Abstract:
MUC1, a tumor-associated antigen overexpressed in many carcinomas, represents a candidate of choice for cancer immunotherapy. Flagella-based MUC1 vaccines were tested in therapeutic setting in two aggressive breast cancer models, comprising the implantation of the 4T1-MUC1 cell line in either Balb/c, or Human MUC1 transgenic mice in which spontaneous metastases occurs. Recombinant flagella carrying only 7 amino acid of MUC1 elicited therapeutic activity, affecting both the growth of established growing tumors and the number of metastases. Higher therapeutic activity was achieved with an additional recombinant flagella designed with the SYFPEITHI algorithm. The vaccines triggered a Th1 response against MUC1 with no evident autoimmune response towards healthy MUC1-expressing tissues. Recombinant flagella carrying a 25-residue fragment of MUC1, induced the most effective response, as evidenced by a significant reduction of both the size and growth rate of the tumor as well as by the lower number of metastases, and expanding life span of vaccinated mice.

Abstract:
The phenomenon in the essence of classical uncertainty principles is well known since the thirties of the last century. We introduce a new phenomenon which is in the essence of a new notion that we introduce: "Generalized Uncertainty Principles". We show the relation between classical uncertainty principles and generalized uncertainty principles. We generalized "Landau-Pollak-Slepian" uncertainty principle. Our generalization relates the following two quantities and two scaling parameters: 1) The weighted time spreading $\int_{-\infty}^\infty |f(x)|^2w_1(x)dx$, ($w_1(x)$ is a non-negative function). 2) The weighted frequency spreading $\int_{-\infty}^\infty |\hat{f}(\omega)|^2w_2(\omega)d\omega$. 3) The time weight scale $a$, ${w_1}_a(x)=w_1(xa^{-1})$ and 4) The frequency weight scale $b$, ${w_2}_b(\omega)=w_2(\omega b^{-1})$. "Generalized Uncertainty Principle" is an inequality that summarizes the constraints on the relations between the two spreading quantities and two scaling parameters. For any two reasonable weights $w_1(x)$ and $w_2(\omega)$, we introduced a three dimensional set in $R^3$ that is in the essence of many uncertainty principles. The set is called "possibility body". We showed that classical uncertainty principles (such as the Heiseneberg-Pauli-Weyl uncertainty principle) stem from lower bounds for different functions defined on the possibility body. We investigated qualitative properties of general uncertainty principles and possibility bodies. Using this approach we derived new (quantitative) uncertainty principles for Landau-Pollak-Slepian weights. We found the general uncertainty principles related to homogeneous weights, $w_1(x)=w_2(x)=x^k$, $k\in N$, up to a constant.

Abstract:
It is now evident that solid tumors beyond a given volume are dependent on the supply of oxygen and nutrients from the vascular system, which has to grow concomitantly with the tumor, similar to embryonic development. This process, of newly developed blood capillaries and blood vessels from pre-existing ones, has been termed angiogenesis and enables the tumor not only to increase its size but also its aggressiveness and its ability to metastasize [1-4]. The process of angiogenesis is implicated not only in the pathology of tumors but also in many other diseases including psoriasis [5,6], age-related macular degeneration[7,8] and rheumatoid arthritis [9].Some of the most deadly malignancies that depend on the angiogenic process for their growth are primary brain tumors[10], among which glioblastoma multiforme (GBM) represents 40% of all cases. GBM has been targeted with many inhibitors of angiogenesis including tissue inhibitors of matrix metalloproteinases[11-13], chemokines [14-16], tyrosine kinase inhibitors [17-20], interleukins [21,22], and naturally occurring proteolytic fragments of large precursor molecules such as endostatin, vasostatin, canstatin, angiostatin and others [23-29]. These molecules exert their inhibitory functions on endothelial cells by multiple mechanisms including proliferation, migration, protease activity, as well as the induction of apoptosis [30]. Although such angiogenesis inhibitors hold great promise, the ones that reached clinical trials for brain tumor patients have failed to achieve significant therapeutic outcome. One possible explanation for this outcome that is supported by many researchers is the lack of combinatory treatment with standard chemo and radiotherapy [31]. Another obstacle which may hamper the therapeutic outcome of anti-angiogenic therapy is the blood brain barrier (BBB, although destabilized in high grade GBM patients) which therapeutics need to bypass to exert a significant brain tumor inhibitory effect. The brai

Abstract:
For (E) being one of the three sets: the whole real axis, a finite symmetric interval and the positive semiaxis, we discuss the simplest differential operators of the second order which commute with the truncated Fourier operator (\mathscr{F}_E).

Abstract:
Let (\mathscr{F}) be the one dimensional Fourier-Plancherel operator and (E) be a subset of the real axis. The truncated Fourier operator is the operator (\mathscr{F}_E) of the form (\mathscr{F}_E=P_E\mathscr{F}P_E), where ((P_Ex)(t)=\chi_E(t)x(t)), and (\chi_E(t)) is the indicator function of the set (E). In the presented first part of the work, the basic properties of the operator (\mathscr{F}_E) according to the set (E) are discussed. Among these properties there are the following one. The operator (\mathscr{F}_E): 1. has a not-trivial null-space; 2. is strictly contractive; 3. is a normal operator; 4. is a Hilbert-Schmidt operator; 5. is a trace class operator.

Abstract:
We consider the formal prolate spheroid differential operator on a finite symmetric interval and describe all its self-adjoint boundary conditions. Only one of these boundary conditions corresponds to a self-operator differential operator which commutes with the Fourier operator truncated on the considered finite symmetric interval.

Abstract:
The spectral theory of the Fourier operator (non-truncated) is expounded. The known construction of basis of eigenvectors consisting of the Hermite functions is presented. The detail description of the eigenspaces in the spirit of a work by Hardy and Titchmarsh is done.

Abstract:
The Fourier operator truncated on a finite symmetric interval is considered. The limiting behavior of its spectrum is discussed as the length of the interval tends to infinity.