minority class contains the most relevant samples to the research purposes. A novel experimental framework is designed to show the advantages of our approach when compared to the standard use of well-established measures, demonstrating its consistency and validity. [Copyright &y& Elsevier]

Abstract: A topological space G is said to be a rectifiable space provided that there are a surjective homeomorphism and an element such that and for every we have , where is the projection to the first coordinate. We firstly define the concept of rectifiable completion of rectifiable spaces and study some properties of rectifiable complete spaces, and then we mainly show that: (1) Each pseudocompact rectifiable space G is a Suslin space, which gives an affirmative answer to V.V. Uspenskijʼs question (Uspenskij, 1989 [29]); (2) Each pseudocompact infinite rectifiable space contains a non-closed countable set; (3) Each pseudocompact rectifiable space G is sequentially pseudocompact; (4) Each infinite pseudocompact rectifiable space with a continuous weak selection is homeomorphic to the Cantor set; (5) Each first-countable ω-narrow rectifiable space has a countable base. Moreover, some examples of rectifiable spaces are given and some questions concerning pseudocompactness on rectifiable spaces are posed. [Copyright &y& Elsevier]

Abstract: A -group is a group such that all its minimal generating sets (with respect to inclusion) have the same size. We prove that the class of finite -groups is closed under taking quotients and that every finite -group is solvable. Via a complete classification of Frattini-free finite -groups we obtain a general structure theorem for finite -groups. Applications include new proofs for the characterization of finite matroid groups. [Copyright &y& Elsevier]

Abstract: In the context of fuzzy metrics in the sense of George and Veeramani, we study when certain families of open balls centered at a point are local bases at this point. This question is related to p-convergence and s-convergence. [Copyright &y& Elsevier]