A Mountain Pass to the Jacobian Conjecture

AbstractThis paper presents an approach to injectivity theorems via the Mountain Pass Lemma and raises an open question. The main result of this paper (Theorem 1.1) is proved by means of the Mountain Pass Lemma and states that if the eigenvalues of are uniformly bounded away from zero for x ∊ Rn, where is a class C1 map, then F is injective. This was discovered in a joint attempt by the authors to prove a stronger result conjectured by the first author: Namely, that a sufficient condition for injectivity of class C1 maps F of Rn into itself is that all the eigenvalues of F′(x) are bounded away from zero on Rn. This is stated as Conjecture 2.1. If true, it would imply (via Reduction-of-Degree) injectivity of polynomial mapssatisfying the hypothesis, det F′(x) ≡ 1, of the celebrated Jacobian Conjecture (JC) of Ott-Heinrich Keller. The paper ends with several examples to illustrate a variety of cases and known counterexamples to some natural questions.

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Ont he monodromy representation of polynomial maps in n variables

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.39.2002.3-4.7 ◽

2002 ◽

Vol 39 (3-4) ◽

pp. 361-367

Author(s):

A. Némethi ◽

I. Sigray

Keyword(s):

Cohomology Group ◽

Jacobian Conjecture ◽

Natural Basis ◽

Generic Fiber ◽

Monodromy Representation ◽

Polynomial Maps ◽

Polynomial Map

For a non-constant polynomial map f: Cn?Cn-1 we consider the monodromy representation on the cohomology group of its generic fiber. The main result of the paper determines its dimension and provides a natural basis for it. This generalizes the corresponding results of [2] or [10], where the case n=2 is solved. As applications, we verify the Jacobian conjecture for (f,g) when the generic fiber of f is either rational or elliptic. These are generalizations of the corresponding results of [5], [7], [8], [11] and [12], where the case n=2 is treated.

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Stability of On-Line Bin Packing with Random Arrivals and Long-Run-Average Constraints

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800001753 ◽

1990 ◽

Vol 4 (4) ◽

pp. 447-460 ◽

Cited By ~ 7

Author(s):

Coastas Courcobetis ◽

Richard Weber

Keyword(s):

Bin Packing ◽

Random Processes ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Expected Number ◽

Long Run ◽

Different Types ◽

On Line ◽

Necessary And Sufficient ◽

Uniformly Bounded

Items of various types arrive at a bin-packing facility according to random processes and are to be combined with other readily available items of different types and packed into bins using one of a number of possible packings. One might think of a manufacturing context in which randomly arriving subassemblies are to be combined with subassemblies from an existing inventory to assemble a variety of finished products. Packing must be done on-line; that is, as each item arrives, it must be allocated to a bin whose configuration of packing is fixed. Moreover, it is required that the packing be managed in such a way that the readily available items are consumed at predescribed rates, corresponding perhaps to optimal rates for manufacturing these items. At any moment, some number of bins will be partially full. In practice, it is important that the packing be managed so that the expected number of partially full bins remains uniformly bounded in time. We present a necessary and sufficient condition for this goal to be realized and describe an algorithm to achieve it.

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Polynomial rings over NLI rings need not be NLI

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.52.2015.1.1305 ◽

2015 ◽

Vol 52 (1) ◽

pp. 129-133 ◽

Cited By ~ 1

Author(s):

Weixing Chen

Keyword(s):

Polynomial Ring ◽

Polynomial Rings ◽

Sufficient Condition ◽

Open Question

It is proved that there exists an NI ring R over which the polynomial ring R[x] is not an NLI ring. This answers an open question of Qu and Wei (Stud. Sci. Math. Hung., 51(2), 2014) in the negative. Moreover a sufficient condition of R[x] to be an NLI ring is included for an NLI ring R.

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Decaying Solutions Of 2mth Order Elliptic Problems

Canadian Journal of Mathematics ◽

10.4153/cjm-1991-027-8 ◽

1991 ◽

Vol 43 (3) ◽

pp. 449-460 ◽

Cited By ~ 3

Author(s):

W. Allegretto ◽

L. S. Yu

Keyword(s):

Positive Solution ◽

Elliptic Problem ◽

Mountain Pass Theorem ◽

Elliptic Problems ◽

Mountain Pass ◽

Weighted Spaces ◽

Semilinear Elliptic Problem ◽

Semilinear Elliptic ◽

Regularity Estimates ◽

Open Question

AbstractWe consider a semilinear elliptic problem , (n > 2m). Under suitable conditions on f, we show the existence of a decaying positive solution. We do not employ radial arguments. Our main tools are weighted spaces, various applications of the Mountain Pass Theorem and LP regularity estimates of Agmon. We answer an open question of Kusano, Naito and Swanson [Canad. J. Math. 40(1988), 1281-1300] in the superlinear case: , and improve the results of Dalmasso [C. R. Acad. Sci. Paris 308(1989), 411-414] for the case .

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On the Existence of Nontrivial Solutions for Nonlocal Elliptic Kirchhoff Type Problems with Nonlinear Boundary Conditions

Fasciculi Mathematici ◽

10.1515/fascmath-2015-0022 ◽

2015 ◽

Vol 55 (1) ◽

pp. 183-188

Author(s):

S. H. Rasouli ◽

B. Salehi

Keyword(s):

Boundary Conditions ◽

Nonlinear Boundary ◽

Type Equation ◽

Nonlinear Boundary Conditions ◽

Mountain Pass ◽

Mountain Pass Lemma ◽

Nontrivial Solutions ◽

Kirchhoff Type ◽

Kirchhoff Type Equation ◽

Kirchhoff Type Problems

Abstract In this paper, by using the Mountain Pass Lemma, we study the existence of nontrivial solutions for a nonlocal elliptic Kirchhoff type equation together with nonlinear boundary conditions.

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An effective study of polynomial maps

Journal of Algebra and Its Applications ◽

10.1142/s0219498817501419 ◽

2016 ◽

Vol 16 (08) ◽

pp. 1750141 ◽

Cited By ~ 1

Author(s):

Elżbieta Adamus ◽

Paweł Bogdan ◽

Teresa Crespo ◽

Zbigniew Hajto

Keyword(s):

Affine Space ◽

Finite Sequence ◽

Jacobian Conjecture ◽

Equivalent Condition ◽

Effective Algorithm ◽

Polynomial Maps ◽

Dimension Formula ◽

Polynomial Map ◽

Equivalent Statement

In this paper, using an effective algorithm, we obtain an equivalent statement to the Jacobian Conjecture. For a polynomial map [Formula: see text] on an affine space of dimension [Formula: see text] over a field of characteristic [Formula: see text], we define recursively a finite sequence of polynomial maps. We give an equivalent condition to the invertibility of [Formula: see text] as well as a formula for [Formula: see text] in terms of this finite sequence of polynomial maps. Some examples illustrate the effective aspects of our approach.

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