We study certain DT invariants arising from stable coherent sheaves in a nonsingular projective threefold supported on the members of linear system fixed line bundle. When canonical bundle satisfies positivity conditions, we relate to Carlsson-Okounkov formulas for twisted Euler's number punctual Hilbert schemes surfaces, and conclude they have modular property.

OBJECTIVE To determine the effectiveness of a care bundle, with a novel line maintenance procedure, in reducing the rate of central line-associated bloodstream infection (CLABSI) in the intensive care unit (ICU). DESIGN, PARTICIPANTS AND SETTING Before-and-after study using CLABSI data reported to the Victorian Healthcare Associated Infection Surveillance System (VICNISS), in adult patients a...

We prove that special ample line bundles on toric varieties arising from root systems are projectively normal. Here the maximal cones of the fans correspond to the Weyl chambers, and special means that the bundle is torus-equivariant such that the character of the line bundle that corresponds to a maximal Weyl chamber is dominant with respect to that chamber. Moreover, we prove that the associa...

We consider the quantum mechanics of a charged particle in presence Dirac’s magnetic monopole. Wave functions are sections complex line bundle and potential is connection on bundle. use continuum eigenfunction expansion to find an invariant domain essential self-adjointness for Hamiltonian. This leads proof Feynman–Kac formula expressing solutions imaginary time Schrödinger equation as stochast...

Central venous catheters (CVCs) are life-saving and the majority of patients in intensive care units (ICUs) have them placed in order to receive medicine and fluids. However, the use of these catheters can result in serious bloodstream infections. The rate of Central Line Associated Blood Stream Infection (CLABSI) in Adult Intensive Care Units (ICUs) at King Abdulaziz Medical City Jeddah (KAMC-...

Given a holomorphic vector bundle F over a compact Riemann surface X, and a subspace Sx ⊂ Fx in the fiber over a point x, the Hecke transformation produces a new vector bundle E on X [10, 16]. The vector bundle E is the kernel of the natural quotient map F −→ Fx/Sx. Hecke transformation is a very useful tool to study the moduli space. For instance, they are used in computation of cohomologies o...

Proposition 1.2 (Splitting Principle). Let E → M be a complex vector bundle of rank n over a manifold X. Then there is a manifold F (E) and a smooth fibration π : F (E) → M such that • The homomorphism π∗ : H∗(M, Z) → H∗(F (E), Z) is injective. • The bundle π∗E splits into a direct sum of complex line bundles, i.e. π∗E = l1 ⊕ · · ·⊕ ln Proof. Consider the projectivisation p : PC(E) → M and the ...

For a semistable reflexive sheaf E of rank r and c1 = a on P n and an integer d such that r|ad, we give sufficient conditions so that the restriction of E on a generic rational curve of degree d is balanced, i.e. a twist of the trivial bundle (for instance, if E has balanced restriction on a generic line, or r = 2 or E is an exterior power of the tangent bundle). Assuming this, we give a formul...

Let X → B be a proper flat morphism between smooth quasiprojective varieties of relative dimension n, and L → X a line bundle which is ample on the fibers. We establish formulas for the first two terms in the Knudsen-Mumford expansion for det(π∗L ) in terms of Deligne pairings of L and the relative canonical bundle K. This generalizes the theorem of Deligne [1], which holds for families of rela...

In this paper, by using the regulator map of Beilinson-Deligne, we show that the quantization condition posed by Gukov is true for the SL2(C) character variety of the hyperbolic knot in S. Furthermore, we prove that the corresponding C∗-valued 1-form is a secondary characteristic class (Chern-Simons) arising from the vanishing first Chern class of the flat line bundle over the smooth part of th...