Performance of revised STO(1M)-3G basis set for prediction of 5-fluorocytosine chemical shifts

Nuclear shieldings and chemical shifts of 5-fluorocytosine (5FC) were predicted in the gas phase and DMSO solution modeled by polarizable continuum model using B3LYP density functional and revised STO(1M)-3G basis set. For comparison, eight arbitrary selected basis sets including STO-3G and medium-size Pople-type and larger dedicated Jensen-type ones were applied. The former basis sets were significantly smaller, but the calculated structural parameters, harmonic vibrational frequencies, were very accurate and close to those obtained with larger, polarization-consistent ones. The predicted ¹³C and ¹H chemical shieldings of 5FC and cytosine, selected as parent molecule, were acceptable (root mean square for ¹³C chemical shifts in DMSO of about 5 ppm and less) though less accurate than those calculated with large basis sets, dedicated for prediction of nuclear magnetic resonance parameters.     

Control problems for energy harvester model and interpolation in Hardy space

Three control problems for the system of two coupled differential equations governing the dynamics of an energy harvesting model are studied. The system consists of the equation of an Euler–Bernoulli beam model and the equation representing the Kirchhoff's electric circuit law. Both equations contain coupling terms representing the inverse and direct piezoelectric effects. The system is reformulated as a single evolution equation in the state space of 3-component functions. The control is introduced as a separable forcing term $\text{◂⋅▸}\mathbf{g}(x)f(t)$ on the right-hand side of the operator equation. The first control problem deals with an explicit construction of $f(t)$ that steers an initial state to zero on a time interval [0, T]. The second control problem deals with the construction of $f(t)$ such that the voltage output is equal to some given function $\mathbf{v}(t)$ (with $\mathbf{g}(x)$ being given as well). The third control problem deals with an explicit construction of both the force profile, $\mathbf{g}(x)$, and the control, $f(t)$, which generate the desired voltage output $\mathbf{v}(t)$. Interpolation theory in the Hardy space of analytic functions is used in the solution of the second and third problems.     

An improvised collocation algorithm with specific end conditions for solving modified Burgers equation

In this work, numerical solution of nonlinear modified Burgers equation is obtained using an improvised collocation technique with cubic B‐spline as basis functions. In this technique, cubic B‐splines are forced to satisfy the interpolatory condition along with some specific end conditions. Crank–Nicolson scheme is used for temporal domain and improvised cubic B‐spline collocation method is used for spatial domain discretization. Quasilinearization process is followed to tackle the nonlinear term in the equation. Convergence of the technique is established to be of order O(h⁴ + Δt²) . Stability of the technique is examined using von‐Neumann analysis. L₂ and L_∞ error norms are calculated and are compared with those available in existing works. Results are found to be better and the technique is computationally efficient, which is shown by calculating CPU time.     

Sampling in a Hilbert space

An analog of the Whittaker-Shannon-Kotel'nikov sampling theorem is derived for functions with values in a separable Hilbert space. The proof uses the concept of frames and frame operators in a Hilbert space. One of the consequences of this theorem is that it allows us to derive sampling theorems associated with boundary-value problems and some homogeneous integral equations, which in turn gives us a generalization of another sampling theorem by Kramer.

     

Modulus of Continuity of Piecewise Analytic Functions

We obtain conditions under which the modulus of continuity of a piecewise analytic function given on a closed interval of the real axis is an analytic function in a neighborhood of zero.     

基于基样条局部逼近散乱数据拟合中的Shepard方法

本文针对散乱数据拟合的shepard方法,提出了一种局部逼近的新方法.该方法以局部三次基样条函数作为Shepard公式中的权函数,新的权函数具有良好的衰减性和二阶连续性,从而改进了传统方法的不足之处,使实际应用效果更好.