Search

Silicon vacancy ($\mathrm{V_{Si}}$) color centers in 4H-SiC are optically accessible through their zero-phonon line (ZPL) photoluminescence (PL), which is sensitive to the sample temperature. We report the effects of localized laser-induced heating in 4H-SiC by measuring the PL spectra of $\mathrm{V_{Si}}$ color centers. The effects of laser-induced heating manifest as the decrease in the peak height, redshift, and broadening of the ZPLs in the PL spectrum. The local temperature in the sample can be determined from the center energy of the ZPLs by using the Varshni equation. The sample temperature can be modeled as a system in contact with a thermal reservoir while being heated by a laser beam. This work highlights the importance of considering laser-induced heating in the optical properties of color centers in 4H-SiC and their potential applications. The result also suggests that the sharp and bright ZPLs of color centers can be used as local temperature probes in 4H-SiC devices., Comment: 13 pages, 14 figures

Optical two-dimensional coherent spectroscopy (2DCS) is a powerful ultrafast spectroscopic technique that can greatly benefit from the unique features of a femtosecond laser operating at a kHz repetition rate. However, isolating specific nonlinear signal in the collinear geometry, especially with a kHz laser, presents challenges. We present an experimental implementation of optical 2DCS in a collinear geometry using a femtosecond laser operating at a 5 KHz repetition rate. The desired nonlinear signal is selectively extracted in the frequency domain by lock-in detection. Both pump-probe and optical 2DCS experiments were conducted on a rubidium vapor. The signal-to-noise ratio of pump-probe and 2DCS spectra was characterised under various experimental parameters. The study highlights the importance of the lock-in reference frequency in overcoming the limitations imposed by low repetition rates, thereby paving the way for the application of collinear optical 2DCS in material systems requiring kHz femtosecond lasers., Comment: 5 pages, 3 figures

In 1939 P\'al Tur\'an and J\'anos Er\H{o}d initiated the study of lower estimations of maximum norms of derivatives of polynomials, in terms of the maximum norms of the polynomials themselves, on convex domains of the complex plane. As a matter of normalization they considered the family $\mathcal{P}_n(K)$ of degree $n$ polynomials with all zeros lying in the given convex, compact subset $K\Subset {\mathbb C}$. While Tur\'an obtained the first results for the interval $I:=[-1,1]$ and the disk $D:=\{ z\in {\mathbb C}~:~ |z|\le 1\}$, Er\H{o}d extended investigations to other compact convex domains, too. The order of the optimal constant was found to be $\sqrt{n}$ for $I$ and $n$ for $D$. It took until 2006 to clarify that all compact convex \emph{domains} (with nonempty interior), follow the pattern of the disk, and admit an order $n$ inequality. For $L^q(\partial K)$ norms with any $1\le q <\infty$ we obtained order $n$ results for various classes of domains. Further, in the generality of all convex, compact domains we could show a $c n/\log n$ lower bound together with an $O(n)$ upper bound for the optimal constant. Also, we conjectured that all compact convex domains admit an order $n$ Tur\'an type inequality. Here we prove this for all \emph{polygonal} convex domains and any $0< q <\infty$., Comment: arXiv admin note: substantial text overlap with arXiv:1512.08268

In two previous papers the second author proved some Carlson type density theorems for zeroes in the critical strip for Beurling zeta functions satisfying Axiom A of Knopfmacher. In the first of these invoking two additonal conditions were needed, while in the second an explicit, fully general result was obtained. Subsequently, Frederik Broucke and Gregory Debruyne obtained, via a different method, a general Carlson type density theorem with an even better exponent, and recently Frederik Broucke improved this further, getting $N(\sigma,T) \le T^{a(1-\sigma)}$ with any $a>\dfrac{4}{1-\theta}$. Broucke employed a new mean value estimate of the Beurling zeta function, while he did not use the method of Hal\'asz and Montgomery. Here we elaborate a new approach of the first author, using the classical zero detecting sums coupled with a kernel function technique and Hal\'asz' method, but otherwise arguing in an elementary way avoiding e.g. mean value estimates for Dirichlet polynomials. We will make essential use of the additional assumptions that the Beurling system of integers consists of natural numbers, and that the system satisfies the Ramanujan condition, too. This way we give a new variant of the Carlson type density estimate with similar strength as Tur\'an's 1954 result for the Riemann $\zeta$ function, coming close even to the Density Hypothesis for $\sigma$ close to 1.

Asymptotic uniform upper density, shortened as a.u.u.d., or simply upper density, is a classical notion which was first introduced by Kahane for sequences in the real line. Syndetic sets were defined by Gottschalk and Hendlund. For a locally compact group $G$, a set $S\subset G$ is syndetic, if there exists a compact subset $C\Subset G$ such that $SC=G$. Syndetic sets play an important role in various fields of applications of topological groups and semigroups, ergodic theory and number theory. A lemma in the book of F\"urstenberg says that once a subset $A \subset {\mathbb Z}$ has positive a.u.u.d., then its difference set $A-A$ is syndetic. The construction of a reasonable notion of a.u.u.d. in general locally compact Abelian groups (LCA groups for short) was not known for long, but in the late 2000's several constructions were worked out to generalize it from the base cases of ${\mathbb Z}^d$ and ${\mathbb R}^d$. With the notion available, several classical results of the Euclidean setting became accessible even in general LCA groups. Here we work out various versions in a general LCA group $G$ of the classical statement that if a set $S\subset G$ has positive asymptotic uniform upper density, then the difference set $S-S$ is syndetic., Comment: arXiv admin note: substantial text overlap with arXiv:0904.1567

We investigate the weighted Bojanov-Chebyshev extremal problem for trigonometric polynomials, that is, the minimax problem of minimizing $\|T\|_{w,C({\mathbb T})}$, where $w$ is a sufficiently nonvanishing, upper bounded, nonnegative weight function, the norm is the corresponding weighted maximum norm on the torus ${\mathbb T}$, and $T$ is a trigonometric polynomial with prescribed multiplicities $\nu_1,\ldots,\nu_n$ of root factors $|\sin(\pi(t-z_j))|^{\nu_j}$. If the $\nu_j$ are natural numbers and their sum is even, then $T$ is indeed a trigonometric polynomial and the case when all the $\nu_j$ are 1 covers the Chebyshev extremal problem. Our result will be more general, allowing, in particular, so-called generalized trigonometric polynomials. To reach our goal, we invoke Fenton's sum of translates method. However, altering from the earlier described cases without weight or on the interval, here we find different situations, and can state less about the solutions.

We investigate the existence of well-behaved Beurling number systems, which are systems of Beurling generalized primes and integers which admit a power saving in the error term of both their prime and integer-counting function. Concretely, we search for so-called $[\alpha,\beta]$-systems, where $\alpha$ and $\beta$ are connected to the optimal power saving in the prime and integer-counting functions. It is known that every $[\alpha,\beta]$-system satisfies $\max\{\alpha,\beta\}\ge1/2$. In this paper we show there are $[\alpha,\beta]$-systems for each $\alpha \in [0,1)$ and $\beta \in [1/2, 1)$. Assuming the Riemann hypothesis, we also construct certain families of $[\alpha,\beta]$-systems with $\beta<1/2$., Comment: 24 pages

In 1939 Tur\'{a}n raised the question about lower estimations of the maximum norm of the derivatives of a polynomial $p$ of maximum norm $1$ on the compact set $K$ of the complex plain under the normalization condition that the zeroes of $p$ in question all lie in $K$. Tur\'{a}n studied the problem for the interval $I=[-1,1]$ and the unit disk $D$ and found that with $n$ denoting the degree of $p$ and with $n$ tending to infinity, the precise growth order of the minimal possible derivative norm (oscillation order) is $\sqrt{n}$ for $I$ and $n$ for $D$. Er\H{o}d continued the work of Tur\'{a}n considering other domains. Finally, in 2006, Hal\'{a}sz and R\'{e}v\'{e}sz proved that the growth of the minimal possible maximal norm of the derivative is of order $n$ for all compact convex domains. Although Tur\'{a}n himself gave comments about the above oscillation question in $L^q$ norms, till recently results were known only for $D$ and $I$. Recently, we have found order $n$ lower estimations for several general classes of compact convex domains, and proved that in $L^q$ norm the oscillation order is at least $n/\log n$ for all compact convex domains. In the present paper we prove that the oscillation order is not greater than $n$ for all compact (not necessarily convex) domains $K$ and $L^q$-norm with respect to any measure supported on more than two points on $K$.

Magnesium and its alloys are the most investigated materials for solid-state hydrogen storage in the form of metal hydrides, but there are still unresolved problems with the kinetics and thermodynamics of hydrogenation and dehydrogenation of this group of materials. Severe plastic deformation (SPD) methods, such as equal-channel angular pressing (ECAP), high-pressure torsion (HPT), intensive rolling and fast forging, have been widely used to enhance the activation, air resistance, and hydrogenation/dehydrogenation kinetics of Mg-based hydrogen storage materials by introducing ultrafine/nanoscale grains and crystal lattice defects. These severely deformed materials, particularly in the presence of alloying additives or second-phase nanoparticles, can show not only fast hydrogen absorption/desorption kinetics but also good cycling stability. It was shown that some materials that are apparently inert to hydrogen can absorb hydrogen after SPD processing. Moreover, the SPD methods were effectively used for hydrogen binding-energy engineering and synthesizing new magnesium alloys with low thermodynamic stability for reversible low/room-temperature hydrogen storage, such as nanoglasses, high-entropy alloys, and metastable phases including the high-pressure {\gamma}-MgH2 polymorph. This article reviews recent advances in the development of Mg-based hydrogen storage materials by SPD processing and discusses their potential in future applications.

In previous papers we investigated so-called sum of translates functions $F({\mathbf{x}},t):=J(t)+\sum_{j=1}^n \nu_j K(t-x_j)$, where $J:[0,1]\to \underline{\mathbb{R}}:={\mathbb{R}}\cup\{-\infty\}$ is a "sufficiently nondegenerate" and upper-bounded "field function", and $K:[-1,1]\to \underline{\mathbb{R}}$ is a fixed "kernel function", concave both on $(-1,0)$ and $(0,1)$, and also satisfying the singularity condition $K(0)=\lim_{t\to 0} K(t)=-\infty$. For node systems ${\mathbf{x}}:=(x_1,\ldots,x_n)$ with $x_0:=0\le x_1\le\dots\le x_n\le 1=:x_{n+1}$, we analyzed the behavior of the local maxima vector ${\mathbf{m}}:=(m_0,m_1,\ldots,m_n)$, where $m_j:=m_j({\mathbf{x}}):=\sup_{x_j\le t\le x_{j+1}} F({\mathbf{x}},t)$. Among other results we proved a strong intertwining property: if the kernels are also decreasing on $(-1,0)$ and increasing on $(0,1)$, and the field function is upper semicontinuous, then for any two different node systems there are $i,j$ such that $m_i({\mathbf{x}})m_j({\mathbf{y}})$. Here we partially succeed to extend this even to nonsingular kernels., Comment: The current v3 is a very slightly corrected version with a few updated references (former ArXiv prerints have already appeared or accepted, and this is now signified). Note that a v2 version was uplodaed recently by mistake (that was intended to be an updated new version for another paper) - it was requested that v2 be removed from the records of this paper. arXiv admin note: text overlap with arXiv:2210.04348, arXiv:2112.10169

Following P. Fenton, we investigate sum of translates functions $F(\mathbf{x},t):=J(t)+\sum_{j=1}^n \nu_j K(t-x_j)$, where $J:[0,1]\to {\underline{\mathbb{R}}}:=\mathbb{R}\cup\{-\infty\}$ is a "sufficiently non-degenerate" and upper-bounded "field function", and $K:[-1,1]\to {\underline{\mathbb{R}}}$ is a fixed "kernel function", concave both on $(-1,0)$ and $(0,1)$, $\mathbf{x}:=(x_1,\ldots,x_n)$ with $0\le x_1\le\dots\le x_n\le 1$, and $\nu_1,\dots,\nu_n>0$ are fixed. We analyze the behavior of the local maxima vector $\mathbf{m}:=(m_0,m_1,\ldots,m_n)$, where $m_j:=m_j(\mathbf{x}):=\sup_{x_j\le t\le x_{j+1}} F(\mathbf{x},t)$, with $x_0:=0$, $x_{n+1}:=1$; and study the optimization (minimax and maximin) problems $\inf_{\mathbf{x}}\max_j m_j(\mathbf{x})$ and $\sup_{\mathbf{x}}\min_j m_j(\mathbf{x})$. The main result is the equality of these quantities, and provided $J$ is upper semicontinuous, the existence of extremal configurations and their description as equioscillation points $\mathbf{w}$. In our previous papers we obtained results for the case of singular kernels, i.e., when $K(0)=-\infty$ and the field $J$ was assumed to be upper semicontinuous. In this work we get rid of these assumptions and prove common generalizations of Fenton's and our previous results, arriving at the greatest generality in the setting of concave kernel functions., Comment: v2 differs from v1 only in an added ArXiv preprint reference to a paper listed formerly as "manuscript". v3 is a slightly revised, corrected version with no essential change, but with further updates of bibliographiy items

In a previous paper we proved a Carlson type density theorem for zeroes in the critical strip for Beurling zeta functions satisfying Axiom A of Knopfmacher. There we needed to invoke two additonal conditions, the integrality of the norm (Condition B) and an "average Ramanujan Condition" for the arithmetical function counting the number of different Beurling integers of the same norm (Condition G). Here we implement a new approach of Pintz, using the classic zero detecting sums coupled with Hal\'asz' method, but otherwise arguing in an elementary way avoiding e.g. large sieve type inequalities. This way we give a new proof of a Carlson type density estimate--with explicit constants--avoiding any use of the two additional conditions, needed earlier. Therefore it is seen that the validity of a Carlson-type density estimate does not depend on any extra assumption--neither on the functional equation, present for the Selberg class, nor on growth estimates of coefficients say of the "average Ramanujan type"-but is a general property, presenting itself whenever the analytic continuation is guaranteed by Axiom A.

We prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. These are concerned with the question of Ingham who asked for optimal and explicit order estimates for the error term $\Delta(x):=\psi(x)-x$, given any zero-free region $D(\eta):=\{s=\sigma+it\in\mathbb{C}~:~ \sigma:=\Re s \ge 1-\eta(t)\}$. In the classical case essentially sharp results are due to some 40 years old work of Pintz. Here we consider a given a system of Beurling primes $\mathcal{P}$, the generated arithmetical semigroup $\mathcal{G}$ and the corresponding integer counting function $N(x)$, and the corresponding error term $\Delta_{\mathcal{G}}(x):=\psi_{\mathcal{G}}(x)-x$ in the PNT of Beurling, where $\psi_{\mathcal{G}}(x)$ is the Beurling analog of $\psi(x)$. First we prove that if the Beurling zeta function $\zeta_{\mathcal{G}}$ does not vanish in $D(\eta)$, then the extension of Pintz' result holds: $|\Delta_{\mathcal{G}}(x)| \le x\exp(-(1-\varepsilon)\omega_\eta(x))~(x>x_0(\varepsilon))$, where $\omega_\eta(y)$ is the naturally occurring conjugate function to $\eta(t)$, introduced into the field by Ingham. In the second part we prove a converse: if $\zeta_{\mathcal{G}}$ has an infinitude of zeroes in the given domain, then analogously to the classical case, $|\Delta_{\mathcal{G}}(x)| \ge x\exp(-(1+\varepsilon)\omega_\eta(x))$ holds "infinitely often". This also shows that both main results are sharp apart from the arbitrarily small $\varepsilon>0$., Comment: arXiv admin note: text overlap with arXiv:2202.01837

Optical two-dimensional coherent spectroscopy (2DCS) has become a powerful tool for studying energy level structure, dynamics, and coupling in many systems including atomic ensembles. Various types of two-dimensional (2D) spectra, including the so-called single-quantum, zero-quantum, and double-quantum 2D spectra, of both D lines (D$_1$ and D$_2$ transitions) of potassium (K) atoms have been reported previously. For rubidium (Rb), a major difference is that the D-lines are about 15 nm apart as opposed to only about 3 nm for K. Simultaneously exciting both D-lines of Rb atoms requires a broader laser bandwidth for the experiment. Here, we report a broadband optical 2DCS experiment in an Rb atomic vapor. A complete set of single-quantum, zero-quantum, and double-quantum 2D spectra including both D-lines of Rb atoms were obtained. The experimental spectra were reproduced by simulated 2D spectra based on the perturbation solutions to the optical Bloch equations. This work in Rb atoms complements previous 2DCS studies of K and Rb with a narrower bandwidth that covers two D-lines of K or only a single D-line of Rb. The broadband excitation enables the capability to perform double-quantum and multi-quantum 2DCS of both D-lines of Rb to study many-body interactions and correlations in comparison with K atoms., Comment: 10 pages, 7 figures

Continuing previous study of the Beurling zeta function, here we prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. First, we address the question of Littlewood, who asked for explicit oscillation results provided a zeta-zero is known. We prove that given a zero $\rho_0$ of the Beurling zeta function $\zeta_P$ for a given number system generated by the primes $P$, the corresponding error term $\Delta_P(x):=\psi_{P}(x)-x$, where $\psi_{P}(x)$ is the von Mangoldt summatory function shows oscillation in any large enough interval, as large as $(\pi/2-\varepsilon) x^{\Re \rho_0}/|\rho_0|$. The somewhat mysterious appearance of the constant $\pi/2$ is explained in the study. Finally, we prove as the next main result of the paper the following: given $\varepsilon>0$, there exists a Beurling number system with primes $P$, such that $|\Delta_P(x)| \le (\pi/2+\varepsilon)x^{\Re \rho_0}/|\rho_0|$. In this second part a nontrivial construction of a low norm sine polynomial is coupled by the application of the wonderful recent prime random approximation result of Broucke and Vindas, who sharpened the breakthrough probabilistic construction due to Diamond, Montgomery and Vorhauer.

For a fixed positive integer $n$ consider continuous functions $ K_1,\dots$, $ K_n:[-1,1]\to \mathbb{R}\cup\{-\infty\}$ that are concave and real valued on $[-1,0)$ and on $(0,1]$, and satisfy $K_j(0)=-\infty$. Moreover, let $J:[0,1]\to \mathbb{R}\cup\{-\infty\}$ be upper bounded and such that $[0,1]\setminus J^{-1}(\{-\infty\})$ has at least $n+1$ elements, but it is arbitrary otherwise. For $x_0:=0

Minimax and maximin problems are investigated for a special class of functions on the interval $[0,1]$. These functions are sums of translates of positive multiples of one kernel function and a very general external field function. Due to our very general setting the obtained minimax, equioscillation, and characterization results extend those of Bojanov, Fenton, Hardin, Kendall, Saff and Ambrus, Ball, Erd\'elyi. Moreover, we discover a surprising intertwining phenomenon of interval maxima, which provides new information even in the most classical extremal problem of Chebyshev.

In this paper we work out a Riemann-von Mangoldt type formula for the summatory function $\psi(x):=\sum_{g\in G, |g|\le x} \Lambda_{G}(g)$, where $G$ is an arithmetical semigroup (a Beurling generalized system of integers) and $\Lambda_{G}$ is the corresponding von Mangoldt function attaining $\log|p|$ for $g=p^k$ with a prime element $p\in G$ and zero otherwise. On the way towards this formula, we prove explicit estimates on the Beurling zeta function $\zeta_{G}$, belonging to $G$, to the number of zeroes of $\zeta_G$ in various regions, in particular within the critical strip where the analytic continuation exists, and to the magnitude of the logarithmic derivative of $\zeta_G$, under the sole additional assumption that Knopfmacher's Axiom A is satisfied. We also construct a technically useful broken line contour to which the technic of integral transformation can be well applied. The whole work serves as a first step towards a further study of the distribution of zeros of the Beurling zeta function, providing appropriate zero density and zero clustering estimates, to be presented in the continuation of this paper., Comment: to appear in Mathematica Pannonica. arXiv admin note: text overlap with arXiv:2012.09045

We prove three results on the density resp. local density and clustering of zeros of the Beurling zeta function $\zeta(s)$ close to the one-line $\sigma:=\Re s=1$. The analysis here brings about some news, sometimes even for the classical case of the Riemann zeta function. Theorem 4 provides a zero density estimate, which is a complement to known results for the Selberg class. Note that density results for the Selberg class rely on use of the functional equation of $\zeta$, which we do not assume in the Beurling context. In Theorem 5 we deduce a variant of a well-known theorem of Tur\'an, extending its range of validity even for rectangles of height only $h=2$. In Theorem 6 we will extend a zero clustering result of Ramachandra from the Riemann zeta case. A weaker result -- which, on the other hand, is a strong sharpening of the average result from the classic book \cite{Mont} of Montgomery -- was worked out by Diamond, Montgomery and Vorhauer. Here we show that the obscure technicalities of the Ramachandra paper (like a polynomial with coefficients like $10^8$) can be gotten rid of, providing a more transparent proof of the validity of this clustering phenomenon.

In this paper, we investigate discrete logarithmic energy problems in the unit circle. We study the equilibrium configuration of $n$ electrons and $n-1$ pairs of external protons of charge $+1/2$. It is shown that all the critical points of the discrete logarithmic energy are global minima, and they are the solutions of certain equations involving Blaschke products. As a nontrivial application, we refine a recent result of Simanek, namely, we prove that any configuration of $n$ electrons in the unit circle is in stable equilibrium (that is, they are not just critical points but are of minimal energy) with respect to an external field generated by $n-1$ pairs of protons., Comment: Revised and extended version

Let G be a locally compact Abelian group, and let X, Y be two open sets in G. We investigate the extremal constant C(X,Y) defined to be the supremum of integrals of functions f from the class F(X,Y), where F(X,Y) is the family of positive definite functions f on G such that f(0) = 1, the positive part of f is supported in X, and its negative part is supported in Y. In the case when X=Y, the problem is exactly the so-called Tur\'an problem for the set X. When Y= G, i.e., there is a restriction only on the set of positivity of f, we obtain the Delsarte problem. The Delsarte problem is the sharpest Fourier analytic tool to study packing density by translates of a given "master copy" set, which was studied first in connection with packing densities of Euclidean balls. We give an upper estimate of the constant C(X,Y) in the situation when the set X satisfies a certain packing type condition. This estimate is given in terms of the asymptotic uniform upper density of sets in locally compact Abelian groups., Comment: The November 22 version is a throughout, but all in all slight revision of the original submission. The January 3 version is a further revision with some more literature but is essentially not changed

In 1939 P. Tur\'an started to derive lower estimations on the norm of the derivatives of polynomials of (maximum) norm 1 on $I:=[-1,1]$ (interval) and $D:=\{z\in\mathbb{C}~:~|z|\le 1\}$ (disk), under the normalization condition that the zeroes of the polynomial in question all lie in $I$ or $D$, respectively. For the maximum norm he found that with $n:=\mathop{\mathrm{deg}} p$ tending to infinity, the precise growth order of the minimal possible derivative norm is $\sqrt{n}$ for $I$ and $n$ for $D$. J. Er\H{o}d continued the work of Tur\'an considering other domains. Finally, a decade ago the growth of the minimal possible $\infty$-norm of the derivative was proved to be of order $n$ for all compact convex domains. Although Tur\'an himself gave comments about the above oscillation question in $L^q$ norms, till recently results were known only for $D$ and $I$. Recently, we have found order $n$ lower estimations for several general classes of compact convex domains, and conjectured that even for arbitrary convex domains the growth order of this quantity should be $n$. Now we prove that in $L^q$ norm the oscillation order is at least $n/\log n$ for all compact convex domains., Comment: 2 figures

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.