Describing some characters of serine proteinase using fractal analysis

In this paper we calculated the fractal dimensions of four proteins, chymotrypsin, elastase, trypsin and subtilisin, which are made up of about 220–275 amino acids and belong to the family of serine proteinase by using three definitions of fractal dimension i.e. the chain fractal dimension (D_L), the mass fractal dimension (D_m) and the correlation fractal dimension (D_c). We also analyzed the relationship between fractal dimension and space structure or secondary structure contents of proteins. The results showed that the values of fractal dimensions are almost same for the global mammalian enzymes (chymotrypsin, elastase and trypsin), but different for the global subtilisin. This demonstrated that the more similar structures, the more equal fractal dimensions, and if the fractal dimensions of proteins are different from each other, the three dimensional structures should not be similar. On the other hand, the detailed structures and fractal dimensions of the active sites of four enzymes are extraordinarily similar. Therefore, the fractal method can be applied to the elucidation of the proteins evolution.     

Thermal properties of bodies in fractal and cantorian physics

Fundamental laws describing the heat diffusion in fractal environment are discussed. It is shown that for the three-dimensional space the heat radiation process occur in structures with fractal dimension D 0,1), whereas in structures with D (1,3 heat conduction and convection have the upper hand (generally in the real gases).To describe the heat diffusion a new law has been formulated. Its validity is more general than the Plank’s radiation law based on the quantum heat diffusion theory. The energy density w = f (K, D), where K is the fractal measure and D is the fractal dimension exhibit typical dependency peaking with agreement with Planck’s radiation law and with the experimental data for the absolutely black body in the energy interval kT < Kc. The positions of the energy density maximums (for fractal dimensions D_m < 0.31854) are in a good agreement with the maximums determined by Wien’s displacement law with the help of the Lambert’s W- Function u(A) = A + W[−Aexp(−A)], where A ≈ 1.9510 and u = hc/λ_mkT_m ≈ 1.5275. The agreement of the fractal model with the experimental outcomes is documented for the spectral characteristics of the Sun. The properties of stellar objects (black holes, relict radiation, etc.) and the elementary particles fields and interactions between them (quarks, leptons, mesons, baryons, bosons and their coupling constants) will be discussed with the help of the described mathematic apparatus in our further contributions.The general gas law for real gases in its more applicable form than the widely used laws (e.g. van der Waals, Berthelot, Kammerlingh–Onnes) has been also formulated. The energy density, which is in this case represented by the gas pressure p = f (K, D), can gain generally complex value and represents the behaviour of real (cohesive) gas in interval D (1,3. The gas behaves as the ideal one only for particular values of the fractal dimensions (the energy density is real-valued). Again, it is shown that above the critical temperature (kT > Kc) and for fractal dimension D_m > 2.0269 the results are comparable to the kinetics theory of real (ideal) gas (van der Waals equation of state, compressibility factor, Boyle’s temperature). For the critical temperature (Kc = kT_r) the compressibility factor gains Z = 1 (except for the ideal gas case D = 3) also for the fractal dimension D = 1/ = 1.618033989, where is the golden mean value of the El Naschie’s golden mean field theory. To determine the minimum it is also possible to employ the Lambert’s W− Function u(A) = A + W[−Aexp(−A)], whereA ≈ 0.6779 and u ≈ −0.7330. The thermal properties of fractal structures (thermal capacity, thermal conductivity, diffusivity) and additional parameters (enthalpy, entropy, etc.) will be defined using the mathematic apparatus in the future. Good agreement of the fractal model with experimental data is documented on the compressibility factor of various gases.     

On the finitistic dimension conjecture II: Related to finite global dimension

In this paper, we study the finitistic dimensions of artin algebras by establishing a relationship between the global dimensions of the given algebras, on the one hand, and the finitistic dimensions of their subalgebras, on the other hand. This is a continuation of the project in [J. Pure Appl. Algebra 193 (2004) 287-305]. For an artin algebra A we denote by gl.dim(A), fin.dim(A) and rep.dim(A) the global dimension, finitistic dimension and representation dimension of A, respectively. The Jacobson radical of A is denoted by rad(A). The main results in the paper are as follows: Let B be a subalgebra of an artin algebra A such that rad(B) is a left ideal in A. Then (1) if gl.dim(A)?4 and rad(A)=rad(B)A, then fin.dim(B)<∞. (2) If rep.dim(A)?3, then fin.dim(B)<∞. The results are applied to pullbacks of algebras over semi-simple algebras. Moreover, we have also the following dual statement: (3) Let ?:B?A be a surjective homomorphism between two algebras B and A. Suppose that the kernel of ? is contained in the socle of the right B-module B_B. If gl.dim(A)?4, or rep.dim(A)?3, then fin.dim(B)<∞. Finally, we provide a class of algebras with representation dimension at most three: (4) If A is stably hereditary and rad(B) is an ideal in A, then rep.dim(B)?3.     

Backbone fractal dimension and fractal hybrid orbital of protein structure

Fractal geometry analysis provides a useful and desirable tool to characterize the configuration and structure of proteins. In this paper we examined the fractal properties of 750 folded proteins from four different structural classes, namely (1) the α-class (dominated by α-helices), (2) the β-class (dominated by β-pleated sheets), (3) the (α/β)-class (α-helices and β-sheets alternately mixed) and (4) the (α + β)-class (α-helices and β-sheets largely segregated) by using two fractal dimension methods, i.e. “the local fractal dimension” and “the backbone fractal dimension” (a new and useful quantitative parameter). The results showed that the protein molecules exhibit a fractal behavior in the range of 1 ? N ? 15 (N is the number of the interval between two adjacent amino acid residues), and the value of backbone fractal dimension is distinctly greater than that of local fractal dimension for the same protein. The average value of two fractal dimensions decreased in order of α > α/β > α + β > β. Moreover, the mathematical formula for the hybrid orbital model of protein based on the concept of backbone fractal dimension is in good coincidence with that of the similarity dimension. So it is a very accurate and simple method to analyze the hybrid orbital model of protein by using the backbone fractal dimension.     

DIMENSION RESULTS FOR ORNSTEIN-UHLENBECK TYPE MARKOV PROCESS

Let {X_t, t ≥ 0} be an Ornstein-Uhlenbeck type Markov process with Levy process A_t, the authors consider the fractal properties of its ranges, give the upper and lower bounds of the Hausdorff dimensions of the ranges and the estimate of the dimensions of the level sets for the process. The existence of local times and occuption times of X_t are considered in some special situations.     

Exact Computation and Approximation of Stochastic and Analytic Parameters of Generalized Sierpinski Gaskets

The interplay of fractal geometry, analysis and stochastics on the one-parameter sequence of self-similar generalized Sierpinski gaskets is studied. An improved algorithm for the exact computation of mean crossing times through the generating graphs SG(m) of generalized Sierpinski gaskets sg(m) for m up to 37 is presented and numerical approximations up to m?=?100 are shown. Moreover, an alternative method for the approximation of the mean crossing times, the walk and the spectral dimensions of these fractal sets based on quasi-random so-called rotor walks is developed, confidence bounds are calculated and numerical results are shown and compared with exact values (if available) and with known asymptotic formulas.     

Finitistic dimension and restricted injective dimension

We study the relations between finitistic dimensions and restricted injective dimensions. Let R be a ring and T a left R-module with A = End _R T. If _R T is selforthogonal, then we show that rid(T _A ) ? findim(A _A ) ? findim( _R T) + rid(T _A ). Moreover, if R is a left noetherian ring and T is a finitely generated left R-module with finite injective dimension, then rid(T _A ) ? findim(A _A ) ? fin.inj.dim( _R R)+rid(T _A ). Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.     

The Terwilliger algebra of a distance-regular graph of negative type

Let Γ denote a distance-regular graph with diameter D?3. Assume Γ has classical parameters (D,b,α,β) with b<-1. Let X denote the vertex set of Γ and let A∈Mat_X(C) denote the adjacency matrix of Γ. Fix x∈X and let A^∗∈Mat_X(C) denote the corresponding dual adjacency matrix. Let T denote the subalgebra of Mat_X(C) generated by A,A^∗. We call T the Terwilliger algebra of Γ with respect to x. We show that up to isomorphism there exist exactly two irreducible T-modules with endpoint 1; their dimensions are D and 2D-2. For these T-modules we display a basis consisting of eigenvectors for A^∗, and for each basis we give the action of A.     

Tangent circle graphs and ‘orders’

Consider a horizontal line in the plane and let γ(A) be a collection of n circles, possibly of different sizes all tangent to the line on the same side. We define the tangent circle graph associated to γ(A) as the intersection graph of the circles. We also define an irreflexive and asymmetric binary relation P on A; the pair (a,b) representing two circles of γ(A) is in P iff the circle associated to a lies to the right of the circle associated to b and does not intersect it. This defines a new nontransitive preference structure that generalizes the semi-order structure. We study its properties and relationships with other well-known order structures, provide a numerical representation and establish a sufficient condition implying that P is transitive. The tangent circle preference structure offers a geometric interpretation of a model of preference relations defined by means of a numerical representation with multiplicative threshold; this representation has appeared in several recently published papers.     

Long-time behavior of the difference solutions to the 2D GKS equation

We study the long-time behavior of the finite difference solution to the generalized Kuramoto-Sivashinsky equation in two space dimensions with periodic boundary conditions. The unique solvability of numerical solution is shown. It is proved that there exists a global attractor of the discrete dynamical system and the upper semicontinuity d(A_h,τ,A)→0. Finally, we obtain the long-time stability and convergence of the difference scheme. Our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems.     

Max-algebraic attraction cones of nonnegative irreducible matrices

It is known that the max-algebraic powers A^r of a nonnegative irreducible matrix are ultimately periodic. This leads to the concept of attraction cone Attr(A, t), by which we mean the solution set of a two-sided system λ^t(A)A^r⊗x=A^r+t⊗x, where r is any integer after the periodicity transient T(A) and λ(A) is the maximum cycle geometric mean of A. A question which this paper answers, is how to describe Attr(A,t) by a concise system of equations without knowing T(A). This study requires knowledge of certain structures and symmetries of periodic max-algebraic powers, which are also described. We also consider extremals of attraction cones in a special case, and address the complexity of computing the coefficients of the system which describes attraction cone.     

Periodic resolutions and self-injective algebras of finite type

We say that an algebra A is periodic if it has a periodic projective resolution as an (A,A)-bimodule. We show that any self-injective algebra of finite representation type is periodic. To prove this, we first apply the theory of smash products to show that for a finite Galois covering B→A, B is periodic if and only if A is. In addition, when A has finite representation type, we build upon results of Buchweitz to show that periodicity passes between A and its stable Auslander algebra. Finally, we use Asashiba’s classification of the derived equivalence classes of self-injective algebras of finite type to compute bounds for the periods of these algebras, and give an application to stable Calabi-Yau dimensions.     

On orthogonal systems of matrix algebras

In this work it is shown that certain interesting types of orthogonal system of subalgebras (whose existence cannot be ruled out by the trivial necessary conditions) cannot exist. In particular, it is proved that there is no orthogonal decomposition of M_n(C)⊗M_n(C)≡M_n²(C) into a number of maximal abelian subalgebras and factors isomorphic to M_n(C) in which the number of factors would be 1 or 3.In addition, some new tools are introduced, too: for example, a quantity c(A,B), which measures “how close” the subalgebras A,B⊂M_n(C) are to being orthogonal. It is shown that in the main cases of interest, c(A^′,B^′) - where A^′ and B^′ are the commutants of A and B, respectively - can be determined by c(A,B) and the dimensions of A and B. The corresponding formula is used to find some further obstructions regarding orthogonal systems.     

On the Gröbner complexity of matrices

In this paper we show that if for an integer matrix A the universal Gröbner basis of the associated toric ideal I_A coincides with the Graver basis of A, then the Gröbner complexity u(A) and the Graver complexity g(A) of its higher Lawrence liftings agree, too. In fact, if the universal Gröbner basis of I_A coincides with the Graver basis of A, then also the more general complexities u(A,B) and g(A,B) agree for arbitrary B. We conclude that for the matrices A_3×3 and A_3×4, defining the 3×3 and 3×4 transportation problems, we have u(A_3×3)=g(A_3×3)=9 and u(A_3×4)=g(A_3×4)≥27. Moreover, we prove that u(A_a,b)=g(A_a,b)=2(a+b)/gcd(a,b) for positive integers a,b and .     

Non-conformal Repellers and the Continuity of Pressure for Matrix Cocycles

The pressure function P(A, s) plays a fundamental role in the calculation of the dimension of “typical” self-affine sets, where A = (A ₁, …,A _k ) is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on A. As a consequence, we show that the dimension of “typical” self-affine sets is a continuous function of the defining maps. This resolves a folklore open problem in the community of fractal geometry. Furthermore we extend the continuity result to more general sub-additive pressure functions generated by the norm of matrix products or generalized singular value functions for matrix cocycles, and obtain applications on the continuity of equilibrium measures and the Lyapunov spectrum of matrix cocycles.     

On regular graphs,V

Let Γ₃ be an infinite regular tree of valence 3. There exist subgroups B of Aut (Γ₃) which are 5-regular on Γ₃, i.e., sharply transitive on the set of 5-arcs of Γ₃. We prove that any two such subgroups are conjugate in Aut (Γ₃). The pair (Γ₃, B) is a universal 5-regular action in the sense that if (G, A) is a pair consisting of a cubical graph G and a 5-regular subgroup A of automorphisms of G then (G, A) can be “covered” by (Γ₃, B) in a certain natural way.     

The second law of thermodynamics and multifractal distribution functions: Bin counting,pair correlations,and the Kaplan–Yorke conjecture

We explore and compare numerical methods for the determination of multifractal dimensions for a doubly-thermostatted harmonic oscillator. The equations of motion are continuous and time-reversible. At equilibrium the distribution is a four-dimensional Gaussian, so that all the dimension calculations can be carried out analytically. Away from equilibrium the distribution is a surprisingly isotropic multifractal strange attractor, with the various fractal dimensionalities in the range 1 < D < 4. The attractor is relatively homogeneous, with projected two-dimensional information and correlation dimensions which are nearly independent of direction. Our data indicate that the Kaplan–Yorke conjecture (for the information dimension) fails in the full four-dimensional phase space. We also find no plausible extension of this conjecture to the projected fractal dimensions of the oscillator. The projected growth rate associated with the largest Lyapunov exponent is negative in the one-dimensional coordinate space.     

On non-linear partial differential equations with an infinite-dimensional conditional symmetry

The invariance of non-linear partial differential equations under a certain infinite-dimensional Lie algebra AN(z) in N spatial dimensions is studied. The special case A₁(2) was introduced in [J. Stat. Phys. 75 (1994) 1023] and contains the Schrödinger Lie algebra sch₁ as a Lie subalgebra. It is shown that there is no second-order equation which is invariant under the massless realizations of AN(z). However, a large class of strongly non-linear partial differential equations is found which are conditionally invariant with respect to the massless realization of AN(z) such that the well-known Monge-Ampère equation is the required additional condition. New exact solutions are found for some representatives of this class.