QUASI-STATIC AND DYNAMICAL ANALYSIS FOR VISCOELASTICTIMOSHENKO BEAM WITH FRACTIONAL DERIVATIVECONSTITUTIVE RELATION

The equations of motion governing the quasi-static and dynamical behavior of a viscoelastic Timoshenko beam are derived. The viscoelastic material is assumed to obey a three-dimensional fractional derivative constitutive relation. ne quasi-static behavior of the viscoelastic Timoshenko beam under step loading is analyzed and the analytical solution is obtained. The influence of material parameters on the deflection is investigated. The dynamical response of the viscoelastic Timoshenko beam subjected to a periodic excitation is studied by means of mode shape functions. And the effect of both transverse shear and rotational inertia on the vibration of the beam is discussed.

DYNAMICAL STABILITY OF VISCOELASTIC COLUMN WITH FRACTIONAL DERIVATIVE CONSTITUTIVE RELATION

The dynamic stability of simple supported viscoelastic column, subjected to a periodic axial force, is investigated. The viscoelastic material was assumed to obey the fractional derivative constitutive relation. The governing equation of motion was derived as a weakly singular Volterra integro-partial-differential equation, and it was simplified into weakly singular Volterra integro-ordinary-differential equation by the Galerkin method. In terms of the averaging method, the dynamical stability was analyzed. A new numerical method is proposed to avoid storing all history data. Numerical examples are presented and the numerical results agree with the analytical ones.

A NUMERICAL METHOD FOR FRACTIONAL INTEGRAL WITH APPLICATIONS

A new numerical method for the fractional integral that only stores part history data is presented, and its discretization error is estimated. The method can be used to solve the integro_differential equation including fractional integral or fractional derivative in a long history. The difficulty of storing all history data is overcome and the error can be controlled. As application,motion equations governing the dynamical behavior of a viscoelastic Timoshenko beam with fractional derivative constitutive relation are given. The dynamical response of the beam subjected to a periodic excitation is studied by using the separation variables method. Then the new numerical method is used to solve a class of weakly singular Volterra integro_differential equations which are applied to describe the dynamical behavior of viscoelastic beams with fractional derivative constitutive relations. The analytical and unmerical results are compared. It is found that they are very close.

Relative Left Derived Functors of Tensor Product Functors

We introduce and study the relative lett derive functor Torn（￡,￡1） category, which unifies several related left derived functors. Then we give some criteria for computing the -resolution dimensions of modules in terms of the properties of Torn（￡,￡1） . We also construct a complete and hereditary cotorsion pair relative to balanced pairs. Some known results are obtained as corollaries.

Categorical resolutions of a class of derived categories

We clarify the relation between the subcategory D_(hf)~b(A) of homological finite objects in D^b(A)and the subcategory K^b(P) of perfect complexes in D^b(A), by giving two classes of abelian categories A with enough projective objects such that D_(hf)~b(A) = K^b(P), and finding an example such that D_(hf)~b(A)≠K^b(P). We realize the bounded derived category D^b(A) as a Verdier quotient of the relative derived category D_C^b(A), where C is an arbitrary resolving contravariantly finite subcategory of A. Using this relative derived categories, we get categorical resolutions of a class of bounded derived categories of module categories of infinite global dimension.We prove that if an Artin algebra A of infinite global dimension has a module T with inj.dimT <∞ such that ~⊥T is finite, then D^b(modA) admits a categorical resolution; and that for a CM(Cohen-Macaulay)-finite Gorenstein algebra, such a categorical resolution is weakly crepant.

Relative Derived Equivalences and Relative Homological Dimensions

Let A be a small abelian category.For a closed subbifunctor F of Ext_A^1（-,-）,Buan has generalized the construction of Verdier’s quotient category to get a relative derived category,where he localized with respect to F-acyclic complexes.In this paper,the homological properties of relative derived categories are discussed,and the relation with derived categories is given.For Artin algebras,using relative derived categories,we give a relative version on derived equivalences induced by F-tilting complexes.We discuss the relationships between relative homological dimensions and relative derived equivalences.

Optical theorem,crossing property,and derivative dispersion relations:implications on the asymptotic behavior of σ_(tot)(s) and ρ(s)

In this paper,we present some results on the behavior of the total cross section and p-parameter at asymptotic energies in proton-proton(pp) and antiproton-proton(pp) collisions.Hence,we consider three of the main theoretical results in high energy physics:the crossing property,derivative dispersion relation,and optical theorem.The use of such machinery facilitates the derivation of analytic formulas for a wide set of the measured global scattering parameters and some important relations between them.The suggested parameterizations approximate the energy dependence for the total cross section and ρ-parameter for pp and pp with a statistically acceptable quality in the multi-TeV region.Additionally,the qualitative description is obtained for important interrelations,namely difference,sum,and ratio of the antiparticle-particle and particle-particle total cross sections.Despite the reduced number of experimental data for the total cross section and p-parameter at the TeV-scale,which complicates any prediction for the beginning of the asymptotic domain,the fitting procedures indicates that asymptotia occur in the energy range 25.5-130 TeV.Moreover,in the asymptotic regime,we obtain α_(P)=1.A detailed quantitative study of the energy behavior of the measured scattering parameters and their combinations in the ultra-high energy domain indicates that the scenario with the generalized formulation of the Pomeranchuk theorem is more favorable with respect to the original formulation of this theorem.

Applications of balanced pairs

Let(X, Y) be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to(X, Y), and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the X-resolution dimension of Y(resp. Y-coresolution dimension of X)is finite, then the bounded homotopy category of Y(resp. X) is contained in that of X(resp. Y). As a consequence, we get that the right X-singularity category coincides with the left Y-singularity category if the X-resolution dimension of Y and the Y-coresolution dimension of X are finite.