Expected Number of Critical Points of Random Holomorphic Sections Over Complex Projective Space

Tumor volume doubling time (TVDT) has been shown to be a potential surrogate marker of biological tumor activity. However, its availability in clinics is strongly limited due to ethical and practical reasons, as its assessment requires at least two subsequent tumor volume measurements in untreated patients. Here, a translational modeling framework to predict TVDT distributions in untreated cancer patient populations from tumor growth data in patient-derived xenograft (PDX) mice is proposed. Eleven solid cancer types were considered. For each of them, a set of tumor growth studies in PDX mice was selected and analyzed through a mathematical model to characterize the distribution of the exponential tumor growth rate in mice. Then, assuming an exponential growth of the tumor mass in humans, the growth rates were scaled from PDX mice to humans through an allometric scaling approach and used to predict TVDTs in untreated patients. A very good agreement was found between model predicted and clinically observed TVDTs, with 91% of the predicted TVDT medians fell within 1.5-fold of observations. Further, exploiting the intrinsic relationship between tumor growth dynamics and progression free survival (PFS), the exponential growth rates in humans were used to generate the expected PFS curves in absence of anticancer treatment. Predicted curves were extremely close to published PFS data from studies involving patient cohorts treated with supportive care or low effective therapies. The proposed approach shows promise as a potential tool to increase knowledge about TVDT in humans without the need of directly measuring tumor dimensions in untreated patients, and to predict PFS curves in untreated patients, that could fill the absence of placebo-controlled arms against which to compare treaded arms during clinical trials. However, further validation and refinement are needed to fully assess its effectiveness in this regard.

Minimal 2-spheres in a complex projective space

Pressure and exponential rate over periodic orbits

In the case where $\Sigma$ is a Riemann sphere, the above result was obtained by Siu [S-l, $Z$] (see also [B-R-S]). In the case where $M$ is a complex projective space and deg $f|\geqq m(p-1)/(m+1)$ where $p$ denotes the genus of $\Sigma$ , the above result was obtained by Eells and Wood [E-W]. They used algebraic geometric arguments (theorems of Riemann-Roch and Grothendieck). Recently, by using a twistor space over the domain manifold, Burns and de Bartolomeis have shown the above result in the case where $M$ is a complex projective space (cf. Remark 6 of [B-R-S]). We show the above theorem by a simple argument for the second variations used in [L-S].

Validation and Analysis of Modeled Predictions of Growth of Bacillus cereus Spores in Boiled Rice

Growth of Escherichia coli O157:H7 in Raw Ground Beef Stored at 10° C and the Influence of Competitive Bacterial Flora, Strain Variation, and Fat Level

From the Riemannian geometric point of view, one of the most fundamental problems in the study of Lagrangian submanifolds is the classiflcation of Lagrangian submanifolds with parallel second fundamental form. In 1980’s, H. Naitoh completely classifled the Lagrangian submanifolds with parallel second fundamental form and without Euclidean factor in complex projective space, by using the theory of Lie groups and symmetric spaces. He showed that such a submanifold is always locally symmetric and is one of the symmetric spaces: SO(k + 1)=SO(k)(k ‚ 2), SU(k)=SO(k)(k ‚ 3), SU(k)(k ‚ 3), SU(2k)=Sp(k)(k ‚ 3), E6=F4. In this paper, we completely classify the Lagrangian submanifolds in complex projective space with parallel second fundamental form by an elementary geometrical method. We prove that such a Lagrangian submanifold is either totally geodesic or the Calabi product of a point with a lower dimensional Lagrangian submanifold with parallel second fundamental form, or the Calabi product of two lower dimensional Lagrangian submanifolds with parallel second fundamental form, or one of the standard symmetric spaces: SU(k)=SO(k)(k ‚ 3), SU(k)(k ‚ 3), SU(2k)=Sp(k)(k ‚ 3), E6=F4. As the arguments are of a local nature, at the same time, due to the correspondence between C-parallel Lagrangian submanifolds in Sasakian space forms and parallel Lagrangian submanifolds in complex space forms, we can also give a complete classiflcation of all Cparallel submanifolds of S 2n+1 equipped with its standard Sasakian structure. 2000 Mathematics Subject Classiflcation: Primary 53B25; Secondary 53C42

A real hypersurface M in a complex projective space inherits an almost contact metric structure from the Kählerian structure of the ambient space. This almost contact metric structure allows us to define, for any nonzero real number k, the so-called k-th generalized Tanaka–Webster connection. With this connection and the Levi-Civita one we can associate two tensors of type (1,2) to the structure Jacobi operator R_{xi } of M. We classify real hypersurfaces in complex projective space for which such tensors satisfy a cyclic property.

In Cogdell et al., LMS Lecture Notes Series 459, 393–427 (2020), the authors proved an analogue of Kronecker’s limit formula associated to any divisor D \mathcal D which is smooth in codimension one on any smooth Kähler manifold X X . In the present article, we apply the aforementioned Kronecker limit formula in the case when X X is complex projective space C P n \mathbb {C}\mathbb {P}^n for n ≥ 2 n \geq 2 and D \mathcal D is a hyperplane, meaning the divisor of a linear form P D ( z ) P_D({z}) for z = ( Z j ) ∈ C P n {z} = (\mathcal {Z}_{j}) \in \mathbb {C}\mathbb {P}^n . Our main result is an explicit evaluation of the Mahler measure of P D P_{D} as a convergent series whose each term is given in terms of rational numbers, multinomial coefficients, and the L 2 L^{2} -norm of the vector of coefficients of P D P_{D} .

Lie derivatives and structure Jacobi operator on real hypersurfaces in complex projective spaces

On a real hypersurface M in a complex projective space, we can consider the Levi-Civita connection and for any nonnull constant k the kth g-Tanaka–Webster connection. Therefore, we can also consider their associated Lie derivatives. We classify real hypersurfaces such that both the Lie derivatives associated with the Levi-Civita connection and the kth g-Tanaka–Webster connection either in the direction of the structure vector field ξ or in any direction of the maximal holomorphic distribution coincide when we apply them to the shape operator of M.

This paper gives a classification of real hypersurfaces in a complex projective space under assumptions that the structure vector $\xi$ is principal, the focal map has constant rank, and ${\nabla _\xi }S = 0$, where S is the Ricci tensor of the real hypersurface.

The study of homogeneous and evenly homogeneous complex supermanifolds of dimension whose reduction is the complex projective space is continued. All non-split supermanifolds of this type with retract admitting the full projective group as group of automorphisms are classified. The cohomology with values in the tangent sheaf is calculated for homogeneous supermanifolds satisfying the above-mentioned conditions.

Developmental patterns produced during normal expansion of the leaf of Vitis vinifera cv. Ruby Red are quantitatively characterized from the distribution of relative growth rates and growth velocity vectors and are compared to patterns produced during the development of an abnormally shaped leaf. The bilateral symmetry of the V. vinifera leaf, which is present in the normal leafand absent in the malformed leaf, is shown during growth by the patterns of velocity isolines. Ellipses formed by the isolines around the midrib during normal development are distorted during development of the malformed leaf. During normal growth, tissue elements are displaced in rather straight lines, resulting in streamlines which radiate outward from the petiole. Element motion in the abnormally developing leaf causes curving of streamlines. Relative growth in area of elements located in an area in the normal leaf are higher than those in the malformed leaf. The most frequently observed category of relative area growth in the normal leaf is 40%-59% d~1, while 20%–39% d−-1 predominates in the abnormal leaf. A spatial gradient in growth appears during normal development with lowest relative growth (20%–39% d−-1) present in the tip region, intermediate values (40%–59% d−-1) in the midsection, and maximum growth (> 60% d−-1) appearing in the basal region. During development of the abnormally shaped leaf, the gradient along the midrib is disrupted with low magnitudes of growth (<20% d−-1) appearing in the midsection of the leaf where intermediate values are expected. The theoretical and numerical distinctions between two common expressions for relative growth (relative elemental growth and exponential growth rate) are discussed. Relative elemental growth is shown to become increasingly larger than the exponential growth rate as the magnitude of growth increases relative to initial size of the tissue element. Numerical methods for evaluating relative growth based on finite element areas are compared to methods based on displacement and velocity gradients and are shown to produce similar results.

In this paper, we give a complete classification of Hopf ∗-[Formula: see text]-Ricci–Yamabe soliton on real hypersurfaces in the complex projective space [Formula: see text]. As an application, first we give a complete classification of gradient ∗-[Formula: see text]-Ricci–Yamabe solitons on real hypersurfaces with isometric Reeb flow in the complex projective space [Formula: see text]. Next, we prove that a contact real hypersurface in [Formula: see text] which admits a gradient ∗-[Formula: see text]-Ricci–Yamabe soliton is pseudo-Einstein.