Stability of functions in Boolean models of gene regulatory networks

Boolean networks are used to model large nonlinear systems such as gene regulatory networks. We will present results that can be used to understand how the choice of functions affects the network dynamics. The so called bias-map and its fixed points depict much of the function's dynamical role in the network. We define the concept of stabilizing functions and show that many Post and canalizing functions are also stabilizing functions. Boolean networks constructed using the same type of stabilizing functions are always stable regardless of the average in-degree of network functions. We derive the number of all stabilizing functions and find it to be much larger than the number of Post and canalizing functions. We also discuss the implementation of functions and apply the presented results to biological data that give an approximation of the distribution of regulatory functions in eucaryotic cells. We find that the obtained theoretical results on the number of active genes are biologically plausible. Finally, based on the presented results, we discuss why canalizing and Post regulatory functions seem to be common in cells.     

Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions

Considering the equations for some functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions. The first series applies single Beta functions as expansion functions, while the second one involves a combination of two Beta functions. The coefficients of expansions obey four- and five-term recurrence relations, respectively. It is shown that the proposed technique is potent to produce series solutions in terms of other special functions. Two examples of such expansions in terms of the incomplete Gamma functions are presented.     

泰勒展开公式的新认识③——卷积型级数的Mbius反演

在泰勒展开的基础上理解卷积型级数的Mbius反演,给出了若干对反演系数,并应用于讨论已有的特殊函数,导出了它们的级数反演形式.     

On the representations of the groups SO(3), O(2, 1), and M(2) defined in terms of Lamé, Mathieu,and parabolic functions

Products of Mathieu, parabolic-cylinder and Lamé functions appear as basis functions for the irreducible representations of the groups M(2), O(2,1) and SO(3). Addition formulas concerning the Mathieu functions, parabolic cylinder functions, and Lamé polynomials are recovered or obtained.     

Gluon cascades and amplitudes in light-front perturbation theory

We construct the gluon wave functions, fragmentation functions and scattering amplitudes within the light-front perturbation theory. Recursion relations on the light-front are constructed for the wave functions and fragmentation functions, which in the latter case are the light-front analogs of the Berends–Giele recursion relations. Using general relations between wave functions and scattering amplitudes it is demonstrated how to obtain the maximally-helicity violating amplitudes, and explicit verification of the results is based on simple examples.     

Generalized Value Distribution for Herglotz Functions and Spectral Theory

We generalize the theory of value distribution for a class of functions defined as boundary values of Herglotz functions, by considering other measures than Lebesgue measure. The link with compositions of Herglotz functions is presented, and precise relations for the associated measures are obtained. We also consider uniformly convergent sequences of Herglotz functions on compact subsets of the upper half-plane, and prove that the corresponding sequence of Herglotz measures and the generalized value distribution of these functions also converge.     

Graphical functions in parametric space

Graphical functions are positive functions on the punctured complex plane \({\mathbb C}{\setminus }\{0,1\}\) which arise in quantum field theory. We generalize a parametric integral representation for graphical functions due to Lam, Lebrun and Nakanishi, which implies the real analyticity of graphical functions. Moreover, we prove a formula that relates graphical functions of planar dual graphs.     

Comparing the evolution of canalyzing and threshold networks

We study the influence of the type of update functions on the evolution of Boolean networks under selection for dynamical robustness. The chosen types of functions are canalyzing functions and threshold functions. Starting from a random initial network, we evolve the network by an adaptive walk. During the first time period, where the networks evolve to the plateau of 100 percent fitness, we find that both type of update functions give the same behavior, albeit for different network sizes and connectedness. However, on the long run, as the networks continue to evolve on the fitness plateau, the different types of update functions give rise to different network structure, due to their different mutational robustness. When both types of update functions occur together, none of them is preferred under long-term evolution.     

Analytic continuation and reciprocity relation for collinear splitting in QCD

It is well-known that direct analytic continuation of the DGLAP evolution kernel(splitting functions)from space-like to time-like kinematics breaks down at three loops.We identify the origin of this breakdown as due to splitting functions not being analytic functions of external momenta.However,splitting functions can be constructed from the squares of(generalized)splitting amplitudes.We establish the rules of analytic continuation for splitting amplitudes,and use them to determine the analytic continuation of certain holomorphic and anti-holomorphic part of splitting functions and transverse-momentum dependent distributions.In this way we derive the time-like splitting functions at three loops without ambiguity.We also propose a reciprocity relation for singlet splitting functions,and provide non-trivial evidence that it holds in QCD at least through three loops.     

Molecular electronic distribution functions and correlation holes Theory and application to H2, LiH and BH

The one-electron density functions for the diatomic hydrides H₂, LiH and BH, defined with respect to a limited STO basis set, are partially integrated to yield longitudinal and transverse distribution functions which admit of a simple pictorial representation. The maxima in the longitudinal distribution functions occur in regions conventionally ascribed to core pairs, lone pairs or bond pairs.

The correlation between pairs of electrons of given spin is then analysed in terms of the analogous partially integrated pair density functions and in terms of the associated hole functions. In the case of LiH and BH, the Fermi hole functions are similar in form to the negatives of the one-electron distribution functions for the appropriate singly occupied molecular orbitals; but the differences between the two functions are more marked in BH, where there is increased spatial interpenetration of the valence electron pairs. The results also show, in a simple way, how the electron motion is over-correlated in H₂ when the Heitler-London wavefunction is used.     

Using the Semantic Information G Measure to Explain and Extend Rate-Distortion Functions and Maximum Entropy Distributions

In the rate-distortion function and the Maximum Entropy (ME) method, Minimum Mutual Information (MMI) distributions and ME distributions are expressed by Bayes-like formulas, including Negative Exponential Functions (NEFs) and partition functions. Why do these non-probability functions exist in Bayes-like formulas? On the other hand, the rate-distortion function has three disadvantages: (1) the distortion function is subjectively defined; (2) the definition of the distortion function between instances and labels is often difficult; (3) it cannot be used for data compression according to the labels’ semantic meanings. The author has proposed using the semantic information G measure with both statistical probability and logical probability before. We can now explain NEFs as truth functions, partition functions as logical probabilities, Bayes-like formulas as semantic Bayes’ formulas, MMI as Semantic Mutual Information (SMI), and ME as extreme ME minus SMI. In overcoming the above disadvantages, this paper sets up the relationship between truth functions and distortion functions, obtains truth functions from samples by machine learning, and constructs constraint conditions with truth functions to extend rate-distortion functions. Two examples are used to help readers understand the MMI iteration and to support the theoretical results. Using truth functions and the semantic information G measure, we can combine machine learning and data compression, including semantic compression. We need further studies to explore general data compression and recovery, according to the semantic meaning.     

Acoustics of a flanged cylindrical pipe using singular basis functions

The problem of acoustic radiation from a cylindrical pipe with an infinite flange has been discussed in a number of papers. The most common approach is to decompose the field inside the pipe over a basis of Bessel functions. A very large number of basis functions is usually required, with a large degree of ripple appearing as an artifact in the solution. In this paper it is shown that a close analysis of the velocity field near the corner yields a new family of functions, which are called "edge functions." Using this set of functions as test functions and applying the moment method on the boundary between the waveguide and free space, a solution is obtained with greatly improved convergence properties and no ripple.     

Correlated basis functions for the many-electron problem

The method of correlated basis functions is applied to the electron system in a metal. To overcome the twofold difficulty of large particle number and long-range Coulomb interaction in metals, a new optimal cluster decomposition for arbitrary correlated wave functions is derived. With this method, not only ground-state properties, but also thermal averages and response functions can be calculated from a given set of correlated basis functions. The appropriate synthesis of correlated wave functions including physically expected properties of subsystems as for instance partially filled inner shells in transition metals is discussed in detail.     

The scaling limit and Osterwalder-Schrader axioms for the two-dimensional Ising model

From a Feynman-Kac formula in a Fermion Fock space for the Schwinger functions of the infinite lattice periodic two-dimensional Ising model, scaled and scaling limit Schwinger functions are defined and shown to admit an absolutely convergent series representation. As the critical temperature is attained, it is shown that the scaled Schwinger functions converge and that the resulting scaling limit Schwinger functions obey the Osterwalder-Schrader axioms.     

Explicit Solutions of (2+1)-Dimensional Canonical Generalized KP,KdV, and(2+1)-Dimensional Burgers Equations with Variable Coefficients

In this paper, using the generalized G'/G-expansion method and the auxiliary differential equation method, we discuss the (2+1)-dimensional canonical generalized KP (CGKP), KdV, and (2+1)-dimensional Burgers equations with variable coefficients. Many exact solutions of the equations are obtained in terms of elliptic functions, hyperbolic functions, trigonometric functions, and rational functions.     

一类环状非球谐振子势场中相对论粒子的束缚态解

提出了一种新的环状非球谐振子势, 在标量势与矢量势相等的条件下，给出了其Klein-Gordon方程和Dirac方程的束缚态解. Klein-Gordon方程的θ角向波函数以超几何函数表示，径向波函数可用合流超几何函数或广义拉盖尔多项式表示，能谱方程由径向波函数满足的束缚态边界条件得到. Dirac方程的旋量波函数可用Klein-Gordon方程的解构造. 关键词： 环状非球谐振子势 Klein-Gordon方程 Dirac方程 束缚态     

不同微场分布函数对Stark加宽和频移的影响

分别采用Holtsmark,Neutral Point,Nearest-Neighbor和Mayer模型微场分布函数对Stark线型进行了研究,进而得到相应微场函数下的Stark加宽和频移,研究了4种不同的微场分布函数对Stark加宽和频移的影响。研究结果表明,在电子加宽参数不变时,4种微场分布函数对Stark加宽和频移的影响随离子加宽参数的增加而增加;在离子加宽参数不变时,4种微场分布函数对Stark加宽和频移的影响随电子加宽参数的增加而增加;特别是,当离子加宽参数较大时,Mayer模型微场分布函数对Stark加宽和频移的影响异常明显。这说明,微场分布函数对谱线的加宽和频移的影响在离子与离子碰撞剧烈的等离子体环境中尤其显著, 在这样的等离子体环境中,进行等离子体诊断时,选择合适的微场分布函数非常重要。结果对等离子体诊断有一定参考价值。     

Collective Excitations in Spin-2 Bose-Einstein Condensates

The Green＇s functions and the correlation functions in spin-2 Bose-Einstein condensates at finite temperature are defined and the generalized Dyson-Beliaev equations are introduced. We discuss the spin conservation in z direction and decouple the Green＇s functions and the generalized Dyson-Beliaev equations according to different spin conservations in z direction. The anomalous vertex functions are introduced and the self-energies are separated into the proper self-energies and the improper self-energies. The generalized Dyson-Beliaev equations are decoupled according to separation of the self-energies. We calculate the Green＇s functions step by step in the Bogoliubov approximation and discuss the collective excitations in spin-2 Bose-Einstein condensates in the polar, ferromagnetic, and cyclic cases, respectively.