Modular quintics in ℙ4

We will examine the arithmetic of some of the members of a pencil of symmetric quintics in projective 4‐space. We will give evidence for the modularity of some of the exceptional members (even the non‐rigid ones) and give a proof in one rigid case. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Automorphisms of a Generic Jacobian Kummer Surface

For a generic quartic surface F in ³ with sixteen nodes, we shall prove that every birational automorphism of F is induced by a Cremona transformation of ³.     

ℙmax variations related to slaloms

We prove the iteration lemmata, which are the key lemmata to show that extensions by Pmax variations satisfy absoluteness for Π₂-statements in the structure 〈H (ω ₂), ∈, NS_{ω 1}, R 〉 for some set R of reals in L (ℝ), for the following statements: (1) The cofinality of the null ideal is ℵ₁. (2) There exists a good basis of the strong measure zero ideal. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finding for a Surface Σ of Characteristic −4

For each surface Σ, we define max G is a class two graph of maximum degree that can be embedded in . Hence, Vizing's Planar Graph Conjecture can be restated as if Σ is a sphere. In this article, by applying some newly obtained adjacency lemmas, we show that if Σ is a surface of characteristic . Until now, all known satisfy . This is the first case where .     

Relaxed Hyperelastic Curves on a Non-Degenerate Surface

We study the variational problem belonging to a relaxed hyperelastic curve for non-null curve on a non-degenerate surface in Minkowski three-space \({E_{1}^{3}}\) . Firstly, we derive the intrinsic equations for a relaxed hyperelastic curve and we give the necessary condition for being relaxed hyperelastic curve of any non-null geodesic on the surface in \({E_{1}^{3}}\) . Then, we examine this formulation on non-null geodesics of pseudo-plane, pseudo-sphere \({S_{1}^{2}(r) }\) , hyperbolic space \({H_{0}^{2}(r)}\) and pseudo-cylinder \({C_{1}^{2}(r)}\) .     

On Σ1-definable Functions Provably Total in I ∏

We prove the following theorem: Let φ(x) be a formula in the language of the theory PA^? of discretely ordered commutative rings with unit of the form ?yφ′(x,y) with φ′ and let ∈ Δ₀ and let fφ: ? → ? such that f_φ(x) = y iff φ′(x,y) & (?z < y) φ′(x,z). If I ∏ ∈(?x ≥ 0), φ then there exists a natural number K such that I ∏ ? ?y?x(x > y ? ?φ(x) < x^K). Here I ∏₁^? denotes the theory PA^? plus the scheme of induction for formulas φ(x) of the form ?yφ′(x,y) (with φ′) with φ′ ∈ Δ₀.     

Rational Cuspidal Plane Curves of Type (d,d — 3)

In the previous paper [FlZa 2] we classified the rational cuspidal plane curves C with a cusp of multiplicity deg C — 2. In particular, we showed that any such curve can be transformed into a line by Cremona transformations. Here we do the same for rational cuspidal plane curves C with a cusp of multiplicity deg C — 3.     

Differential Modular Forms on Shimura Curves, I

The quotient of a Shimura curve by the isogeny equivalence relation is not an object of algebraic geometry. The paper shows how this quotient space becomes a geometric object in a more general geometry obtained from 'usual algebraic geometry', by adjoining a new operation; this operation looks like a 'Fermat quotient' and should be viewed as an arithmetic analogue of usual derivations.     

2-Harmonic Curves Isometrially Immersed in a Surface

In this paper,the author proves the necessary and sufficient condition for the existenceof 2-harmonically and isometrically immersed curves in a 2-dimensinonal surface N■IE~3.     

Linear Series on 4 – Gonal Curves

Let C be the general 4 – gonal curve of genus g > 6. We investigate the complete linear series on C and the varieties W^r_d(C), and we study the birational models of C in ℙ^r (r ≥ 2) of minimal degree.     

On first order congruences of lines in ℙ4 with irreducible fundamental surface

In this article we study congruences of lines in ?ⁿ, and in particular of order one. After giving general results, we obtain a complete classification in the case of ?⁴ in which the fundamental surface F is in fact a variety, i.e. it is integral, and the congruence is the irreducible set of the trisecant lines of F. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

W2, p estimates of the heat equation in Ω⊂ℝn and the restrictions on ∂Ω

This is an alternative approach of finding the W^{2, p} estimates of the heat equation in a domain, Ω??ⁿ. Methods used in (Acta Math. Sin. 2003; 19 (2):381–396) are expanded to the case of a bounded domain. As a result, milder restrictions are applied to ?Ω than previously required by using the classical singular integral approach. Copyright © 2005 John Wiley & Sons, Ltd.     

Family of Pearson discrete distributions generated by the univariate hypergeometric function 3F2(α1, α2, α3; γ1, γ2; λ)

In this work we present a study of the Pearson discrete distributions generated by the hypergeometric function ₃F₂(α₁, α₂, α₃;γ₁, γ₂; λ), a univariate extension of the Gaussian hypergeometric function, through a constructive methodology. We start from the polynomial coefficients of the difference equation that lead to such a function as a solution. Immediately after, we obtain the generating probability function and the differential equation that it satisfies, valid for any admissible values of the parameters. We also obtain the differential equations that satisfy the cumulants generating function, moments generating function and characteristic function, From this point on, we obtain a relation in recurrences between the moments about the origin, allowing us to create an equation system for estimating the parameters by the moment method. We also establish a classification of all possible distributions of such type and conclude with a summation theorem that allows us study some distributions belonging to this family. © 1997 by John Wiley & Sons, Ltd.     

ω‐saturated quasi‐minimal models of Th(ℚω, +, σ, 0)

We show that (ℚ^ω, +, σ, 0) is a quasi-minimal torsion-free divisible abelian group. After discussing the axiomatization of the theory of this structure, we present its ω-saturated quasi-minimal model. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finding Δ(Σ) for a surface σ of characteristic χ(Σ) = −5

For each surface Σ, we define Δ(Σ) = max{Δ(G)|Gis a class two graph of maximum degree Δ(G) that can be embedded in Σ}. Hence, Vizing's Planar Graph Conjecture can be restated as Δ(Σ) = 5 if Σ is a plane. In this paper, we show that Δ(Σ) = 9 if Σ is a surface of characteristic χ(Σ) = ?5. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:148‐168, 2011     

Σ‐algebraically compact modules and ‐compact cardinals

We prove that the property characterizes Σ‐algebraically compact modules if is not ω‐measurable. Moreover, under a large cardinal assumption, we show that over any ring R where is not ω‐measurable, any free module M of ω‐measurable rank satisfies , hence the assumption on cannot be dropped in general (e.g., over small non‐right perfect rings). In this way, we extend results from a recent paper by Simion Breaz 4 .     

Difference between irregular chaotic patterns of second-order double-loop ΣΔ modulators and second-order interpolative bandpass ΣΔ modulators

In this paper, we find that, by computing the difference between two consecutive state vectors of second-order double-loop sigma-delta modulators (SDMs) and plotting one component of the subtracted vectors against the other component, irregular chaotic patterns will become two vertical lines. By multiplying a matrix on the subtracted vectors, it can be further transformed to two fixed points. However, second-order interpolative bandpass SDMs still exhibit chaotic behaviors after applying the same transformations. Moreover, it is found that the Lyapunov exponent of state vectors of second-order double-loop SDMs is higher than that of second-order interpolative bandpass SDMs, whereas the Lyapunov exponent of transformed vectors becomes negative infinity for second-order double-loop SDMs and increases for second-order interpolative bandpass SDMs. Hence, by examining the occurrence of chaotic behaviors of the transformed vectors of these two SDMs, these two SDMs can be distinguished from their state vectors and their transformed vectors without solving the state equations and knowing the information of input signals.     

Weak -property in Σ-products

It is shown that each Σ-product of paracompact p-spaces has the weak -property.