Structural Approach to Subset Sum Problems

We discuss results obtained jointly with Van Vu on the length of arithmetic progressions in \(\ell \)-fold sumsets of the form

$$\begin{aligned} \ell \mathcal {A}=\{a_1+\dots +a_\ell ~|~a_i\in \mathcal {A}\} \end{aligned}$$

and

$$\begin{aligned} \ell \mathcal {A}=\{a_1+\dots +a_\ell ~|~a_i\in \mathcal {A}\text { all distinct}\}, \end{aligned}$$

where \(\mathcal {A}\) is a set of integers. Applications are also discussed.

     

Synthesis,characterization of V2O5 nanoparticles and determination of catalase mimetic activity by new colorimetric method

Journal of Thermal Analysis and Calorimetry - This paper deals with an advanced colorimetric method used to determine the catalase mimetic activity of V2O5 nanoparticles by measuring the decrease...     

Electrospun zeolite-templated carbon composite fibres for hydrogen storage applications

Research on Chemical Intermediates - The current study explored the application of the electrospinning technique to produce multi-hierarchical composites for hydrogen storage applications....     

Logical analysis of data: classification with justification

Learning from examples is a frequently arising challenge, with a large number of algorithms proposed in the classification, data mining and machine learning literature. The evaluation of the quality of such algorithms is frequently carried out ex post, on an experimental basis: their performance is measured either by cross validation on benchmark data sets, or by clinical trials. Few of these approaches evaluate the learning process ex ante, on its own merits. In this paper, we discuss a property of rule-based classifiers which we call “justifiability”, and which focuses on the type of information extracted from the given training set in order to classify new observations. We investigate some interesting mathematical properties of justifiable classifiers. In particular, we establish the existence of justifiable classifiers, and we show that several well-known learning approaches, such as decision trees or nearest neighbor based methods, automatically provide justifiable classifiers. We also identify maximal subsets of observations which must be classified in the same way by every justifiable classifiers. Finally, we illustrate by a numerical example that using classifiers based on “most justifiable” rules does not seem to lead to overfitting, even though it involves an element of optimization.     

On the local time of random walk on the 2-dimensional comb

We study the path behaviour of general random walks, and that of their local times, on the 2-dimensional comb lattice C² that is obtained from Z² by removing all horizontal edges off the x-axis. We prove strong approximation results for such random walks and also for their local times. Concentrating mainly on the latter, we establish strong and weak limit theorems, including Strassen-type laws of the iterated logarithm, Hirsch-type laws, and weak convergence results in terms of functional convergence in distribution.     

Magnetoelectric properties of particulate and bi-layer PMN-PT/CoFe2O4 composites

Our studies comprise electrical dielectric and magnetoelectric properties of CoFe₂O₄ (CFO) and Pb(Mg_1/3Nb_2/3)_0.67Ti_0.33O₃ [PMN-PT] magnetoelectric composites. The individual phases were prepared by conventional ceramic method. The particulate composites of ferrite and ferroelectric phases were prepared in ferroelectric rich region. Presence of both the phases in the composites was confirmed using X-ray diffraction techniques. The scanning electron microscopic images recorded in backscattered mode were used to study the microstructure of composites. Lattice constant, dielectric constant, electrical resistivity, ferroelectric, and magnetic properties of individual as well as particulate composites were studied. Further the bi-layer composites were made using the discs obtained from the powders of individual phases where hot press technique was employed to obtain disc of individual phases. CFO phase used in bi-layer composites was obtained using chemical co-precipitation technique. Magnetoelectric (ME) measurements were carried out on both, particulate and layered magnetoelectric composites. Comparison of ME signal obtained from particulate and layered composites revealed that the layered composites gives superior magnetoelectric signal. ME data obtained for layered composites show good agreement with the theoretical model.     

Greedy Approximation of High-Dimensional Ornstein–Uhlenbeck Operators

We investigate the convergence of a nonlinear approximation method introduced by Ammar et?al. (J. Non-Newtonian Fluid Mech. 139:153–176, 2006) for the numerical solution of high-dimensional Fokker–Planck equations featuring in Navier–Stokes–Fokker–Planck systems that arise in kinetic models of dilute polymers. In the case of Poisson’s equation on a rectangular domain in ?², subject to a homogeneous Dirichlet boundary condition, the mathematical analysis of the algorithm was carried out recently by Le Bris, Lelièvre and Maday (Const. Approx. 30:621–651, 2009), by exploiting its connection to greedy algorithms from nonlinear approximation theory, explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173–187, 1996); hence, the variational version of the algorithm, based on the minimization of a sequence of Dirichlet energies, was shown to converge. Here, we extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le Bris et al. to a technically more complicated situation, where the Laplace operator is replaced by an Ornstein–Uhlenbeck operator of the kind that appears in Fokker–Planck equations that arise in bead–spring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a high-dimensional Cartesian product configuration space D=D ₁×?×D _N contained in ? ^Nd , where each set D _i , i=1,…,N, is a bounded open ball in ? ^d , d=2,3.     

How to find groups? (and how to use them in Erdős geometry?)

Geometric questions which involve Euclidean distances often lead to polynomial relations of type F(x, y, z)=0 for some F ∈ ?[x, y, z]. Several problems of Combinatorial Geometry can be reduced to studying such polynomials which have many zeroes on n×n×n Cartesian products. The special case when the relation F = 0 can be re-written as z = f(x, y), for a polynomial or rational function f ∈ ?(x, y), was considered in [8]. Our main goal is to extend the results found there to full generality (and also to show some geometric applications, e.g. one on “circle grids”). The main result of our paper concerns low-degree algebraic sets F which contain “too many” points of a (large) n×n×n Cartesian product. Then we can conclude that, in a neighborhood of almost any point, the set F must have a very special (and very simple) form. More precisely, then either F is a cylinder over some curve, or we find a group behind the scene: F must be the image of the graph of the multiplication function of an appropriate algebraic group (see Theorem 3 for the 3D special case and Theorem 27 in full generality).