Asymptotic Properties of Polynomials and Algebraic Functions of Several Variables

Gorin, E. A.

doi:10.1070/rm1961v016n01abeh004100

Cited by 49 publications

(24 citation statements)

References 3 publications

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“…Now the hypothesis (i 1 ) and [3,Theorem 3.1] show that there are two constants, d positive and v rational such that…”

Section: Technical Preliminarymentioning

confidence: 98%

Time Decay Rates for Undamped Constant Coefficients Linear Partial Differential Equations

Kelemen

1996

Journal of Differential Equations

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this work is dedicated to professor isaac horowitz, on his 75th birthdayThe zeros of the characteristic polynomial of many important equations in mathematical physics (e.g. the wave equation, the Schro dinger equation) are situated on the imaginary axis. This causes a very slow decay in the time variable of the solution driven by initial conditions of such equations. In this article we show that by displacing (by feedback) the zeros to the left of the imaginary axis so that they approach this axis asymptotically, one can change drastically the above situation. Indeed, one can achieve a polynomial decay of arbitrary degree in the time variable of the absolute value of the solution, uniformly in the space variable, provided the initial conditions are smooth enough. For such equations``smoothness in space implies decay in time.'' The relation between smoothness and decay is established in a quantitative way. The systems under investigation are linear undamped partial differential equations with constant coefficients, in multidimensional space. We provide also natural conditions for the exponential decay of the absolute value of the solution.

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“…Now the hypothesis (i 1 ) and [3,Theorem 3.1] show that there are two constants, d positive and v rational such that…”

Section: Technical Preliminarymentioning

confidence: 98%

Time Decay Rates for Undamped Constant Coefficients Linear Partial Differential Equations

Kelemen

1996

Journal of Differential Equations

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“…It is also important that polynomial images of semi-algebraic sets are again semi-algebraic (the Seidenberg-Tarski theorem; see [ l l ] , [5]). More generally, a map f : S -+ R", with S C V, is said to be rational nonsingular if there are finitely many Zariski-open sets Z, c V and rational maps f,: Z, + R " such that f , is nonsingular on Z, (f, = P,/Q, with Q, # 0 in Z,), the 2, cover S, and f , = f on 2, n S .…”

Section: Semi-algebraic Setsmentioning

confidence: 99%

A canonical approach to multiplicity formulas for induced and restricted representations of nilpotent lie groups

Corwin

Greenleaf

1988

Comm Pure Appl Math

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“…As Proposition 5 will show, this formula will also hold when dim x > 1. 6. We now remove the hypothesis that dimx = 1> and prove Theorem 5 in full generality.…”

Section: -1 Je?u)mentioning

confidence: 99%

Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups

Corwin¹,

Greenleaf²,

Grélaud³

1987

Trans. Amer. Math. Soc.

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Let K K be a Lie subgroup of the connected, simply connected nilpotent Lie group G G , and let k \mathfrak {k} , g \mathfrak {g} be the corresponding Lie algebras. Suppose that σ \sigma is an irreducible unitary representation of K K . We give an explicit direct integral decomposition of Ind k → G σ {\operatorname {Ind} _{k \to G}}\sigma into irreducibles. The description uses the Kirillov orbit picture, which gives a bijection between G ∧ G^\wedge and the coadjoint orbits in g ∗ {\mathfrak {g}^{\ast }} (and similarly for K ∧ , k ∗ K^\wedge ,\,{\mathfrak {k}^{\ast }} ). Let P : k ∗ → g ∗ P:{\mathfrak {k}^{\ast }} \to {\mathfrak {g}^{\ast }} be the canonical projection, let O σ ⊂ k ∗ {\mathcal {O}_\sigma } \subset {\mathfrak {k}^{\ast }} be the orbit corresponding to σ \sigma , and, for π ∈ G ∧ \pi \in G^\wedge , let O π ⊂ g ∗ {\mathcal {O}_\pi } \subset {\mathfrak {g}^{\ast }} be the corresponding orbit. The main result of the paper says essentially that π ∈ G ∧ \pi \in G^\wedge appears in the direct integral iff P − 1 ( O σ ) {P^{ - 1}}({\mathcal {O}_\sigma }) meets O π {\mathcal {O}_\pi } ; the multiplicity of π \pi is the number of Ad ∗ ( K ) {\operatorname {Ad} ^{\ast }}(K) -orbits in O π ∩ P − 1 ( O σ ) {\mathcal {O}_\pi } \cap {P^{ - 1}}({\mathcal {O}_\sigma }) . There is also a natural description of the measure class in the integral.

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Asymptotic Properties of Polynomials and Algebraic Functions of Several Variables

Cited by 49 publications

References 3 publications

Time Decay Rates for Undamped Constant Coefficients Linear Partial Differential Equations

Time Decay Rates for Undamped Constant Coefficients Linear Partial Differential Equations

A canonical approach to multiplicity formulas for induced and restricted representations of nilpotent lie groups

Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups

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