Vast numbers of studies and developments in the nanotechnology area have been conducted and many nanomaterials have been utilized to detect cancers at early stages. Nanomaterials have unique physical, optical and electrical properties that have proven to be very useful in sensing. Quantum dots, gold nanoparticles, magnetic nanoparticles, carbon nanotubes, gold nanowires and many other materials have been developed over the years, alongside the discovery of a wide range of biomarkers to lower the detection limit of cancer biomarkers. Proteins, antibody fragments, DNA fragments, and RNA fragments are the base of cancer biomarkers and have been used as targets in cancer detection and monitoring. It is highly anticipated that in the near future, we might be able to detect cancer at a very early stage, providing a much higher chance of treatment.

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We study the denseness or norm of numerical radius attaining multilinear mappings and polynomials between Banach spaces, and examine the relations between norms and numerical radii of such mappings. IntroductionAfter the celebrated paper of E. Bishop and R. Phelps [6], a great deal of attention has been paid to the study of norm attaining operators (see [9,14,16,17,20], for example) and numerical radius attaining operators (see [1,3,5,7,8,10,11,13,18,21], for example). In particular, J. Bourgain [9] proved that a Banach space X has the Bishop-Phelps property if and only if it has the Radon-Nikodym property, and M. Acosta and R. Paya [3] showed that the set NRA (S£{X, X)) of numerical radius attaining operators from Zinto X'\s dense in the space S£{X, X) of bounded operators from X into X, if X has the Radon-Nikodym property. This work is motivated by their results. We shall study norm or numerical radius attaining multilinear mappings and polynomials, and examine their denseness. Our main interest will be the cases X = c 0 , l x , and a Banach space with the Radon-Nikodym property.

We study a Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B. For domain spaces, we study Banach spaces X such that (X, Y ) has the Bishop-Phelps-Bollobás property (BPBp) for every Banach space Y . We show that in this case, there exists a universal function η X (ε) such that for every Y , the pair (X, Y ) has the BPBp with this function. This allows us to prove some necessary isometric conditions for X to have the property. We also prove that if X has this property in every equivalent norm, then X is one-dimensional. For range spaces, we study Banach spaces Y such that (X, Y ) has the Bishop-Phelps-Bollobás property for every Banach space X. In this case, we show that there is a universal function η Y (ε) such that for every X, the pair (X, Y ) has the BPBp with this function. This implies that this property of Y is strictly stronger than Lindenstrauss property B. The main tool to get these results is the study of the Bishop-Phelps-Bollobás property for c 0 -, 1 -and ∞ -sums of Banach spaces.goal here is to introduce and study analogues of properties A and B in the context of vector-valued versions of the Bishop-Phelps-Bollobás theorem.Let us now restart the Introduction, this time giving the necessary background material to help make the paper entirely accessible. The Bishop-Phelps-Bollobás property was introduced in 2008 [2] as an extension of the Bishop-Phelps-Bollobás theorem to the vector-valued case. It can be regarded as a "quantitative version" of the study of norm-attaining operators initiated by J. Lindenstrauss in 1963. We begin with some notation to present its definition. Let X and Y be Banach spaces over the field K = R or C. We will use the common notation S X , B X , X * for the unit sphere, the closed unit ball and the dual space of X respectively, L(X, Y ) for the Banach space of all bounded linear operators from X into Y, and NA(X, Y ) for the subset of all norm-attaining operators. (We say that an operator T ∈ L(X, Y ) attains its norm if T = T x for some x ∈ S X .) We will abbreviate L(X, X), resp. NA(X, X), by L(X), resp. NA(X). Definition 1.1 ([2, Definition 1.1]). A pair of Banach spaces (X, Y ) is said to have the Bishop-Phelps-Bollobás property (BPBp for short) if for every ε ∈ (0, 1) there is η(ε) > 0 such that for every T 0 ∈ L(X, Y ) with T 0 = 1 and every x 0 ∈ S X satisfying T 0 (x 0 ) > 1 − η(ε), there exist S ∈ L(X, Y ) and x ∈ S X such that

(*) For any polynomial P:Y"->K = R o r C , the composition Pog : Bx -* K is uniformly continuous.

DNA origami nanostructures provide a platform where dye molecules can be arranged with nanoscale accuracy allowing to assemble multiple fluorophores without dye-dye aggregation. Aiming to develop a bright and sensitive ratiometric sensor system, we systematically studied the optical properties of nanoarrays of dyes built on DNA origami platforms using a DNA template that provides a high versatility of label choice at minimum cost. The dyes are arranged at distances, at which they efficiently interact by Förster resonance energy transfer (FRET). To optimize array brightness, the FRET efficiencies between the donor fluorescein (FAM) and the acceptor cyanine 3 were determined for different sizes of the array and for different arrangements of the dye molecules within the array. By utilizing nanoarrays providing optimum FRET efficiency and brightness, we subsequently designed a ratiometric pH nanosensor using coumarin 343 as a pH-inert FRET donor and FAM as a pH-responsive acceptor. Our results indicate that the sensitivity of a ratiometric sensor can be improved simply by arranging the dyes into a well-defined array. The dyes used here can be easily replaced by other analyte-responsive dyes, demonstrating the huge potential of DNA nanotechnology for light harvesting, signal enhancement, and sensing schemes in life sciences.

In this paper, we introduce the polynomial numerical index of order k of a Banach space, generalizing to k-homogeneous polynomials the 'classical' numerical index defined by Lumer in the 1970s for linear operators. We also prove some results. Let k be a positive integer. We then have the following:is sharp.(iii) The inequalities(iv) The relation between the polynomial numerical index of c 0 , l 1 , l∞ sums of Banach spaces and the infimum of the polynomial numerical indices of them.(v) The relation between the polynomial numerical index of the space C(K, E) and the polynomial numerical index of E.(vi) The inequality n (k) (E * * ) n (k) (E) for every Banach space E.Finally, some results about the numerical radius of multilinear maps and homogeneous polynomials on C(K) and the disc algebra are given.

Abstract. We study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space X, i.e. when the equality Id + P = 1 + P is satisfied for all weakly compact polynomials P : X → X. We show that this is the case when X = C(K), the real or complex space of continuous functions on a compact space K without isolated points. We also study the alternative Daugavet equationId + ωP = 1 + P for polynomials P : X → X. We show that this equation holds for every polynomial on the complex space X = C(K) (K arbitrary) with values in X. This result is not true in the real case. Finally, we study the Daugavet and the alternative Daugavet equations for k-homogeneous polynomials.In 1963, I. K. Daugavet [13] showed that every compact linear operator T on C[0, 1] satisfies Id + T = 1 + T , a norm equality which has become known as the Daugavet equation. Over the years, the validity of the above equality has been established for many classes of operators on many Banach spaces. For instance, weakly compact linear operators on C(K), K perfect, and L 1 (µ), µ atomless, satisfy the Daugavet equation (see [25] for an elementary approach). We refer the reader to the books [1, 2] and papers [20,26]