A Novel Design Method for Protein-Like Molecules from the Perspective of Sheaf Theory

Proteins perform a variety of functions in living organisms and their functions are largely determined by their shape. In this paper, we propose a novel mathematical method for designing protein-like molecules of a given shape. In the mathematical model, molecules are represented as loops of n-simplices (2-simplices are triangles and 3-simplices are tetrahedra). We design a new molecule of a given shape by patching together a set of smaller molecules that cover the shape. The covering set of small molecules is defined using a binary relation between sets of molecules. A new molecule is then obtained as a sum of the smaller molecules, where addition of molecules is defined using transformations acting on a set of (n + 1)-dimensional cones. Due to page limitations, only the two-dimensional case (i.e., loops of triangles) is considered. No prior knowledge of Sheaf Theory, Category Theory, or Protein Science is required. The author hopes that this paper will encourage further collaboration between Mathematics and Protein Science.

Synthesis and Solution Properties of Comb-like Acrylamide Copolymers

A comb-like acrylamide copolymer （HCJ-1） was synthesized by using aqueous free radical polymerization of acrylamide （AM） as main monomer and ether carboxylate as functional monomer. The copolymers were characterized with FT-IR and SEM. The SEM results show that the molecular structure of copolymer is extended in salt water. It is proved that the copolymer shows good salt resistance.The solution properties of HCJ-1 were studied and compared with those of partially hydrolyzed polyacrylamide （HPAM）. The experimental results show that the obtained HCJ-1, compared with HPAM, exhibits a dramatic enhancement in the salt-resistant properties. The apparent viscosity retention rate of 1.5 g/L HCJ-1 aqueous solution and salt solution after 180 days at 60 ℃ are 70.6% and 64.5%, respectively, exhibiting good thermal stability. In addition, HCJ-1 solution also displays the excellent shearing resistance. In a word, the experimental results show the HCJ-1 is a promising profiling agent for high salinity reservoir.

Electro-rheological Effect of Barium Titanate Particles Coated with Urea and Suspended in Methyl Silicone Oil

Barium titanate nano-powders were synthesized under defined conditions. The surface of these particles was successfully modified by coating with urea. The characteristics of these composite particles were studied by X-ray diffraction,transmission electron microscopy (TEM) and Fourier transform infrared spectroscopy. The electro-rheological (ER) effects of these particles suspended in methyl-silicone oil were measured. The particle,methyl-silicon oil ratio was 30%-35% weight percent. The experimental results indicate that these ER particles exhibit a remarkable ER effect. The ER fluid shows Bingham characteristics and the static shearing stress increases with an increase of the electric field strength. The highest static shearing stress under a 4 MV/m electric field is 13.2 kPa at room temperature,an increase of about 8.7 kPa compared to untreated BaTiO(C2O4)2 powders.

Influence of Ammoniacal Copper Quaternary treatments on mechanical properties of blue-stained Lodgepole Pine wood

Three concentrations （2.8%, 2.0%, 1.2%） of Ammoniacal Copper Quaternary （ACQ） was selected to treat Lodgepole pine wood for evaluating ACQ treatment on mechanical properties of blue-stained wood. The bending modules of elasticity （MOE）, modules of rupture （MOR）, toughness and shearing strength parallel to grain on tangential surface, are tested according to the criteria GB1927-1943-91. Non-treated sample were also tested according to the same procedure. The results showed that the three groups specimen impregnated by different concentrations of ACQ solution met the AWPA standard 2003 of America （UC4A 6.4g/cm^3）. There were significant difference of toughness between treated wood and non-treated wood （p=0.01）, but there were no statistically significant differences among three concentrations in terms of toughness, and toughness of treated wood was approximately 20% lower than non-treated. MOR, MOE as well as sheafing strength parallel to grain were found to be not significantly different between treated wood and non-treated one, and there were no statistically significant difference among three concentrations of ACQ too. Toughness, MOR, MOE and sheafing strength parallel to grain increased with decrease of concentration of ACQ, but they were hardly affected by ACQ preservatives.

THE JUMPING PHENOMENON OF THE DIMENSIONS OF COHOMOLOGY GROUPS OF TANGENT SHEAF

Let X be a compact complex manifold. Consider a small deformation π ： X → B of X, the dimensions of the cohomology groups of tangent sheaf Hq（xt, Txt ） may vary under this deformation. This article studies such phenomena by studying the obstructions to deform a class in Hq（X, 5TX） with parameter t and gets a formula for the obstructions.

Affine A_3^((1)) N=2 Monopole as the D Module and Aftine ADHMN Sheaf

A Higgs-Yang-Mills monopole scattering spherical symmetrically along light cones is given. The left incoming anti-self-dual α plane fields are holomorphic, but the right outgoing SD β plane fields are antiholomorphic, meanwhile the diffeomorphism symmetry is preserved with mutual inverse afiine rapidity parameters μ and μ^-1. The Dirac wave function scattering in this background also factorized respectively into the （anti）holomorphic amplitudes. The holomorphic anomaly is realized by the center term of a quasi Hopf algebra corresponding to an integrable conformal affine massive field. We find explicit Nahm transformation matrix （Fourier Mukai transformation） between the Higgs YM BPS （fiat） bundles （1） modules） and the affinized blow up ADHMN twistors （perverse sheafs）. Thus we establish the algebra for the ＇t Hooft Hecke operators in the Hecke correspondence of the geometric Langlands program.

Some Kinds of Sheaf Control Problems for Control Systems

Recently, the field of differential equations has been studying in a very abstract method. Instead of considering the behaviour of one solution of a differential equation, one studies its sheaf-solutions in many kinds of properties, for example, the problems of existence, comparison,... of sheaf solutions. In this paper we study some of the problems of controllability for sheaf solutions of control systems.

Vertex Algebra Sheaf Structure on Torus

In this paper, we first give a 1-1 corresponds between torus C/Λ and cubic curve C in P_C^2. As complex manifold, they are isomorphic, therefore we can treat C/Λ as a variety and construction a vertex algebra sheaf on it.

L^(2) EXTENSIONS WITH SINGULAR METRICS ON K?HLER MANIFOLDS

In this paper,we give a survey of our recent results on extension theorems on Kähler manifolds for holomorphic sections or cohomology classes of(pluri)canonical line bundles twisted with holomorphic line bundles equipped with singular metrics,and also discuss their applications and the ideas contained in the proofs.

Penrose Transform on <i>D</i>-Modules, Moduli Spaces and Field Theory

We consider a generalization of the Radon-Schmid transform on coherent D-modules of sheaves of holomorphic complex bundles inside a moduli space, with the purpose of establishing the equivalences among geometric objects (vector bundles) and algebraic objects as they are the coherent D-modules, these last with the goal of obtaining conformal classes of connections of the holomorphic complex bundles. The class of these equivalences conforms a moduli space on coherent sheaves that define solutions in field theory. Also by this way, and using one generalization of the Penrose transform in the context of coherent D-modules we find conformal classes of the space-time that include the heterotic strings and branes geometry.

Penrose Transform on Induced <i>D_G/H</i>-Modules and Their Moduli Stacks in the Field Theory

We consider generalizations of the Radon-Schmid transform on coherent DG/H-Modules, with the intention of obtaining the equivalence between geometric objects (vector bundles) and algebraic objects (D-Modules) characterizing conformal classes in the space-time that determine a space moduli [1] on coherent sheaves for the securing solutions in field theory [2]. In a major context, elements of derived categories like D-branes and heterotic strings are considered, and using the geometric Langlands program, a moduli space is obtained of equivalence between certain geometrical pictures (non-conformal world sheets [3]) and physical stacks (derived sheaves), that establishes equivalence between certain theories of super symmetries of field of a Penrose transform that generalizes the implications given by the Langlands program. With it we obtain extensions of a cohomology of integrals for a major class of field equations to corresponding Hecke category.

Concavity of Minimal L^(2) Integrals Related to Multiplier Ideal Sheaves

In this article,we present the concavity of the minimal L^(2) integrals related to multiplier ideals sheaves on Stein manifolds.As applications,we obtain a necessary condition for the concavity degenerating to linearity,a characterization for 1-dimensional case,and a characterization for the equality in 1-dimensional optimal L^(2) extension problem to hold.

L^(2) Extension and Effectiveness of L^(p) Strong Openness Property

In this note,we present an L^(2) extension approach to the effectiveness result of L^(P) strong openness property of multiplier ideal sheaves.

Review of self-referenced measurement algorithms： Bridging lateral shearing interferometry and multi-probe error separation

With the development of new materials and ultra-precision processing technology, the sizes of mea- sured objects increase, and the requirements for machining accuracy and surface quality become more exacting. The traditional measurement method based on reference datum is inadequate for measuring a high-precision object when the quality of the reference datum is approximately within the same order as that of the object. Self-referenced measurement techniques provide an effective means when the direct reference-based method cannot satisfy the required measurement or calibration accuracy. This paper discusses the reconstruction algorithms for self-referenced measurement and connects lateral shearing interferometry and multi-probe error separation. In lateral shearing interferometry, the reconstruction algorithms are generally categorized into modal or zonal methods. The multi-probe error separation techniques for straightness measurement are broadly divided into two-point and three-point methods. The common features of the lateral sheafing interferometry method and the multi-probe error separation method are identified. We conclude that the reconstruction principle in lateral shearing interferometry is similar to the two-point method in error separation on the condition that no yaw error exists. This similarity may provide a basis or inspiration for the development of both classes of methods.

Multiplier ideal sheaves in complex and algebraic geometry

The application of the method of multiplier ideal sheaves to effective problems in algebraic geometry is briefly discussed. Then its application to the deformational invariance of plurigenera for general compact algebraic manifolds is presented and discussed.Finally its application to the conjecture of the finite generation of the canonical ring is explored, and the use of complex algebraic geometry in complex Neumann estimates is discussed.

A general extension theorem for cohomology classes on non reduced analytic subspaces

The main purpose of this paper is to generalize the celebrated L^2 extension theorem of Ohsawa and Takegoshi in several directions: The holomorphic sections to extend are taken in a possibly singular hermitian line bundle, the subvariety from which the extension is performed may be non reduced, the ambient manifold is K¨ahler and holomorphically convex, but not necessarily compact.

Generalized Derivations with Nilpotent Values on Semiprime Rings

Let R be a semiprime ring, R F be its left Martindale quotient ring and I be an essential ideal of R. Then every generalized derivation μ defined on I can be uniquely extended to a generalized derivation of R F . Furthermore, if there exists a fixed positive integer n such that μ(x) n = 0 for all x ∈I, then μ = 0.

Strong openness of multiplier ideal sheaves and optimal L^2 extension

In this paper, we reveal that our solution of Demailly's strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Koll′ar and Jonsson-Mustat?a implies the truth of twisted versions of the strong openness conjecture; our optimal L^2 extension implies Berndtsson's positivity of vector bundles associated to holomorphic fibrations over a unit disc.

On Tubular Tilting Objects in the Stable Category of Vector Bundles

The present paper focuses on the study of the stable category of vector bundles for the weighted projective lines of weight triple. We find some important triangles in this category and use them to construct tilting objects with tubular endomorphism algebras for the case of genus one via cluster tilting theory.

A Construction of the Rational Function Sheaves on Elliptic Curves

Abstract The authors introduce an effective method to construct the rational function sheaf κ on an elliptic curve E, and further study the relationship between κ and any coherent sheaf on E. Finally, it is shown that the category of all coherent sheaves of finite length on E is completely characterized by κ.