Rapid decay and polynomial growth for bicrossed products

In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in the state variable. We deal with the case of a fixed terminal time, as well as the case of random terminal time. The need for this type of extension of the classical existence and uniqueness results comes from the desire to provide a probabilistic representation of the solutions of semilinear partial differential equations in the spirit of a nonlinear Feynman-Kac formula. Indeed, in many applications of interest, the nonlinearity is polynomial, e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.

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A Generalization of Strassen’s Theorem on Preordered Semirings

Order ◽

10.1007/s11083-021-09570-7 ◽

2021 ◽

Author(s):

Péter Vrana

Keyword(s):

Compact Hausdorff Space ◽

Polynomial Growth ◽

Hausdorff Space ◽

Continuous Functions ◽

Equivalent Condition ◽

Real Numbers ◽

Asymptotic Spectrum ◽

Ring Of Continuous Functions ◽

Archimedean Property ◽

Locally Compact Hausdorff Space

AbstractGiven a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.

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Role of the adenovirus 72-kDa DNA binding protein in the rapid decay of early viral mRNA

Virology ◽

10.1016/0042-6822(88)90587-9 ◽

1988 ◽

Vol 165 (2) ◽

pp. 438-445 ◽

Cited By ~ 6

Author(s):

Ianis Lazaridis ◽

Alexander Babich ◽

Joseph R. Nevins

Keyword(s):

Dna Binding ◽

Binding Protein ◽

Dna Binding Protein ◽

Viral Mrna ◽

Rapid Decay

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Odour Memory and Odour Hedonics in Children

Perception ◽

10.1068/p180391 ◽

1989 ◽

Vol 18 (3) ◽

pp. 391-396 ◽

Cited By ~ 24

Author(s):

Loredana Hvastja ◽

Lucia Zanuttini

Keyword(s):

Recognition Test ◽

Age Groups ◽

Stimulus Presentation ◽

Emotional Memory ◽

Point Scale ◽

Olfactory Memory ◽

Rapid Decay ◽

Memory Decay ◽

Olfactory Stimuli

The characteristics of olfactory memory during development were investigated and the hypothesis that the pleasantness of smells may be affected by previous associations with pleasant or unpleasant objects or events was tested. This type of emotional memory was compared in the immediate and long-term recognition of olfactory stimuli. Children from three different age groups (mean ages: 6 years 6 months; 8 years 9 months; and 10 years 5 months) were subdivided into two groups. One group was presented with six different odours, each with a slide depicting a pleasant picture. The other group was presented with the odours accompanied by unpleasant pictures. Immediately after stimulus presentation the subjects underwent a recognition test. One month later the subjects underwent a second recognition test, at the end of which they were required to give an evaluation of the pleasantness of each odour on a nine-point scale. At no age level did the pictures matched to the odours affect the recognition score. Olfactory memory varied with age, chiefly because memory decay increased with age, perhaps because of greater proactive interference. With increasing age more rapid decay was set against better immediate recognition. The hypothesis that the hedonic characteristics of odours are partially learned and are affected by events experienced in other modalities was supported.

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Heat equation with singular potential and singular data

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210505000442 ◽

2005 ◽

Vol 135 (4) ◽

pp. 863-886 ◽

Cited By ~ 1

Author(s):

M. Nedeljkov ◽

S. Pilipović ◽

D. Rajter-Ćirić

Keyword(s):

Polynomial Growth ◽

Singular Potential ◽

Function Algebra ◽

Energy Estimates ◽

Generalized Function ◽

Type Condition ◽

Classical Energy ◽

The Cauchy Problem ◽

White Noise Calculus ◽

Singular Data

Nets of Schrödinger C0-semigroups (Sε)ε with the polynomial growth with respect to ε are used for solving the Cauchy problem (∂t − Δ)U + VU = f(t, U), U(0, x) = U0(x) in a suitable generalized function algebra (or space), where V and U0 are singular generalized functions while f satisfies a Lipschitz-type condition. The existence of distribution solutions is proved in appropriate cases by the means of white noise calculus as well as classical energy estimates.

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On stable solutions of the biharmonic problem with polynomial growth

Pacific Journal of Mathematics ◽

10.2140/pjm.2014.270.79 ◽

2014 ◽

Vol 270 (1) ◽

pp. 79-93 ◽

Cited By ~ 13

Author(s):

Hatem Hajlaoui ◽

Abdellaziz Harrabi ◽

Dong Ye

Keyword(s):

Polynomial Growth ◽

Biharmonic Problem ◽

Stable Solutions

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On exponential sums for coefficients of general L-functions

International Journal of Number Theory ◽

10.1142/s1793042121500664 ◽

2021 ◽

pp. 1-22

Author(s):

Fei Hou

Keyword(s):

Functional Equation ◽

General Formula ◽

Fourier Coefficients ◽

Exponential Sums ◽

Exponential Functions ◽

Maass Forms ◽

Maass Form ◽

Rapid Decay

We investigate the order of exponential sums involving the coefficients of general [Formula: see text]-functions satisfying a suitable functional equation and give some new estimates, including refining certain results in preceding works [X. Ren and Y. Ye, Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for [Formula: see text], Sci. China Math. 58(10) (2015) 2105–2124; Y. Jiang and G. Lü, Oscillations of Fourier coefficients of Hecke–Maass forms and nonlinear exponential functions at primes, Funct. Approx. Comment. Math. 57 (2017) 185–204].

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