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Dimensions of level sets of generic continuous functions and generic H\"older functions defined on a fractal $F$ encode information about the geometry, ``the thickness" of $F$. While in the continuous case this quantity is related to a reasonably tame dimension notion which is called the topological Hausdorff dimension of $F$, the H\"older case seems to be highly nontrivial. A number of earlier papers attempted to deal with this problem, carrying out investigation in the case of Hausdorff dimension and box dimension. In this paper we continue our study of the Hausdorff dimension of almost every level set of generic $1$-H\"older-$ {\alpha}$ functions, denoted by $D_{*}( {\alpha}, F)$. We substantially improve previous lower and upper bounds on $D_{*}( {\alpha}, \Delta)$, where $\Delta$ is the Sierpi\'nski triangle, achieving asymptotically equal bounds as $\alpha\to 0+$. Using a similar argument, we also give an even stronger lower bound on the generic lower box dimension of level sets. Finally, we construct a connected fractal $F$ on which there is a phase transition of $D_{*}( {\alpha}, F)$, thus providing the first example exhibiting this behavior., Comment: Revised version after referee's report. Readability of the paper was improved especially in Section 4

Hausdorff dimensions of level sets of generic continuous functions defined on fractals can give information about the "thickness/narrow cross-sections'' of a "network" corresponding to a fractal set, $F$. This lead to the definition of the topological Hausdorff dimension of fractals. In this paper we continue our study of the level sets of generic $1$-H\"older-$\alpha$ functions. While in a previous paper we gave the initial definitions and established some properties of these generic level sets, in this paper we provide numerical estimates in the case of the Sierpi\'nski triangle. These calculations give better insight and illustrate why can one think of these generic $1$-H\"older-$\alpha$ level sets as something measuring "thickness/narrow cross-sections/conductivity'' of a fractal "network". We also give an example for the phenomenon which we call phase transition for $D_{*}(\alpha, F)$. This roughly means that for a certain lower range { of $\alpha$s } only the geometry of $F$ determines $D_{*}(\alpha, F)$ while for larger values the H\"older exponent, $\alpha$ also matters., Comment: Updated version after referee's suggestions

Hausdorff dimensions of level sets of generic continuous functions defined on fractals were considered in two papers by R. Balka, Z. Buczolich and M. Elekes. In those papers the topological Hausdorff dimension of fractals was defined. In this paper we start to study level sets of generic $1$-H\"older-$\alpha$ functions defined on fractals. This is related to some sort of "thickness", "conductivity" properties of fractals. The main concept of our paper is $D_{*}(\alpha, F)$ which is the essential supremum of the Hausdorff dimensions of the level sets of a generic $1$-H\"older-$\alpha$ function defined on the fractal $F$. We prove some basic properties of $D_{*}(\alpha, F)$, we calculate its value for an example of a "thick fractal sponge", we show that for connected self similar sets $D_{*}(\alpha, F)$ it equals the Hausdorff dimension of almost every level in the range of a generic $1$-H\"older-$\alpha$ function., Comment: Revised version after referee's report. A new example was added, introduction was modified, a Main results section was also added

In this paper we give an example of a closed, strongly one-sided dense set which is not of uniform density type. We also show that there is a set of uniform density type which is not of strong uniform density type., Comment: Revised version after referee's report. Period Math Hung (2022)

We denote the local ``little" Lipschitz constant of a function $f: {{\mathbb R}}\to { {\mathbb R}}$ by $ {\mathrm{lip}}f$. In this paper we settle the following question: For which sets $E {\subset} { {\mathbb R}}$ is it possible to find a continuous function $f$ such that $ {\mathrm{lip}}f=\mathbf{1} _E$? In an earlier paper we introduced the concept of strongly one-sided dense sets. Our main result characterizes $ {\mathrm{lip}}1$ sets as countable unions of closed sets which are strongly one-sided dense. We also show that a stronger statement is not true i.e. there are strongly one-sided dense $F _\sigma$ sets which are not $ {\mathrm{lip}}1$.

We denote the local "little" and "big" Lipschitz functions of a function $f: {{\mathbb R}}\to {{\mathbb R}}$ by $ {\mathrm {lip}}f$ and $ {\mathrm {Lip}}f$. In this paper we continue our research concerning the following question. Given a set $E {\subset} {{\mathbb R}}$ is it possible to find a continuous function $f$ such that $ {\mathrm {lip}}f=\mathbf{1}_E$ or $ {\mathrm {Lip}}f=\mathbf{1}_E$? In giving some partial answers to this question uniform density type (UDT) and strong uniform density type (SUDT) sets play an important role. In this paper we show that modulo sets of zero Lebesgue measure any measurable set coincides with a ${\mathrm {Lip}} 1$ set. On the other hand, we prove that there exists a measurable SUDT set $E$ such that for any $G_\delta$ set $\widetilde{E}$ satisfying $|E\Delta\widetilde{E}|=0$ the set $\widetilde{E}$ does not have UDT. Combining these two results we obtain that there exists ${\mathrm {Lip}} 1$ sets not having UDT, that is, the converse of one of our earlier results does not hold., Comment: This is the version accepted to appear in Mathematica Slovaca

Given a continuous function $f: {{\mathbb R}}\to {{\mathbb R}}$ we denote the so-called "big Lip" and "little lip" functions by $ {{\mathrm {Lip}}} f$ and $ {{\mathrm {lip}}} f$ respectively}. In this paper we are interested in the following question. Given a set $E {\subset} {{\mathbb R}}$ is it possible to find a continuous function $f$ such that $ {{\mathrm {lip}}} f=\mathbf{1}_E$ or $ {{\mathrm {Lip}}} f=\mathbf{1}_E$? For monotone continuous functions we provide the rather straightforward answer. For arbitrary continuous functions the answer is much more difficult to find. We introduce the concept of uniform density type (UDT) and show that if $E$ is $G_\delta$ and UDT then there exists a continuous function $f$ satisfying $ {{\mathrm {Lip}}} f =\mathbf{1}_E$, that is, $E$ is a $ {{\mathrm {Lip}}} 1$ set. In the other direction we show that every ${{\mathrm {Lip}}} 1$ set is $G_\delta$ and weakly dense. We also show that the converse of this statement is not true, namely that there exist weakly dense $G_{{\delta}}$ sets which are not $ {{\mathrm {Lip}}} 1$. We say that a set $E\subset \mathbb{R}$ is ${{\mathrm {lip}}} 1$ if there is a continuous function $f$ such that ${{\mathrm {lip}}} f=\mathbf{1}_E$. We introduce the concept of strongly one-sided density and show that every ${{\mathrm {lip}}} 1$ set is a strongly one-sided dense $F_\sigma$ set., Comment: This is the final preprint version accepted to appear in European Journal of Mathematics

Suppose $\Lambda$ is a discrete infinite set of nonnegative real numbers. We say that $ {\Lambda}$ is type $1$ if the series $s(x)=\sum_{\lambda\in\Lambda}f(x+\lambda)$ satisfies a zero-one law. This means that for any non-negative measurable $f: {{\mathbb R}}\to [0,+ {\infty})$ either the convergence set $C(f, {\Lambda})=\{x: s(x)<+ {\infty} \}= {{\mathbb R}}$ modulo sets of Lebesgue zero, or its complement the divergence set $D(f, {\Lambda})=\{x: s(x)=+ {\infty} \}= {{\mathbb R}}$ modulo sets of measure zero. If $ {\Lambda}$ is not type $1$ we say that $ {\Lambda}$ is type 2. The exact characterization of type $1$ and type $2$ sets is not known. In this paper we continue our study of the properties of type $1$ and $2$ sets. We discuss sub and supersets of type $1$ and $2$ sets and we give a complete and simple characterization of a subclass of dyadic type $1$ sets. We discuss the existence of type $1$ sets containing infinitely many algebraically independent elements. Finally, we consider unions and Minkowski sums of type $1$ and $2$ sets.

Suppose $\Lambda$ is a discrete infinite set of nonnegative real numbers. We say that $ {\Lambda}$ is type $2$ if the series $s(x)=\sum_{\lambda\in\Lambda}f(x+\lambda)$ does not satisfy a zero-one law. This means that we can find a non-negative measurable "witness function" $f: {\mathbb R}\to [0,+ {\infty})$ such that both the convergence set $C(f, {\Lambda})=\{x: s(x)<+ {\infty} \}$ and its complement the divergence set $D(f, {\Lambda})=\{x: s(x)=+ {\infty} \}$ are of positive Lebesgue measure. If $ {\Lambda}$ is not type $2$ we say that $ {\Lambda}$ is type $1$. The main result of our paper answers a question raised by Z. Buczolich, J-P. Kahane, and D. Mauldin. By a random construction we show that one can always choose a witness function which is the characteristic function of a measurable set. We also consider the effect on the type of a set $ {\Lambda}$ if we randomly delete its elements. Motivated by results concerning weighted sums $\sum c_n f(nx)$ and the Khinchin conjecture, we also discuss some results about weighted sums $\sum_{n=1}^{\infty}c_n f(x+\lambda_n)$.

Suppose $\Lambda$ is a discrete infinite set of nonnegative real numbers. We say that $ {\Lambda}$ is of type 1 if the series $s(x)=\sum_{\lambda\in\Lambda}f(x+\lambda)$ satisfies a zero-one law. This means that for any non-negative measurable $f: { {\mathbb R}}\to [0,+ {\infty})$ either the convergence set $C(f, {\Lambda})=\{x: s(x)<+ {\infty} \}= { {\mathbb R}}$ modulo sets of Lebesgue zero, or its complement the divergence set $D(f, {\Lambda})=\{x: s(x)=+ {\infty} \}= { {\mathbb R}}$ modulo sets of measure zero. If $ {\Lambda}$ is not of type 1 we say that $ {\Lambda}$ is of type 2. In this paper we show that there is a universal $ {\Lambda}$ with gaps monotone decreasingly converging to zero such that for any open subset $G \subset { {\mathbb R}}$ one can find a characteristic function $f_{G}$ such that $G \subset D(f_G, {\Lambda})$ and $C(f_G, {\Lambda})= { {\mathbb R}} {\setminus} G$ modulo sets of measure zero. We also consider the question whether $C(f, {\Lambda})$ can contain non-degenerate intervals for continuous functions when $D(f, {\Lambda})$ is of positive measure. The above results answer some questions raised in a paper of Z. Buczolich, J-P. Kahane, and D. Mauldin.

Graphene covered metal nanoparticles constitute a novel type of hybrid materials, which provide a unique platform to study plasmonic effects, surface-enhanced Raman scattering (SERS), and metal-graphene interactions at the nanoscale. Such a hybrid material is fabricated by transferring graphene grown by chemical vapor deposition onto closely spaced gold nanoparticles produced on a silica wafer. The morphology and physical properties of nanoparticle-supported graphene is investigated by atomic force microscopy, optical reflectance spectroscopy, scanning tunneling microscopy and spectroscopy (STM/STS), and confocal Raman spectroscopy. This study shows that the graphene Raman peaks are enhanced by a factor which depends on the excitation wavelength, in accordance with the surface plasmon resonance of the gold nanoparticles, and also on the graphene-nanoparticle distance which is tuned by annealing at moderate temperatures. The observed SERS activity is correlated to the nanoscale corrugation of graphene. STM and STS measurements show that the local density of electronic states in graphene is modulated by the underlying gold nanoparticles.

The effects produced by the condensation of water vapours from the ambient in the various intricate nanoarchitectures occurring in the wing scales of several Lepidoptera species were investigated by controlled cooling (from room temperature to -5 - -10 {\deg}C) combined with in situ measurement of changes in the reflectance spectra. It was determined that, due to this procedure, all photonic nanoarchitectures giving a reflectance maximum in the visible range and having an open nanostructure exhibited alteration of the position of the reflectance maximum associated with the photonic nanoarchitectures. The photonic nanoarchitectures with a closed structure exhibited little to no alteration in colour. Similarly, control specimens coloured by pigments did not exhibit a colour change under the same conditions. Hence, this effect can be used to identify species with open photonic nanoarchitectures in their scales. For certain species, an almost complete disappearance of the reflectance maximum was found. All specimens recovered their original colours following warming and drying. Cooling experiments using thin copper wires demonstrated that colour alterations could be limited to a millimetre, or below. Dried museum specimens do not exhibit colour changes when cooled in the absence of a heat sink due to the low heat capacity of the wings., Comment: 18 pages, 9 figures, including supplement

Mechanically exfoliated graphene layers deposited on SiO2 substrate were irradiated with Ar+ ions in order to experimentally study the effect of atomic scale defects and disorder on the low-energy electronic structure of graphene. The irradiated samples were investigated by scanning tunneling microscopy and spectroscopy measurements, which reveal that defect sites, besides acting as scattering centers for electrons through local modification of the on-site potential, also induce disorder in the hopping amplitudes. The most important consequence of the induced disorder is the substantial reduction in the Fermi velocity, revealed by bias-dependent imaging of electron-density oscillations observed near defect sites.

The optical properties of the inflorescence of the high-altitude ''Leontopodium nivale'' subsp. ''alpinum'' (edelweiss) is investigated, in relation with its submicrometer structure, as determined by scanning electron microscopy. The filaments forming the hair layer have been found to exhibit an internal structure which may be one of the few examples of a photonic structure found in a plant. Measurements of light transmission through a self-supported layer of hair pads taken from the bracts supports the idea that the wooly layer covering the plant absorbs near-ultraviolet radiation before it reaches the cellular tissue. Calculations based on a photonic-crystal model provides insight on the way radiation can be absorbed by the filamentary threads., Comment: 9 pages, 13 figures. Published paper

The dipolar-random field Ising model (DRFIM) recently introduced displays a behaviour that can be connected to the magnetization of bidimensional magnetic media. Epitaxial magnetic garnet films seem to be the ideal test material for such a model. In this work the results of the measurements performed on garnet samples are presented, as well as the comparisons with simulation results obtained by the DRFIM. The results prove that a variety of hysteresis loops are well described by the DRFIM. This capability does not derive from the fine tuning of a great number of parameters, but by the interplay of exchange and dipolar interactions., Comment: HTML paper, GIF images. To appear on Phys.Rev.B (Brief Reports). Application of the model published on Phys.Rev.B 59 (1999) 985, cond-mat/9907074