A Course in Multivariable Calculus and Analysis

Ghorpade, Sudhir R.; Limaye, Balmohan V.

doi:10.1007/978-1-4419-1621-1

Cited by 64 publications

(36 citation statements)

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“…One can prove that the objective function in (16) is monotonically decreasing in α and C x [23]. Furthermore, one can show that the first constraint in (16) can be equivalently expressed as α ≥ a (C x ), where a (C x ) is defined as…”

Section: B Improper Gaussian Signaling Based Designmentioning

confidence: 97%

Overlay Spectrum Sharing using Improper Gaussian Signaling

Amin

Abediseid

Alouini

2016

IEEE J. Select. Areas Commun.

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Abstract-Improper Gaussian signaling (IGS) scheme has been recently shown to provide performance improvements in interference limited networks as opposed to the conventional proper Gaussian signaling (PGS) scheme. In this paper, we implement the IGS scheme in overlay cognitive radio system, where the secondary transmitter broadcasts a mixture of two different signals. The first signal is selected from the PGS scheme to match the primary message transmission. On the other hand, the second signal is chosen to be from the IGS scheme in order to reduce the interference effect on the primary receiver. We then optimally design the overlay cognitive radio to maximize the secondary link achievable rate while satisfying the primary network quality of service requirements. In particular, we consider full and partial channel knowledge scenarios and derive the feasibility conditions of operating the overlay cognitive radio systems. Moreover, we derive the superiority conditions of the IGS schemes over the PGS schemes supported with closed form expressions for the corresponding power distribution and the circularity coefficient and parameters. Simulation results are provided to support our theoretical derivations.

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Section: B Improper Gaussian Signaling Based Designmentioning

confidence: 97%

Overlay Spectrum Sharing using Improper Gaussian Signaling

Amin

Abediseid

Alouini

2016

IEEE J. Select. Areas Commun.

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“…We now show that in the current situation this implies that χ • det | C 1 K l ′ = 1 by showing that det(C 1 K l ) ⊇ det(C 1 K l ′ ). To prove the latter inclusion, we first prove that (14) det(C l ′ K l ) = 1 + p l ′ .…”

Section: 4mentioning

confidence: 99%

“…In fact, we have only found two sources containing this (in the two-variable case), but both of them misstate the result. Indeed, [7,Section 32 (2)] and [14,Proposition 7.57] state that given a non-negative decreas-…”

Section: Introductionmentioning

confidence: 99%

Representation growth of compact linear groups

Häsä

Stasinski

2019

Trans. Amer. Math. Soc.

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We study the representation growth of simple compact Lie groups and of SLn(O), where O is a compact discrete valuation ring, as well as the twist representation growth of GLn(O). This amounts to a study of the abscissae of convergence of the corresponding (twist) representation zeta functions.We determine the abscissae for a class of Mellin zeta functions which include the Witten zeta functions. As a special case, we obtain a new proof of the theorem of Larsen and Lubotzky that the abscissa of Witten zeta functions is r/κ, where r is the rank and κ the number of positive roots.We then show that the twist zeta function of GLn(O) exists and has the same abscissa of convergence as the zeta function of SLn(O), provided n does not divide char O. We compute the twist zeta function of GL 2 (O) when the residue characteristic p of O is odd, and approximate the zeta function when p = 2 to deduce that the abscissa is 1. Finally, we construct a large part of the representations of SL 2 (Fq[[t]]), q even, and deduce that its abscissa lies in the interval [1, 5/2]. that the abscissa of ζ SL2(O) (s) is 1 whenever char O = 0. Our computations of the abscissa ofζ GL2(O) (s) in this case, together with Proposition 3.4, give a new proof of this fact. We also show that the abscissa ofζwith q even. This does not follow from any previously known results and our computation is substantially harder than in the cases where char O = 2.The zeta function of SL 2 (F q [[t]]), q even. In Section 5, we assume that char O = 2, that is, O = F q [[t]] with q even. We give a Clifford theory construction of the representations of SL 2 (F q [[t]]/(t r )) for r even, which is completely explicit apart from the order of certain finite groups V (β, θ) (see Definition 5.7) and certain integers c ∈ {1, 2, 3} (see Lemma 4.21). We use this construction to approximate the zeta function ζ SL 2 (O) (s), and show that its abscissa lies between 1 and 5/2 (Theorem 5.9). The lower bound 1 follows from a general result of Larsen and Lubotzky [21, Proposition 6.6], but we give an independent proof of this.Section 6 is devoted to a proof of Lemma 4.21, which is crucial for our results aboutζ GL 2 (Fq[[t]]) (s) and ζ SL 2 (Fq[[t]]) (s) when q is even. The lemma gives the number of solutions, up to a factor c ∈ {1, 2, 3}, in F q [[t]]/(t i ), i ≥ 1, to the equation∆ τ ] mod (t) is a scalar plus a regular nilpotent matrix. The number of solutions depends in a delicate way on a new invariant, which we call the odd depth, of the twist orbit (i.e., orbit modulo scalars) of the matrix [ 0 1∆ τ ] (see Definition 4.19). Remark. After the present paper had been accepted for publication, Hassain M and Pooja Singla [24] announced results about the representations of SL 2 (O), p = 2, which in particular imply that the abscissa of ζ SL2(Fq[[t]]) (s) is 1. Notation.We let N stand for the set of natural numbers, not including 0.In Sections 4 and 5, we will use the Vinogradov notation f (r) ≪ g(r) for two functions f (r), g(r) of r (or of l = ⌈r/2⌉). Note that f (r) ≪ g(r) is equiv...

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“…In order to determine the absolute convergence of double series, one can use the ratio test for double series (see, for example, [7] (Corollary 7.35)): Proposition 5. Let (a k,l ) be a double sequence of nonzero real numbers such that either |a k,l+1 |/|a k,l | → a or |a k+1,l |/|a k,l | →ã as (k, l) → (∞, ∞), where a,ã ∈ R ∪ ∞.…”

Section: Propositionmentioning

confidence: 99%

On a Reduction Formula for a Kind of Double q-Integrals

Liu

2016

Symmetry

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Using the q-integral representation of Sears' nonterminating extension of the q-Saalschütz summation, we derive a reduction formula for a kind of double q-integrals. This reduction formula is used to derive a curious double q-integral formula, and also allows us to prove a general q-beta integral formula including the Askey-Wilson integral formula as a special case. Using this double q-integral formula and the theory of q-partial differential equations, we derive a general q-beta integral formula, which includes the Nassrallah-Rahman integral as a special case. Our evaluation does not require the orthogonality relation for the q-Hermite polynomials and the Askey-Wilson integral formula.

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A Course in Multivariable Calculus and Analysis

Cited by 64 publications

References 0 publications

Overlay Spectrum Sharing using Improper Gaussian Signaling

Overlay Spectrum Sharing using Improper Gaussian Signaling

Representation growth of compact linear groups

On a Reduction Formula for a Kind of Double q-Integrals

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