Single cell discretization of O(kh2 + h4) for the estimates of ${\partial u}\over{\partial n}$ for the two‐space dimensional quasi‐linear parabolic equation

In this article, new stable two‐level explicit difference methods of O(kh² + h⁴) for the estimates of for the two‐space dimensional quasi‐linear parabolic equation are derived, where k > 0 and h > 0 are grid sizes in time and space directions, respectively. We use a single computational cell for the methods, which are applicable to the problems both in cartesian and polar coordinates. The proposed methods have the simplicity in nature and employ the same marching type technique of solution. Numerical results obtained by the proposed methods for several different problems were compared with the exact solutions. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 250–261, 2001     

Single cell finite difference approximations of O(kh2 + h4) for ∂u/∂x for one space dimensional nonlinear parabolic equation

We report a new two‐level explicit finite difference method of O(kh² + h⁴) using three spatial grid points for the numerical solution of for the solution of one‐space dimensional nonlinear parabolic partial differential equation subject to appropriate initial and Dirichlet boundary conditions. The method is shown to be unconditionally stable when applied to a linear equation. The proposed method is applicable to the problems both in cartesian and polar coordinates. Numerical examples are provided to demonstrate the efficiency and accuracy of the method discussed. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 408–415, 2000     

Single‐cell discretization of O(kh2 + h4) for ∂u/∂n for three‐space dimensional mildly quasi‐linear parabolic equation

In this article, using a single computational cell, we report some stable two‐level explicit finite difference approximations of O(kh² + h⁴) for ?u/?n for three‐space dimensional quasi‐linear parabolic equation, where h > 0 and k > 0 are mesh sizes in space and time directions, respectively. When grid lines are parallel to x‐, y‐, and z‐coordinate axes, then ?u/?n at an internal grid point becomes ?u/?x, ?u/?y, and ?u/?z, respectively. The proposed methods are also applicable to the polar coordinates problems. The proposed methods have the simplicity in nature and use the same marching type of technique of solution. Stability analysis of a linear difference equation and computational efficiency of the methods are discussed. The results of numerical experiments are compared with exact solutions. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 327–342, 2003.     

Categorical Abstract Algebraic Logic: Algebraic Semantics for (\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\bf{\pi }$\end{document})‐Institutions

Various aspects of the work of Blok and Rebagliato on the algebraic semantics for deductive systems are studied in the context of logics formalized as π‐institutions. Three kinds of semantics are surveyed: institution, matrix (system) and algebraic (system) semantics, corresponding, respectively, to the generalized matrix, matrix and algebraic semantics of the theory of sentential logics. After some connections between matrix and algebraic semantics are revealed, it is shown that every (finitary) N‐rule based extension of an N‐rule based π‐institution possessing an algebraic semantics also possesses an algebraic semantics. This result abstracts one of the main theorems of Blok and Rebagliato. An attempt at a Blok‐Rebagliato‐style characterization of those π‐institutions with a mono‐unary category of natural transformations on their sentence functors having an algebraic semantics is also made. Finally, a necessary condition for a π‐institution to possess an algebraic semantics is provided.     

The Quasi-Canonical Solution Operator to $${\bar \partial }$$ Restricted to the Fock-Space

We consider the solution operator S: ℱ_μ,(p,q) → L ²(μ)_{(p, q)} to the -operator restricted to forms with coefficients in ℱ_μ = {f: f is entire and ∫_ℂn |f(z)|² dμ(z) < ∞}. Here ℱ_μ,(p,q) denotes (p,q)-forms with coefficients in ℱ_μ, L ²(μ) is the corresponding L ²-space and μ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula S to . This solution operator will have the property Sv ⊥ ℱ_(p,q) ∀v ∈ ℱ_(p,q+1). As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness of commutators of Toeplitz-operators : ℱ_μ → L ²(μ).     

Topological invariants from quantum group $$\mathcal {U}_{\xi }\mathfrak {sl}(2|1)$$ at roots of unity

In this article we construct link invariants and 3-manifold invariants from the quantum group associated with the Lie superalgebra \(\mathfrak {sl}(2|1)\). The construction is based on nilpotent irreducible finite dimensional representations of quantum group \(\mathcal {U}_{\xi }\mathfrak {sl}(2|1)\) where \(\xi \) is a root of unity of odd order. These constructions use the notion of modified trace and relative \( G \)-modular category of previous authors.     

Global solution with \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$ \mathcal {C}^{k}$\end{document}‐estimates for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\bar{\partial }$\end{document}‐equation on q‐convex intersections

We construct a global solution with $\mathcal {C}^{k}$‐estimates for the $\bar{\partial }$‐equation on q‐convex intersections.     

Explicit sparse almost‐universal graphs for ${\bf {{\cal G}(n, {k \over n})}}$

For any integer n, let be a probability distribution on the family of graphs on n vertices (where every such graph has nonzero probability associated with it). A graph Γ is ‐almost‐universal if Γ satisifies the following: If G is chosen according to the probability distribution , then G is isomorphic to a subgraph of Γ with probability 1 ‐ . For any p ∈ [0,1], let (n,p) denote the probability distribution on the family of graphs on n vertices, where two vertices u and v form an edge with probability p, and the events {u and v form an edge}; u,v ∈ V (G) are mutually independent. For k ≥ 4 and n sufficiently large we construct a ‐almost‐universal‐graph on n vertices and with O(n)polylog(n) edges, where q = ? ? for such k ≤ 6, and where q = ? ? for k ≥ 7. The number of edges is close to the lower bound of Ω( ) for the number of edges in a universal graph for the family of graphs with n vertices and maximum degree k. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Geometric Optimization Problems Likely Not Contained in $$\mathbb{A}\mathbb{P}\mathbb{X}$$

   Abstract. Maximizing geometric functionals such as the classical l _p -norms over polytopes plays an important role in many applications, hence it is desirable to know how efficiently the solutions can be computed or at least approximated. While the maximum of the l _∞ -norm over polytopes can be computed in polynomial time, for 2≤ p < ∞ the l _p -norm-maxima cannot be computed in polynomial time within a factor of 1.090 , unless P=NP. This result holds even if the polytopes are centrally symmetric parallelotopes. Quadratic Programming is a problem closely related to norm-maximization, in that in addition to a polytope P ⊂ R ⁿ , numbers c _ij , 1≤ i≤ j≤ n , are given and the goal is to maximize Σ _{1≤ i≤ j≤ n} c _ij x _i x _j over P . It is known that Quadratic Programming does not admit polynomial-time approximation within a constant factor, unless P=NP. Using the observation that the latter result remains true even if the existence of an integral optimal point is guaranteed, in this paper it is proved that analogous inapproximability results hold for computing the l _p -norm-maximum and various l _p -radii of centrally symmetric polytopes for 2≤ p < ∞.     

Single‐cell compact finite‐difference discretization of order two and four for multidimensional triharmonic problems

In this article, we discuss finite‐difference methods of order two and four for the solution of two‐and three‐dimensional triharmonic equations, where the values of u,(?²u/?n²) and (?⁴u/?n⁴) are prescribed on the boundary. For 2D case, we use 9‐ and for 3D case, we use 19‐ uniform grid points and a single computational cell. We introduce new ideas to handle the boundary conditions and do not require to discretize the boundary conditions at the boundary. The Laplacian and the biharmonic of the solution are obtained as byproduct of the methods. The resulting matrix system is solved by using the appropriate block iterative methods. Computational results are provided to demonstrate the fourth‐order accuracy of the proposed methods. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Non‐well‐founded extensions of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {V}$\end{document}

This paper describes a new and user‐friendly method for constructing models of non‐well‐founded set theory. Given a sufficiently well‐behaved system θ of non‐well‐founded set‐theoretic equations, we describe how to construct a model M_θ for $\mathsf {ZFC}^-$ in which θ has a non‐degenerate solution. We shall prove that this M_θ is the smallest model for $\mathsf {ZFC}^-$ which contains $\mathbf {V}$ and has a non‐degenerate solution of θ.     

Actions of $$\mathbf {SAut}\varvec{(F_n)}$$

We study actions of SAut\((F_n)\), the unique subgroup of index two in the automorphism group of a free group of rank n, and obtain rigidity results for its representations. In particular, we show that every smooth action of SAut\((F_n)\) on a low dimensional torus is trivial.     

Global boundary regularity for the $ \bar \partial $‐equation on q‐pseudoconvex domains

We introduce a notion of q ‐pseudoconvex domain of new type for a bounded domain of ?ⁿ and prove that for given a ‐closed (p, r)‐form, r ≥ q, that is smooth up to the boundary, there exists a (p, r – 1)‐form smooth up to the boundary which is a solution of ‐equation on a bounded q ‐pseudoconvex domain. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On unit power integral bases of $$\mathbb{Z}\left[ {\sqrt[4]{m}} \right]$$

Let m ≠ 0 be an integer which is not a perfect square and consider number fields of the form \(\mathbb{Q}\left[ {\sqrt[4]{m}} \right]\). We characterize all orders of the form \(\mathbb{Z}\left[ {\sqrt[4]{m}} \right]\) which admit a unit power integral basis, i.e., there exists a unit ε such that 1, ε, ε ² and ε ³ is an integral basis of \(\mathbb{Z}\left[ {\sqrt[4]{m}} \right]\).     

Existence of multiple positive solutions for - \Delta u = \lambda u + u^{\frac{{N + 2}}{{N - 2}}} + \mu f\left( x \right)*

In this paper, we discuss the existence and nonexistence of solutions for the problem311-2 where Ω is a bounded smoothness domain inR ^N, γ ε R¹, μ>-0,f(x) is a given non-negative function. Some interesting resultus have been obtained. This work was completed in Institute of Math. Academia Sinica as a visiting scholar     

$${\overline{\partial}}$$ -problem for an overdetermined system with two higher dimensional variables

Our main purpose is to describe necessary and sufficient conditions for the solvability of the \({{\overline\partial}}\) -problem for biregular complex Clifford algebra valued functions of two higher dimensional variables in even dimensional Euclidean spaces. Since the biregular function theory falls naturally from the so-called isotonic Clifford analysis, the main idea of the proof is based on the notion of biregular-conjugate harmonic functions to be introduced here. Besides, whenever the \({{\overline\partial}}\) -problem is solvable, we give the general solution of it in a quite explicit form.     

On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$

Let f be an endomorphism of \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} and ν be an f-invariant measure with positive Lyapunov exponents (λ ₁, . . . , λ _k ). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f, the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k = 2, it implies that the Hausdorff dimension of μ is estimated by dim_Hm 3 [(log d)/(l₁)] + [(log d)/(l₂)]{{\rm dim}_\mathcal{H}\mu \geq {{\rm log} d \over \lambda_1} + {{\rm log} d \over \lambda_2}}, which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the ν-generic inverse branches of f ⁿ in \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} and a normalization theorem for the ν-generic inverse branches of f ⁿ .     

Automorphisms of higher‐dimensional Hadamard matrices

This article derives from first principles a definition of equivalence for higher‐dimensional Hadamard matrices and thereby a definition of the automorphism group for higher‐dimensional Hadamard matrices. Our procedure is quite general and could be applied to other kinds of designs for which there are no established definitions for equivalence or automorphism. Given a two‐dimensional Hadamard matrix H of order ν, there is a Product Construction which gives an order ν proper n‐dimensional Hadamard matrix P⁽ⁿ⁾(H). We apply our ideas to the matrices P⁽ⁿ⁾(H). We prove that there is a constant c > 1 such that any Hadamard matrix H of order ν > 2 gives rise via the Product Construction to cν inequivalent proper three‐dimensional Hadamard matrices of order ν. This corrects an erroneous assertion made in the literature that ”P⁽ⁿ⁾(H) is equivalent to “P⁽ⁿ⁾(H′) whenever H is equivalent to H′.” We also show how the automorphism group of P⁽ⁿ⁾(H) depends on the structure of the automorphism group of H. As an application of the above ideas, we determine the automorphism group of P⁽ⁿ⁾(Hk) when Hk is a Sylvester Hadamard matrix of order 2^k. For ν = 4, we exhibit three distinct families of inequivalent Product Construction matrices P⁽ⁿ⁾(H) where H is equivalent to H₂. These matrices each have large but non‐isomorphic automorphism groups. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 507–544, 2008     

Markov subshifts and partial representation of $$\mathbb{F}_{\text n}$$

In this paper we fix a set * of positive elements of the free group (e. g. the set of finite words occurring in a Markov subshift) as well as n partial isometries on a Hilbert space H. Based on these we define a map S : which we prove to be a partial representation of on H under certain conditions studied by Matsumoto.*Supported by Capes.