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In this paper we investigate the following questions. Let $\mu, \nu$ be two regular Borel measures of finite total variation. When do we have a constant $C$ satisfying $$\int f d\nu \le C \int f d\mu$$ whenever $f$ is a continuous nonnegative positive definite function? How the admissible constants $C$ can be characterized, and what is their optimal value? We first discuss the problem in locally compact abelian groups. Then we make further specializations when the Borel measures $\mu, \nu$ are both either purely atomic or absolutely continuous with respect to a reference Haar measure. In addition, we prove a duality conjecture posed in our former paper., Comment: 26 pages

We consider random walks on the square lattice of the plane along the lines of Heyde (1982, 1993) and den Hollander (1994), whose studies have in part been inspired by the so-called transport phenomena of statistical physics. Two-dimensional anisotropic random walks with anisotropic density conditions a' la Heyde (1982, 1993) yield fixed column configurations and nearest-neighbour random walks in a random environment on the square lattice of the plane as in den Hollander (1994) result in random column configurations. In both cases we conclude simultaneous weak Donsker and strong Strassen type invariance principles in terms of appropriately constructed anisotropic Brownian motions on the plane, with self-contained proofs in both cases. The style of presentation throughout will be that of a semi-expository survey of related results in a historical context., Comment: 25 pages

The giant magnetoresistance (GMR) was investigated for electrodeposited Co/Cu multilayers. In order to better understand the formation of individual layers and their influence on GMR, multilayers produced by two different deposition strategies were compared. One series of Co(2 nm)/Cu(tCu) multilayers with tCu ranging from 0.5 nm to 6 nm was produced with the conventional two-pulse plating by using a galvanostatic/potentiostatic (G/P) pulse combination for the magnetic/non-magnetic layer deposition, respectively, whereby the Cu layer deposition was carried out at the electrochemically optimized potential. Another Co(2 nm)/Cu(tCu) multilayer series with the same tCu range was prepared with the help of a G/P/G pulse combination. In this latter case, first a bilayer of Co(2 nm)/Cu(6 nm) was deposited in each cycle as in the G/P mode after which a third G pulse was applied with a small anodic current to dissolve part of the 6 nm thick Cu layer in order to ensure the targeted tCu value. The comparison of the two series revealed that the G/P/G pulse combination yields multilayers for which GMR can be obtained even at such low nominal Cu layer thicknesses where G/P multilayers already exhibit bulk-like anisotropic magnetoresistance only. Surface roughness measurements by atomic force microscopy revealed that the two kinds of pulse combination yield different surface roughness values which correlate with the structural quality of the multilayers as indicated by the absence or presence of multilayer satellite reflections in the X-ray diffraction patterns. The results of multilayer chemical analysis revealed that mainly an increased Cu content of the magnetic layer is responsible for the onset of SPM regions in the form of Co segregations in the G/P/G multilayers with small Cu layer thicknesses., Comment: 32 pages, 10 figures

Although electrodeposition from aqueous media has been widely used to obtain metallic deposits, there are cases where the application of non-aqueous solutions offers advantages over the traditional baths or even represents the only way to electrodeposit some metals. For this reason, a study of the electrodeposition of Ni from various alcoholic solutions was performed. Apart from a detailed cyclic voltammetry study of these solutions, the surface morphology, crystal structure and texture as well magnetic properties of the deposits have also been investigated. The best results were obtained with methanol as solvent, so results on Ni deposits prepared from this solution will be presented in more detail whereas the case of the other alcoholic solutions investigated will be summarized only briefly. Structural and magnetic properties of the deposits obtained have been compared to literature data on Ni samples obtained from various non-aqueous solvents., Comment: 27 pages, 9 figures

Let $\ell>0$ be arbitrary. We introduce the extremal quantities $$ G(\ell):=\frac{\sup_{f} \int_{-\ell}^{\ell} f\,dx}{\int_{-1}^1 f\,dx},\quad C(\ell):=\frac{\sup_{f} \sup_{a\in {\mathbb R}} \int_{a-\ell}^{a+\ell} f\,dx}{\int_{-1}^1 f\,dx}, $$ where the supremum is taken over all not identically zero non-negative positive definite functions. We are interested in the question: how large can the above extremal quantities be? This problem was originally posed by Yu. Shteinikov and S. Konyagin for the case $\ell=2$. In this note we obtain exact values for the right limits $G(k+0)$ and $C(k+0)$ at natural numbers $k$, and sufficiently close bounds for other values of $\ell$. We point out that the problem provides an extension of the classical problem of Wiener.

P. Tur\'an was the first to derive lower estimations on the uniform norm of the derivatives of polynomials $p$ of uniform norm $1$ on the interval I:=[-1,1] and the disk D:=$\{z \in C~:~|z| \le 1\}$, under the normalization condition that the zeroes of the polynomial p in question all lie in I or D, resp. Namely, in 1939 he proved that with n:=deg p tending to infinity, the precise growth order of the minimal possible derivative norm is $\sqrt{n}$ for I and n for D. Already the same year J. Er\H{o}d considered the problem on other domains. In his most general formulation, he extended Tur\'an's order n result on D to a certain general class of piecewise smooth convex domains. Finally, a decade ago the growth order of the minimal possible norm of the derivative was proved to be n for all compact convex domains. Tur\'an himself gave comments about the above oscillation question in $L^q$ norm on D. Nevertheless, till recently results were known only for I, D and so-called R-circular domains. Continuing our recent work, also here we investigate the Tur\'an-Er\H{o}d problem on general classes of domains., Comment: arXiv admin note: text overlap with arXiv:1512.08268

No systematic studies have been carried out on the giant magnetoresistance (GMR) of electrodeposited (ED) Fe-Co/Cu multilayers since the elaboration of a method for the optimization of the Cu layer deposition potential. In this paper, we present results on the electrochemical optimization of the Cu layer deposition potential which was found to depend on the relative iron concentration in the bath. An X-ray diffraction study of ED Fe5Co95(1.5 nm)/Cu(dCu) multilayers with dCu ranging from 0.8 nm to 10 nm revealed an fcc structure. For most of the multilayers, a weak superlattice satellite reflection could be identified. The room-temperature magnetoresistance was studied in detail as a function of the individual layer thicknesses. Multilayers with Cu layer thicknesses above about 1.5 nm were found to exhibit a GMR behavior with a maximum GMR of about 5 % and a typical saturation field of 1 kOe. The GMR magnitude decreased with increasing Fe-content in the magnetic layer. The spacer layer thickness evolution of the MR data was established in detail after separating the ferromagnetic and superparamagnetic GMR contributions and no oscillatory GMR was found. A comparison with literature data on both physically deposited and ED Fe-Co/Cu multilayers is also made., Comment: 22 pages, 9 figures

A simple random walk and a Brownian motion are considered on a spider that is a collection of half lines (we call them legs) joined in the origin. We give a strong approximation of these two objects and their local times. For fixed number of legs we establish limit theorems on local and occupation times in n steps., Comment: 22 pages,contains small corrections,and some detailed explanations than the previous version

It has been shown previously for electrodeposited Co/Cu multilayers that the single-bath electrodeposition process can be optimized from an electrochemical point of view in order to avoid unwanted Co dissolution and incorporation of Co in the non-magnetic layer during the Cu deposition pulse. In the present work, electrodeposition of Ni-Co/Cu multilayers has been studied to clarify if the same optimization method is appropriate when two magnetic elements are present and if this potential results in the largest giant magnetoresistance (GMR) for the particular alloy system studied. For this purpose, several Ni-Co/Cu multilayers were prepared by varying the deposition potential of the Cu layer. The composition analysis of the deposits showed that the Ni:Co ratio exhibits a minimum as a function of the Cu deposition potential, which can be explained by considering both the dissolution of Co and the mass transport of the reactants. Both the saturation GMR value and the intensity of the satellite peaks in the X-ray diffractograms were highly correlated with the resulting surface roughness of the deposits which was strongly varying with the Cu deposition potential. Higher GMR values, lower saturation fields and more perfect multilayer structure were observed for sufficiently positive Cu deposition potentials only which enabled a partial Co dissolution resulting in a reduced surface roughness. The results draw attention to the complexity of the optimization procedure of the deposition of multilayers with several alloying components., Comment: 26 pages, 10 figures

The electrical resistivity and the anisotropic magnetoresistance (AMR) was investigated for Ni Co alloys at and below room temperature. The Ni Co alloy layers having a thickness of about 2 um were prepared by electrodeposition on Si wafers with evaporated Cr and Cu underlayers. The alloy composition was varied in the whole concentration range by varying the ratio of Ni sulfate and Co sulfate in the electrolyte. The Ni Co alloy deposits were investigated first in the as deposited state on the substrates and then, by mechanically stripping them from the substrates, as self supporting layers both without and after annealing. According to an X ray diffraction study, a strongly textured face centered cubic (fcc) structure was formed in the as deposited state with an average grain size of about 10 nm. Upon annealing, the crystal structure was retained whereas the grain size increased by a factor of 3 to 5, depending on alloy composition. The zero field resistivity decreased strongly by annealing due to the increased grain size. The annealing hardly changed the AMR below 50 at.% Co but strongly decreased it above this concentration. The composition dependence of the resistivity and the AMR of the annealed Ni Co alloy deposits was in good quantitative agreement with the available literature data both at 13 K and at room temperature. Both transport parameters were found to exhibit a pronounced maximum in the composition range between 20 and 30 at.% Co and the data of the Ni Co alloys fit well to the limiting values of the pure component metals (fcc Ni and fcc Co). The only theoretical calculation reported formerly on fcc Ni Co alloys yielded at T=0K a resistivity value smaller by a factor of 5 and an AMR value larger by a factor of about 2 than the corresponding low temperature experimental data, although the theoretical results properly reproduced the composition dependence of both quantities., Comment: 27 pages, 10 figures

We consider two or more simple symmetric walks on some graphs, e.g. the real line, the plane or the two dimensional comb lattice, and investigate the properties of the distance among the walkers., Comment: 27 pages

We investigate an extension of an equilibrium-type result, conjectured by Ambrus, Ball and Erd\'elyi, and proved recently by Hardin, Kendall and Saff. These results were formulated on the torus, hence we also work on the torus, but one of the main motivations for our extension comes from an analogous setup on the unit interval, investigated earlier by Fenton. Basically, the problem is a minimax one, i.e. to minimize the maximum of a function $F$, defined as the sum of arbitrary translates of certain fixed "kernel functions", minimization understood with respect to the translates. If these kernels are assumed to be concave, having certain singularities or cusps at zero, then translates by $y_j$ will have singularities at $y_j$ (while in between these nodes the sum function still behaves realtively regularly). So one can consider the maxima $m_i$ on each subintervals between the nodes $y_j$, and look for the minimization of $\max F = \max_i m_i$. Here also a dual question of maximization of $\min_i m_i$ arises. This type of minimax problems were treated under some additional assumptions on the kernels. Also the problem is normalized so that $y_0=0$. In particular, Hardin, Kendall and Saff assumed that we have one single kernel $K$ on the torus or circle, and $F=\sum_{j=0}^n K(\cdot-y_j)= K + \sum_{j=1}^n K(\cdot-y_j)$. Fenton considered situations on the interval with two fixed kernels $J$ and $K$, also satisfying additional assumptions, and $F= J + \sum_{j=1}^n K(\cdot-y_j)$. Here we consider the situation (on the circle) when \emph{all the kernel functions can be different}, and $F=\sum_{j=0}^n K_j(\cdot- y_j) = K_0 + \sum_{j=1}^n K_j(\cdot-y_j)$. Also an emphasis is put on relaxing all other technical assumptions and give alternative, rather minimal variants of the set of conditions on the kernel.

Some 76 years ago P. Tur\'an was the first to establish lower estimations of the ratio of the maximum norm of the derivatives of polynomials and the maximum norm of the polynomials themselves on the interval I:=[-1,1] and on the unit disk D:={z : |z| <= 1} under the normalization condition that the zeroes of the polynomial all lie in the interval or in the disk, respectively. He proved that with n:=deg p tending to infinity, the precise growth order of the minimal possible ratio of the derivative norm and the norm is square-root{n} for I and n for D. J. Er\"od continued the work of Tur\'an and extended his results to several other domains. The growth of the minimal possible ratio of the infinity norm of the derivative and the polynomial itself was proved to be of order n for all compact convex domains a decade ago. Although Tur\'an himself gave comments about the above oscillation question in Lq norms, till recently results were known only for D and I. Here we prove that in Lq norm the oscillation order is again n for a certain class of convex domains, including all smooth convex domains and also convex polygonal domains having no acute angles at their vertices.

The century old extremal problem, solved by Carath\'eodory and Fej\'er, concerns a nonnegative trigonometric polynomial normalized by a0 = 1, and the quantity to be maximized is the coefficient a1. In the complex exponential form, the coefficient sequence (ck) will be supported in [-n; n] and normalized by c0 =1. Reformulating, nonnegativity of T translates to positive definiteness of the sequence (ck), and the extremal problem becomes a maximization problem for the value at 1 of a normalized positive definite function c : Z --> C, supported in [-n; n]. Boas and Katz, Arestov, Berdysheva and Berens, Kolountzakis and R\'ev\'esz and recently Krenedits and R\'ev\'esz investigated the problem in increasing generality, reaching analogous results for all locally compact Abelian groups. We prove an extension to all the known results in not necessarily commutative locally compact groups.

This is an expository paper on the asymptotic results concerning path behaviour of the anisotropic random walk on the two-dimensional square lattice Z^2. In recent years Mikl\'os and the authors of the present paper investigated the properties of this random walk concerning strong approximations, local times and range. We give a survey of these results together with some further problems., Comment: 20 pages

A simple symmetric random walk is considered on a spider that is a collection of half lines (we call them legs) joined at the origin. We establish a strong approximation of this random walk by the so-called Brownian spider. Transition probabilities are studied, and for a fixed number of legs we investigate how high the walker can go on the legs in $n$ steps. The heights on the legs are also investigated when the number of legs goes to infinity., Comment: arXiv admin note: text overlap with arXiv:1402.5682

We consider a simple symmetric random walk on a spider, that is a collection of half lines (we call them legs) joined at the origin. Our main question is the following: if the walker makes $n$ steps how high can he go up on all legs. This problem is discussed in two different situations; when the number of legs are increasing, as $n$ goes to infinity and when it is fixed., Comment: 14 pages

If a function $f:\mathbb{R}\to\mathbb{R}$ can be represented as the sum of $n$ periodic functions as $f=f_1+\dots+f_n$ with $f(x+\alpha_j)=f(x)$ ($j=1,\dots,n$), then it also satisfies a corresponding $n$-order difference equation $\Delta_{\alpha_1}\dots\Delta_{\alpha_n} f=0$. The periodic decomposition problem asks for the converse implication, which may hold or fail depending on the context (on the system of periods, on the function class in which the problem is considered, etc.). The problem has natural extensions and ramifications in various directions, and is related to several other problems in real analysis, Fourier and functional analysis. We give a survey about the available methods and results, and present a number of intriguing open problems.

We work out the optimization problem, initiated by K. Soundararajan, for the choice of the underlying polynomial P used in the construction of the weight function in the Goldston--Pintz--Yildirim method for finding small gaps between primes. First we reformulate to a maximization problem on L^2[0,1] for a self-adjoint operator T, the norm of which is then the maximal eigenvalue of T. To find eigenfunctions and eigenvalues, we derive a differential equation which can be explicitly solved. The aimed maximal value is S(k)=4/(k+ck^{1/3}), achieved by the (k-1)st integral of x^{1-k/2}J_{k-2}(a_1\sqrt{x}), where a_1 is the first positive root of the (k-2)nd Bessel function J_{k-2} and as such, is asymptotically ck^{1/3} with a well-known constant c. As this naturally gives rise to a number of technical problems in the application of the GPY method, we also construct a polynomial P which is a simpler function yet it furnishes an approximately optimal extremal quantity, 4/(k+Ck^{1/3}) with some other constant C. In the forthcoming paper of J. Pintz (also to appear in the Tur\'an-100 Memorial Volume by de Gruyter), it is indeed shown how this quasi-optimal choice of the polynomial in the weight finally can exploit the GPY method to its theoretical limits., Comment: This work was done during the last months with the aim of finding the limit of the GPY method. It will appear in the Tur\'an-100 Memorial Volume by de Gruyter and it was placed also on the last author's homepage at the Renyi Institute. However, due to the recent breakthrough of Zhang it can be used now to optimize the numerical value of the maximal gaps between consecutive primes

We consider the extremal problem of maximizing a point value jf(z)j at a given point z 2 G by some positive definite and continuous function f on an Abelian group G, where for a given symmetric open set 3 z, f vanishes outside and is normalized by f(0) = 1. Denote the extremal value as CG(; z). This extremal problem was investigated in R and Rd and for a 0-symmetric convex body in a paper of Boas and Kac in 1943. Arestov and Berdysheva extended the investigation to Td, where T := R=Z. Kolountzakis and R?ev?esz gave a more general setting, considering arbitrary open sets, in all the classical groups above. Also they observed, that such extremal problems occurred in certain special cases and in a different, but equivalent formulation already a century ago in the work of Carath?eodory and Fej?er. Moreover, following observations of Boas and Kac, Kolountzakis and R?ev?esz showed how the general problem can be reduced to equivalent discrete problems of "Carath?eodory-Fej?er type" on Z or Zm := Z=mZ. We extend the results of Kolountzakis and R?ev?esz to locally compact Abelian groups.

We consider the classical Wiener-Ikehara Tauberian theorem, with a generalized condition of slow decrease and some additional poles on the boundary of convergence of the Laplace transform. In this generality, we prove the otherwise known asymptotic evaluation of the transformed function, when the usual conditions of the Wiener-Ikehara theorem hold. However, our version also provides an effective error term, not known thus far in this generalality. The crux of the proof is a proper, asymptotic variation of the lemmas of Ganelius and Tenenbaum, also constructed for the sake of an effective version of the Wiener-Ikehara Theorem.

We prove that the number gamma(N) of the zeros of a two-parameter simple random walk in its first N-by-N time steps is almost surely equal to N to the power 1+o(1) as N goes to infinity. This is in contrast with our earlier joint effort with Z. Shi [4]; that work shows that the number of zero crossings in the first N-by-N time steps is N to the power (3/2)+o(1) as N goes to infinity. We prove also that the number of zeros on the diagonal in the first N time steps is (c+o(1)) log N as N goes to infinity, where c is 2\pi., Comment: 14 pages

Starting out from results known for the most classical cases of N, Z^d, R^d or for sigma-finite abelian groups, here we define the notion of asymptotic uniform upper density in general locally compact abelian groups. Even if a bit surprising, the new notion proves to be the right extension of the classical cases of Z^d, R^d. The new notion is used to extend some analogous results previously obtained only for classical cases or sigma-finite abelian groups. In particular, we show the following extension of a well-known result for Z of Furstenberg: if in a general locally compact Abelian group G a subset S of G has positive uniform asymptotic upper density, then S-S is syndetic.

We study the path behaviour of a simple random walk on the 2-dimensional comb lattice ${\mathbb C}^2$ that is obtained from ${\mathbb Z}^2$ by removing all horizontal edges off the x-axis. In particular, we prove a strong approximation result for such a random walk which, in turn, enables us to establish strong limit theorems, like the joint Strassen type law of the iterated logarithm of its two components, as well as their marginal Hirsch type behaviour., Comment: 30 pages, 1 figure

This is a companion paper of a recent one, entitled {\sl Integral concentration of idempotent trigonometric polynomials with gaps}. New results of the present work concern $L^1$ concentration, while the above mentioned paper deals with $L^p$-concentration. Our aim here is two-fold. At the first place we try to explain methods and results, and give further straightforward corollaries. On the other hand, we push forward the methods to obtain a better constant for the possible concentration (in $L^1$ norm) of an idempotent on an arbitrary symmetric measurable set of positive measure. We prove a rather high level $\gamma_1>0.96$, which contradicts strongly the conjecture of Anderson et al. that there is no positive concentration in $L^1$ norm. The same problem is considered on the group $\mathbb{Z}/q\mathbb{Z}$, with $q$ say a prime number. There, the property of absolute integral concentration of idempotent polynomials fails, which is in a way a positive answer to the conjecture mentioned above. Our proof uses recent results of B. Green and S. Konyagin on the Littlewood Problem., Comment: 23 pages

We consider transient nearest neighbor random walks on the positive part of the real line. We give criteria for the finiteness of the number of cutpoints and strong cutpoints. Examples and open problems are presented., Comment: 17 pages