that represents the algorithm of Taylor’s formula in a generalized form. Now, if \(\alpha =1\), Eq. (12) turns to Taylor’s series formula in a classical form such as $$\begin{aligned} \mathfrak {S}(\aleph , \wp )=\sum _{n=0}^{\infty } \frac{\partial ^m \mathfrak {...
Van der Waals heterostructures have opened new opportunities to develop atomically thin (opto)electronic devices with a wide range of functionalities. The recent focus on manipulating the interlayer twist angle has led to the observation of out-of-plane room temperature ferroelectricity in twisted rhombo...
where the densitythe velocity, the magnetic fieldand the pressureare unknown functions of the spacial variablesand the timet. HereSis the entropy,is the total energy andstands for the internal energy. By using the state equation of gas,and the first principle of thermodynamics, we have that (...
We present a high-dimensional measurement device-independent (MDI) quantum key distribution (QKD) protocol employing biphotons to encode information. We exploit the biphotons as qutrits to improve the tolerance to error rate. Qutrits have a larger quantum system; hence they carry more bits of clas...
The eta invariant for even dimensional pin c manifolds - Gilkey - 1985 () Citation Context ...itrary isomorphism of vector bundles π∗W and π∗ SF over the cotangent sphere bundle, which exists by condition (47). The index formula for even boundary value problems leads to a similar...
which is known as the Rodrigues formula for \hat{s}=0,1,2,\ldots . We define next the so-called shifted version of JPs on the interval [0,L], where L>0. With the aid of new change of variable x\rightarrow (-1+2x/L), we get the shifted JPs (SJPs) denoted by \mathca...
Using the Green’s formula, we obtain \begin{aligned} \int _{ B_{2}} \bigl(\Delta ^{2} u_{2} \bigr) \nabla u_{2} ={}&{-} \int _{ B_{2}}(\Delta u_{2})\nabla \bigl( \Delta (u_{2}) \bigr)- \int _{ B_{2}} \nabla \bigl(\nabla (\Delta u_{2}).\...
We prove \({|x|^{-2}}\) decay of the critical two-point function for the continuous-time weakly self-avoiding walk on \({\mathbb{Z}^{d}}\), in the upper critical dimension d = 4. This is a statement that the critical exponent \({\eta}\) exists and is equal to zero. Result...
The electric field was in turn calculated using the relation \(\overrightarrow{E}=-\overrightarrow{\nabla }\phi \left(x,y\right)=-\overrightarrow{\nabla }\left(\zeta (x)+\eta \left(y\right)\right)\). Having reviewed the preliminary underlying formulation of the single-region pseudo-...
(4). While formula (10) does not seem amenable to detailed interpretation under misspecification, it serves to illustrate complicated dependence on key aspects of the formulation. Table 4 of Sect. 6.2 shows that the loss of efficiency in the gamma model for random effects can be severe when ...