Bruckner--Garg-type results with respect to Haar null sets in C[0,1]C[0,1]

A set $\mathcal{A}\subset C[0,1]$ is \emph{shy} or \emph{Haar null } (in the sense of Christensen) if there exists a Borel set $\mathcal{B}\subset C[0,1]$ and a Borel probability measure $\mu$ on $C[0,1]$ such that $\mathcal{A}\subset \mathcal{B}$ and $\mu\left(\mathcal{B}+f\right) = 0$ for all $f \in C[0,1]$. The complement of a shy set is called a \emph{prevalent} set. We say that a set is \emph{Haar ambivalent} if it is neither shy nor prevalent. The main goal of the paper is to answer the following question: What can we say about the topological properties of the level sets of the prevalent/non-shy many $f\in C[0,1]$? The classical Bruckner--Garg Theorem characterizes the level sets of the generic (in the sense of Baire category) $f\in C[0,1]$ from the topological point of view. We prove that the functions $f\in C[0,1]$ for which the same characterization holds form a Haar ambivalent set. In an earlier paper we proved that the functions $f\in C[0,1]$ for which positively many level sets with respect to the Lebesgue measure $\lambda$ are singletons form a non-shy set in $C[0,1]$. The above result yields that this set is actually Haar ambivalent. Now we prove that the functions $f\in C[0,1]$ for which positively many level sets with respect to the occupation measure $\lambda\circ f^{-1}$ are not perfect form a Haar ambivalent set in $C[0,1]$. We show that for the prevalent $f\in C[0,1]$ for the generic $y\in f([0,1])$ the level set $f^{-1}(y)$ is perfect. Finally, we answer a question of Darji and White by showing that the set of functions $f \in C[0,1]$ for which there exists a perfect $P_f\subset [0,1]$ such that $f'(x) = \infty$ for all $x \in P_f$ is Haar ambivalent.Comment: 12 page

Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps

The notions of shyness and prevalence generalize the property of being zero and full Haar measure to arbitrary (not necessarily locally compact) Polish groups. The main goal of the paper is to answer the following question: What can we say about the Hausdorff and packing dimension of the fibers of prevalent continuous maps? Let $K$ be an uncountable compact metric space. We prove that the prevalent $f\in C(K,\mathbb{R}^d)$ has many fibers with almost maximal Hausdorff dimension. This generalizes a theorem of Dougherty and yields that the prevalent $f\in C(K,\mathbb{R}^d)$ has graph of maximal Hausdorff dimension, generalizing a result of Bayart and Heurteaux. We obtain similar results for the packing dimension. We show that for the prevalent $f\in C([0,1]^m,\mathbb{R}^d)$ the set of $y\in f([0,1]^m)$ for which $\dim_H f^{-1}(y)=m$ contains a dense open set having full measure with respect to the occupation measure $\lambda^m \circ f^{-1}$, where $\dim_H$ and $\lambda^m$ denote the Hausdorff dimension and the $m$-dimensional Lebesgue measure, respectively. We also prove an analogous result when $[0,1]^m$ is replaced by any self-similar set satisfying the open set condition. We cannot replace the occupation measure with Lebesgue measure in the above statement: We show that the functions $f\in C[0,1]$ for which positively many level sets are singletons form a non-shy set in $C[0,1]$. In order to do so, we generalize a theorem of Antunovi\'c, Burdzy, Peres and Ruscher. As a complementary result we prove that the functions $f\in C[0,1]$ for which $\dim_H f^{-1}(y)=1$ for all $y\in (\min f,\max f)$ form a non-shy set in $C[0,1]$. We also prove sharper results in which large Hausdorff dimension is replaced by positive measure with respect to generalized Hausdorff measures, which answers a problem of Fraser and Hyde.Comment: 42 page

Rolewicz-type chaotic operators

In this article we introduce a new class of Rolewicz-type operators in l_p, $1 \le p < \infty$. We exhibit a collection F of cardinality continuum of operators of this type which are chaotic and remain so under almost all finite linear combinations, provided that the linear combination has sufficiently large norm. As a corollary to our main result we also obtain that there exists a countable collection of such operators whose all finite linear combinations are chaotic provided that they have sufficiently large norm.Comment: 15 page

Nanomedicine-based Immunochemotherapy for the MR Imaging and Treatment of Triple Negative Breast Cancer

Cancer is known to be a detrimental disease and it accounts for many deaths every year. In women Triple negative breast cancer (TNBC) known to be a very aggressive type of tumor. Current treatment approaches are abortive and have many side effects. With the recent advances in nanotechnology and immunotherapy for cancer treatment, we have designed new nanomedicine that combines both the aspects of cancer treatment. In this work, we have developed an anti-PD-L1-conjugated, Doxo-SS-Gd MR imaging agent encapsulating iron oxide nanoparticles (IONPs) to form IONP-Doxo-SS-Gd-PD-L1 nanomedicine for the targeted MR imaging and treatment of TNBC. The anti-PD-L1 acts as a checkpoint inhibitor in the PD-1 & PD-L1 interaction, helping in generating an immune response against the cancer cells. Furthermore, the Gd-DTPA functionalized doxorubicin prodrug (Doxo-SS-Gd), is known as a chemotherapeutic drug and a strong T1 MR agent, which would provide bright T1 MR contrast for the imaging of tumor. To assess the therapeutic potential of the designed nanomedicine in treating cancer, various cell bases experiments were carried out. The bright and dark MR contrast of the IONP-Doxo-SS-Gd- PD-L1 nanomedicine was also evaluated using clinical MRI instrument (B = 9.3T). Therefore, the developed immunochemotherapeutic-nanomedicine provides combination approaches for the synergistic (immunotherapy and chemotherapy) treatment of TNBC and further has MR imaging functionality for diagnosis and treatment monitoring

A Novel Design of Multi-Chambered Biomass Battery

In this paper, a novel design of biomass battery has been introduced for providing electricity to meet the lighting requirements of rural household using biomass. A biomass battery is designed, developed and tested using cow dung as the raw material. This is done via anaerobic digestion of the cow dung, and power generation driven by the ions produced henceforth. The voltage and power output is estimated for the proposed system. It is for the first time that such a high voltage is obtained from cow dung fed biomass battery. The output characteristics of this novel battery design have also been compared with the previously designed battery

Li-Yorke Chaos for Composition Operators on LpL^p-Spaces

Li-Yorke chaos is a popular and well-studied notion of chaos. Several simple and useful characterizations of this notion of chaos in the setting of linear dynamics were obtained recently. In this note we show that even simpler and more useful characterizations of Li-Yorke chaos can be given in the special setting of composition operators on $L^p$ spaces. As a consequence we obtain a simple characterization of weighted shifts which are Li-Yorke chaotic. We give numerous examples to show that our results are sharp