Patrick Jeronne Martinez - Academia.edu (original) (raw)
Papers by Patrick Jeronne Martinez
SIAM Journal on Control and Optimization, 2019
For α ∈ (0, 2) we study the null controllability of the parabolic operator P u = ut − (|x| α ux)x... more For α ∈ (0, 2) we study the null controllability of the parabolic operator P u = ut − (|x| α ux)x (1 < x < 1), which degenerates at the interior point x = 0, for locally distributed controls acting only one side of the origin (that is, on some interval (a, b) with 0 < a < b < 1). Our main results guarantees that P is null controllable if and only if it is weakly degenerate, that is, α ∈ (0, 1). So, in order to steer the system to zero, one needs controls to act on both sides of the point of degeneracy in the strongly degenerate case α ∈ [1, 2). Our approach is based on spectral analysis and the moment method. Indeed, we completely describe the eigenvalues and eigenfunctions of the associated stationary operator in terms of Bessel functions and their zeroes for both weakly and strongly degenerate problems. Hence, we obtain lower L 2 bounds for the eigenfunctions on the control region in the case α ∈ [0, 1) and deduce the lack of observability in the case of α ∈ [1, 2). We also provide numerical evidence to illustrate our theoretical results.
Le Journal de Physique Colloques, 1988
The ternary compound HoRhySnz shows a frequency dependence of the susceptibility like a spin glas... more The ternary compound HoRhySnz shows a frequency dependence of the susceptibility like a spin glass. The spin-glass behaviour is confirmed by the "viscous" response of AC and DC susceptibility to a steplike variation of magnetic field or temperature.
ESAIM: Proceedings, 2000
We consider the problem of the wave equation with Neumann boundary condition damped by a locally ... more We consider the problem of the wave equation with Neumann boundary condition damped by a locally distributed linear damping a(x)u. When the damping region ω := {x, a(x) ≥ α > 0} contains a neighborhood of the boundary of the domain, E. Zuazua proved that the energy decays exponentially to zero. Using a piecewise multiplier method introduced by K. Liu, we prove that the energy decays exponentially to zero under weaker geometrical conditions. We give explicit examples when the domain is a polyhedron, and in the case of a disc. The proof is based on the construction of multipliers adapted to the geometrical conditions.
Portugaliae Mathematica, 2011
ABSTRACT In this paper, we study the null controllability of degenerate semilinear cas-cade parab... more ABSTRACT In this paper, we study the null controllability of degenerate semilinear cas-cade parabolic systems with one control force. The key tool is the Carleman estimates developed recently for degenerate one dimension parabolic equations. We develop a Carle-man estimate for these systems and then an observability inequality for the linear adjoint system. We conclude by linearization and fixed point arguments. Portugaliae Mathemat-ica.
Transactions of the American Fisheries Society, 2007
Nonnative fishes represent a significant impediment to the recovery of imperiled fishes, includin... more Nonnative fishes represent a significant impediment to the recovery of imperiled fishes, including those endemic to the Colorado River in the southwestern United States. Efforts to control nonindigenous fish abundance in the upper Colorado River basin have been unsuccessful owing in part to lack of knowledge regarding nonnative fish recruitment sources. We determined the source habitat (floodplain pond versus riverine habitats) for nonnative centrarchid fishes (largemouth bass Micropterus salmoides, green sunfish Lepomis cyanellus, bluegill L. macrochirus, and black crappie Pomoxis nigromaculatus) in the upper Colorado River using stable hydrogen isotopic composition (dD) and strontium : calcium (Sr:Ca) ratios in fish otoliths as natural markers of environmental history. Stable hydrogen isotope analysis revealed that 59% of centrarchids exhibited the otolith core signatures expected for riverine-origin fish, while 22% had emigrated from floodplain ponds and 19% were of uncertain origin. Strontium : calcium ratio data were consistent with the dD assays and indicated that relatively few fish immigrated to the river from high-salinity habitats. Black crappie was the only species that originated primarily from floodplain ponds. Efforts to control the abundance of most of the fishes included in this study should be concentrated in riverine habitats given the hydrologic conditions (below-average river discharge) present during our study. However, the proportion of pond-origin fish increased with fish age, which, coupled with historical river discharge data, suggested that floodplain pond contributions to riverine nonnative fish populations fluctuate with the interannual variations in flow regime and river-pond connectivity. Our results are the first to demonstrate the utility of dD as a natural marker of fish environmental history that will probably provide valuable insights into the management of fish in other environments.
SIAM Journal on Control and Optimization, 2000
Abstract. We consider the wave equation damped by a nonlinear boundary velocity feedback q(ut). F... more Abstract. We consider the wave equation damped by a nonlinear boundary velocity feedback q(ut). First we consider the case where q has a linear growth at infinity. We prove that the usual decay rate estimates proved by Nakao [Differential Integral Equations, 8 (1995), pp. 681 ...
SIAM Journal on Control and Optimization, 2003
We are interested in controllability problems of equations coming from a boundary layer model. We... more We are interested in controllability problems of equations coming from a boundary layer model. We simplify the problem by considering only equations with constant coefficients. The problem is described by a degenerate parabolic equation (a linearized Crocco-type equation) where phenomena of diffusion and transport are coupled. First we give a geometric characterization of the influence domain of a locally distributed con- trol. Then we prove regional null controllability results on this domain. The proof is based on an adequate observability inequality for the homogeneous adjoint problem. This inequality is obtained by decomposition of the space-time domain and Carleman-type estimates along characteristics. In the second part of this paper, we treat the case of a boundary control.
SIAM Journal on Control and Optimization, 2008
Given α ∈ [0, 2) and f ∈ L 2 ((0, T) × (0, 1)), we derive new Carleman estimates for the degenera... more Given α ∈ [0, 2) and f ∈ L 2 ((0, T) × (0, 1)), we derive new Carleman estimates for the degenerate parabolic problem wt + (x α wx)x = f , where (t, x) ∈ (0, T) × (0, 1), associated to the boundary conditions w(t, 1) = 0 and w(t, 0) = 0 if 0 ≤ α < 1 or (x α wx)(t, 0) = 0 if 1 ≤ α < 2. The proof is based on the choice of suitable weighted functions and Hardy-type inequalities. As a consequence, for all 0 ≤ α < 2 and ω ⊂⊂ (0, 1), we deduce null controllability results for the degenerate one-dimensional heat equation ut − (x α ux)x = hχω with the same boundary conditions as above.
SIAM Journal on Control and Optimization, 2005
Motivated by several works on ordinary differential equations, we are interested in the asymptoti... more Motivated by several works on ordinary differential equations, we are interested in the asymptotic stability of intermittently controlled partial differential equations. We give a condition of asymptotic stability for second-order evolution equations uniformly damped by an on/off feedback. This result extends to the case of partial differential equations a previous result of R. A. Smith concerning ordinary differential equations.
North American Journal of Fisheries Management, 2009
The Smith-Root (S-R) series of GPP (gas-powered pulsator) electrofishers are widely used in boat-... more The Smith-Root (S-R) series of GPP (gas-powered pulsator) electrofishers are widely used in boat-mounted electrofishing systems. Sampling in waters having different conductivities requires adjustment of the electrofisher controls to maintain consistent output power. However, the current meter supplied on GPP electrofishers provides an average rather than a peak measurement, and because no voltmeter is included, determining peak power output is not possible. As part of an overall effort to promote standardization of electrofishing operations, we used static electrical loads to measure the output characteristics of the commonly used S-R GPP 5.0 electrofishers under simulated electrofishing conditions. The range of resistance values to be simulated with static loads was extrapolated from in-water measurements of five electrode configurations consisting of paired, half-submerged, spherical anodes in combination with a 5.5-m-long flat-bottom aluminum-hull boat serving as the cathode. We measured the power output of GPP 5.0 electrofishers while they were connected to static loads of 114, 19, 9.5, and 5.7 X to simulate a wide range of ambient water conductivity values (approximately 100-1,000 lS/cm). These measurements of GPP 5.0 power output will provide electrofishing fleets with an improved understanding of electrofisher operational controls and performance and allow for a more consistent method of selecting electrofisher settings.
North American Journal of Fisheries Management, 2008
Because of its relatively natural hydrograph, the Yampa River, Colorado, is considered the crown ... more Because of its relatively natural hydrograph, the Yampa River, Colorado, is considered the crown jewel of native fish habitat in the upper basin of the Colorado River and has supported a relatively intact native fish assemblage. Nonnative fishes are thought to pose the greatest threat to native fishes in this system. Removal programs for nonnative northern pike Esox lucius and channel catfish Ictalurus punctatus have highlighted managers' perception of the threat posed by each species. Recent expansion of nonnative smallmouth bass Micropterus dolomieu in the Yampa River attracted an avid angling clientele but also coincided with a precipitous decline in native fishes, necessitating a rigorous assessment of the relative impact of all three nonnative predators on the native fishes. We used abundance, growth, and diet estimates for each predator species to quantify consumptive demand using bioenergetics models. Despite a low abundance of small-bodied fishes and thus a low number of fish in the smallmouth bass diet, total fish consumption by smallmouth bass (mean ¼ 15.2 kg km À1 year À1 ; 95% confidence interval [CI] ¼ 13.3-17.1 kg km À1 year À1) was similar to that estimated for northern pike (mean ¼ 13.7 kg km À1 year À1 , 95% CI ¼ 11.4-16.0 kg km À1 year À1) and was about 65 times higher than the estimate for channel catfish (mean ¼ 0.22 kg km À1 year À1 ; 95% CI ¼ 0.05-0.40 kg km À1 year À1). Diet data from the upper Colorado River, where small-bodied fish were plentiful, suggested that piscivory by smallmouth bass in the Yampa River could be 10 times the piscivory by northern pike and channel catfish, or about 168.5 kg km À1 year À1 (95% CI ¼ 147.0-189.9 kg km À1 year À1), if prey fish were more available. This level of piscivory suggested that smallmouth bass presented the greatest predatory threat to native fishes of the Yampa River. As environmental conditions change, use of field monitoring together with bioenergetics modeling will be an effective framework to assist managers in adapting their nonnative fish control efforts to maximize the likelihood of native fish recovery.
North American Journal of Fisheries Management, 2013
ABSTRACT
North American Journal of Fisheries Management, 1995
Abstract The population of kokanees Oncorhynchus nerka in Lake Granby, Colorado, is expected to s... more Abstract The population of kokanees Oncorhynchus nerka in Lake Granby, Colorado, is expected to satisfy the competing demands of providing summer harvest for anglers, kokanee eggs for restocking. and prey for trophy lake trout Salvelinus namaycush In the ...
Nonlinear Analysis: Theory, Methods & Applications, 2008
The linear viscoelastic equation is considered. We prove uniform decay rates of the energy by ass... more The linear viscoelastic equation is considered. We prove uniform decay rates of the energy by assuming a nonlinear feedback acting on the boundary, without imposing any restrictive growth assumption on the damping term and strongly weakening the usual assumptions on the relaxation function. Our estimate depends both on the behavior of the damping term near zero and on the behavior of the relaxation function at infinity. The proofs are based on the multiplier method and on a general lemma about convergent and divergent series for obtaining the uniform decay rates.
Neuromuscular Disorders, 2009
Journal of Molecular Biology, 1989
A nuclear gene encoding cytosolic glyceraldehyde-3-phosphate dehydrogenase from maize (subunit GA... more A nuclear gene encoding cytosolic glyceraldehyde-3-phosphate dehydrogenase from maize (subunit GAPCl, gene Gpcl) and 2.2 x lo3 base-pairs of its 5' flanking region have been cloned and sequenced. The structure of the maize Gpcl gene (10 introns) is different from that of the maize gene encoding subunit GAPA of chloroplast glyceraldehyde-3-phosphate dehydrogenase (1 intron) and relatively similar to that of the chicken gene (11 introns). Introns in the Gpcl gene show a positional polarity; the more 3' their position, the more they are displaced relative to introns in the chicken gene. The Gpcl gene and other nuclear genes from maize are associated with CpG islands, the relative size of which determines the degree of codon bias in the gene. The promoter of the maize Gpcl gene contains an anaerobic regulatory element and a pyrimidine box upstream from the TATA box and within intron 1. Southern blotting analyses and Northern hybridizations suggest that there are three functional Gpc genes in maize whose transcript levels are controlled differentially by anaerobiosis. In spite of its "typical" anaerobic promoter, the Gpcl gene does not seem t,o be an anaerobic gene in vivo.
Journal of Mathematical Analysis and Applications, 2001
We study the asymptotic behavior of a family of sequences defined by the following nonlinear indu... more We study the asymptotic behavior of a family of sequences defined by the following nonlinear induction relation c 0 = 1 and c n = k j=1 r j c n/m j + k j=k+1 r j c n+1 1/m j −1 for n ≥ 1, where the r j are real positive numbers and m j are integers greater than or equal to 2. Depending on the fact that k j=1 r j is greater or lower than 1, we prove that c n /n α or c n / ln n α goes to some finite limit for some explicit α. Our study is based on Tauberian theorems and extends a result of Erdös et al.
Journal of Evolution Equations, 2006
In this paper, we are interested in controllability properties of parabolic equations degeneratin... more In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain. We derive new Carleman estimates for the degenerate parabolic equation w t + (a(x)w x) x = f, (t, x) ∈ (0, T) × (0, 1), where the function a mainly satisfies a ∈ C 0 ([0, 1]) ∩ C 1 ((0, 1)), a > 0 on (0, 1) and 1 √ a ∈ L 1 (0, 1). We are mainly interested in the situation of a degenerate equation at the boundary i.e. in the case where a(0) = 0 and/or a(1) = 0. A typical example is a(x) = x α (1 − x) β with α, β ∈ [0, 2). As a consequence, we deduce null controllability results for the degenerate one dimensional heat equation u t − (a(x)u x) x = hχ ω , (t,x) ∈ (0, T) × (0, 1), ω ⊂⊂ (0, 1). The present paper completes and improves previous works [7, 8] where this problem was solved in the case a(x) = x α with α ∈ [0, 2).
SIAM Journal on Control and Optimization, 2019
For α ∈ (0, 2) we study the null controllability of the parabolic operator P u = ut − (|x| α ux)x... more For α ∈ (0, 2) we study the null controllability of the parabolic operator P u = ut − (|x| α ux)x (1 < x < 1), which degenerates at the interior point x = 0, for locally distributed controls acting only one side of the origin (that is, on some interval (a, b) with 0 < a < b < 1). Our main results guarantees that P is null controllable if and only if it is weakly degenerate, that is, α ∈ (0, 1). So, in order to steer the system to zero, one needs controls to act on both sides of the point of degeneracy in the strongly degenerate case α ∈ [1, 2). Our approach is based on spectral analysis and the moment method. Indeed, we completely describe the eigenvalues and eigenfunctions of the associated stationary operator in terms of Bessel functions and their zeroes for both weakly and strongly degenerate problems. Hence, we obtain lower L 2 bounds for the eigenfunctions on the control region in the case α ∈ [0, 1) and deduce the lack of observability in the case of α ∈ [1, 2). We also provide numerical evidence to illustrate our theoretical results.
Le Journal de Physique Colloques, 1988
The ternary compound HoRhySnz shows a frequency dependence of the susceptibility like a spin glas... more The ternary compound HoRhySnz shows a frequency dependence of the susceptibility like a spin glass. The spin-glass behaviour is confirmed by the "viscous" response of AC and DC susceptibility to a steplike variation of magnetic field or temperature.
ESAIM: Proceedings, 2000
We consider the problem of the wave equation with Neumann boundary condition damped by a locally ... more We consider the problem of the wave equation with Neumann boundary condition damped by a locally distributed linear damping a(x)u. When the damping region ω := {x, a(x) ≥ α > 0} contains a neighborhood of the boundary of the domain, E. Zuazua proved that the energy decays exponentially to zero. Using a piecewise multiplier method introduced by K. Liu, we prove that the energy decays exponentially to zero under weaker geometrical conditions. We give explicit examples when the domain is a polyhedron, and in the case of a disc. The proof is based on the construction of multipliers adapted to the geometrical conditions.
Portugaliae Mathematica, 2011
ABSTRACT In this paper, we study the null controllability of degenerate semilinear cas-cade parab... more ABSTRACT In this paper, we study the null controllability of degenerate semilinear cas-cade parabolic systems with one control force. The key tool is the Carleman estimates developed recently for degenerate one dimension parabolic equations. We develop a Carle-man estimate for these systems and then an observability inequality for the linear adjoint system. We conclude by linearization and fixed point arguments. Portugaliae Mathemat-ica.
Transactions of the American Fisheries Society, 2007
Nonnative fishes represent a significant impediment to the recovery of imperiled fishes, includin... more Nonnative fishes represent a significant impediment to the recovery of imperiled fishes, including those endemic to the Colorado River in the southwestern United States. Efforts to control nonindigenous fish abundance in the upper Colorado River basin have been unsuccessful owing in part to lack of knowledge regarding nonnative fish recruitment sources. We determined the source habitat (floodplain pond versus riverine habitats) for nonnative centrarchid fishes (largemouth bass Micropterus salmoides, green sunfish Lepomis cyanellus, bluegill L. macrochirus, and black crappie Pomoxis nigromaculatus) in the upper Colorado River using stable hydrogen isotopic composition (dD) and strontium : calcium (Sr:Ca) ratios in fish otoliths as natural markers of environmental history. Stable hydrogen isotope analysis revealed that 59% of centrarchids exhibited the otolith core signatures expected for riverine-origin fish, while 22% had emigrated from floodplain ponds and 19% were of uncertain origin. Strontium : calcium ratio data were consistent with the dD assays and indicated that relatively few fish immigrated to the river from high-salinity habitats. Black crappie was the only species that originated primarily from floodplain ponds. Efforts to control the abundance of most of the fishes included in this study should be concentrated in riverine habitats given the hydrologic conditions (below-average river discharge) present during our study. However, the proportion of pond-origin fish increased with fish age, which, coupled with historical river discharge data, suggested that floodplain pond contributions to riverine nonnative fish populations fluctuate with the interannual variations in flow regime and river-pond connectivity. Our results are the first to demonstrate the utility of dD as a natural marker of fish environmental history that will probably provide valuable insights into the management of fish in other environments.
SIAM Journal on Control and Optimization, 2000
Abstract. We consider the wave equation damped by a nonlinear boundary velocity feedback q(ut). F... more Abstract. We consider the wave equation damped by a nonlinear boundary velocity feedback q(ut). First we consider the case where q has a linear growth at infinity. We prove that the usual decay rate estimates proved by Nakao [Differential Integral Equations, 8 (1995), pp. 681 ...
SIAM Journal on Control and Optimization, 2003
We are interested in controllability problems of equations coming from a boundary layer model. We... more We are interested in controllability problems of equations coming from a boundary layer model. We simplify the problem by considering only equations with constant coefficients. The problem is described by a degenerate parabolic equation (a linearized Crocco-type equation) where phenomena of diffusion and transport are coupled. First we give a geometric characterization of the influence domain of a locally distributed con- trol. Then we prove regional null controllability results on this domain. The proof is based on an adequate observability inequality for the homogeneous adjoint problem. This inequality is obtained by decomposition of the space-time domain and Carleman-type estimates along characteristics. In the second part of this paper, we treat the case of a boundary control.
SIAM Journal on Control and Optimization, 2008
Given α ∈ [0, 2) and f ∈ L 2 ((0, T) × (0, 1)), we derive new Carleman estimates for the degenera... more Given α ∈ [0, 2) and f ∈ L 2 ((0, T) × (0, 1)), we derive new Carleman estimates for the degenerate parabolic problem wt + (x α wx)x = f , where (t, x) ∈ (0, T) × (0, 1), associated to the boundary conditions w(t, 1) = 0 and w(t, 0) = 0 if 0 ≤ α < 1 or (x α wx)(t, 0) = 0 if 1 ≤ α < 2. The proof is based on the choice of suitable weighted functions and Hardy-type inequalities. As a consequence, for all 0 ≤ α < 2 and ω ⊂⊂ (0, 1), we deduce null controllability results for the degenerate one-dimensional heat equation ut − (x α ux)x = hχω with the same boundary conditions as above.
SIAM Journal on Control and Optimization, 2005
Motivated by several works on ordinary differential equations, we are interested in the asymptoti... more Motivated by several works on ordinary differential equations, we are interested in the asymptotic stability of intermittently controlled partial differential equations. We give a condition of asymptotic stability for second-order evolution equations uniformly damped by an on/off feedback. This result extends to the case of partial differential equations a previous result of R. A. Smith concerning ordinary differential equations.
North American Journal of Fisheries Management, 2009
The Smith-Root (S-R) series of GPP (gas-powered pulsator) electrofishers are widely used in boat-... more The Smith-Root (S-R) series of GPP (gas-powered pulsator) electrofishers are widely used in boat-mounted electrofishing systems. Sampling in waters having different conductivities requires adjustment of the electrofisher controls to maintain consistent output power. However, the current meter supplied on GPP electrofishers provides an average rather than a peak measurement, and because no voltmeter is included, determining peak power output is not possible. As part of an overall effort to promote standardization of electrofishing operations, we used static electrical loads to measure the output characteristics of the commonly used S-R GPP 5.0 electrofishers under simulated electrofishing conditions. The range of resistance values to be simulated with static loads was extrapolated from in-water measurements of five electrode configurations consisting of paired, half-submerged, spherical anodes in combination with a 5.5-m-long flat-bottom aluminum-hull boat serving as the cathode. We measured the power output of GPP 5.0 electrofishers while they were connected to static loads of 114, 19, 9.5, and 5.7 X to simulate a wide range of ambient water conductivity values (approximately 100-1,000 lS/cm). These measurements of GPP 5.0 power output will provide electrofishing fleets with an improved understanding of electrofisher operational controls and performance and allow for a more consistent method of selecting electrofisher settings.
North American Journal of Fisheries Management, 2008
Because of its relatively natural hydrograph, the Yampa River, Colorado, is considered the crown ... more Because of its relatively natural hydrograph, the Yampa River, Colorado, is considered the crown jewel of native fish habitat in the upper basin of the Colorado River and has supported a relatively intact native fish assemblage. Nonnative fishes are thought to pose the greatest threat to native fishes in this system. Removal programs for nonnative northern pike Esox lucius and channel catfish Ictalurus punctatus have highlighted managers' perception of the threat posed by each species. Recent expansion of nonnative smallmouth bass Micropterus dolomieu in the Yampa River attracted an avid angling clientele but also coincided with a precipitous decline in native fishes, necessitating a rigorous assessment of the relative impact of all three nonnative predators on the native fishes. We used abundance, growth, and diet estimates for each predator species to quantify consumptive demand using bioenergetics models. Despite a low abundance of small-bodied fishes and thus a low number of fish in the smallmouth bass diet, total fish consumption by smallmouth bass (mean ¼ 15.2 kg km À1 year À1 ; 95% confidence interval [CI] ¼ 13.3-17.1 kg km À1 year À1) was similar to that estimated for northern pike (mean ¼ 13.7 kg km À1 year À1 , 95% CI ¼ 11.4-16.0 kg km À1 year À1) and was about 65 times higher than the estimate for channel catfish (mean ¼ 0.22 kg km À1 year À1 ; 95% CI ¼ 0.05-0.40 kg km À1 year À1). Diet data from the upper Colorado River, where small-bodied fish were plentiful, suggested that piscivory by smallmouth bass in the Yampa River could be 10 times the piscivory by northern pike and channel catfish, or about 168.5 kg km À1 year À1 (95% CI ¼ 147.0-189.9 kg km À1 year À1), if prey fish were more available. This level of piscivory suggested that smallmouth bass presented the greatest predatory threat to native fishes of the Yampa River. As environmental conditions change, use of field monitoring together with bioenergetics modeling will be an effective framework to assist managers in adapting their nonnative fish control efforts to maximize the likelihood of native fish recovery.
North American Journal of Fisheries Management, 2013
ABSTRACT
North American Journal of Fisheries Management, 1995
Abstract The population of kokanees Oncorhynchus nerka in Lake Granby, Colorado, is expected to s... more Abstract The population of kokanees Oncorhynchus nerka in Lake Granby, Colorado, is expected to satisfy the competing demands of providing summer harvest for anglers, kokanee eggs for restocking. and prey for trophy lake trout Salvelinus namaycush In the ...
Nonlinear Analysis: Theory, Methods & Applications, 2008
The linear viscoelastic equation is considered. We prove uniform decay rates of the energy by ass... more The linear viscoelastic equation is considered. We prove uniform decay rates of the energy by assuming a nonlinear feedback acting on the boundary, without imposing any restrictive growth assumption on the damping term and strongly weakening the usual assumptions on the relaxation function. Our estimate depends both on the behavior of the damping term near zero and on the behavior of the relaxation function at infinity. The proofs are based on the multiplier method and on a general lemma about convergent and divergent series for obtaining the uniform decay rates.
Neuromuscular Disorders, 2009
Journal of Molecular Biology, 1989
A nuclear gene encoding cytosolic glyceraldehyde-3-phosphate dehydrogenase from maize (subunit GA... more A nuclear gene encoding cytosolic glyceraldehyde-3-phosphate dehydrogenase from maize (subunit GAPCl, gene Gpcl) and 2.2 x lo3 base-pairs of its 5' flanking region have been cloned and sequenced. The structure of the maize Gpcl gene (10 introns) is different from that of the maize gene encoding subunit GAPA of chloroplast glyceraldehyde-3-phosphate dehydrogenase (1 intron) and relatively similar to that of the chicken gene (11 introns). Introns in the Gpcl gene show a positional polarity; the more 3' their position, the more they are displaced relative to introns in the chicken gene. The Gpcl gene and other nuclear genes from maize are associated with CpG islands, the relative size of which determines the degree of codon bias in the gene. The promoter of the maize Gpcl gene contains an anaerobic regulatory element and a pyrimidine box upstream from the TATA box and within intron 1. Southern blotting analyses and Northern hybridizations suggest that there are three functional Gpc genes in maize whose transcript levels are controlled differentially by anaerobiosis. In spite of its "typical" anaerobic promoter, the Gpcl gene does not seem t,o be an anaerobic gene in vivo.
Journal of Mathematical Analysis and Applications, 2001
We study the asymptotic behavior of a family of sequences defined by the following nonlinear indu... more We study the asymptotic behavior of a family of sequences defined by the following nonlinear induction relation c 0 = 1 and c n = k j=1 r j c n/m j + k j=k+1 r j c n+1 1/m j −1 for n ≥ 1, where the r j are real positive numbers and m j are integers greater than or equal to 2. Depending on the fact that k j=1 r j is greater or lower than 1, we prove that c n /n α or c n / ln n α goes to some finite limit for some explicit α. Our study is based on Tauberian theorems and extends a result of Erdös et al.
Journal of Evolution Equations, 2006
In this paper, we are interested in controllability properties of parabolic equations degeneratin... more In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain. We derive new Carleman estimates for the degenerate parabolic equation w t + (a(x)w x) x = f, (t, x) ∈ (0, T) × (0, 1), where the function a mainly satisfies a ∈ C 0 ([0, 1]) ∩ C 1 ((0, 1)), a > 0 on (0, 1) and 1 √ a ∈ L 1 (0, 1). We are mainly interested in the situation of a degenerate equation at the boundary i.e. in the case where a(0) = 0 and/or a(1) = 0. A typical example is a(x) = x α (1 − x) β with α, β ∈ [0, 2). As a consequence, we deduce null controllability results for the degenerate one dimensional heat equation u t − (a(x)u x) x = hχ ω , (t,x) ∈ (0, T) × (0, 1), ω ⊂⊂ (0, 1). The present paper completes and improves previous works [7, 8] where this problem was solved in the case a(x) = x α with α ∈ [0, 2).