Abstract In this article, the recently discovered phenomenon of delayed Hopf bifurcations (DHB) in reaction–diffusion partial differential equations (PDEs) is analysed in the cubic Complex Ginzburg–Landau equation, as an equation in its own right, with a slowly varying parameter. We begin by using the classical asymptotic methods of stationary phase and steepest descents on the linearized PDE to show that solutions, which have approached the attracting quasi-steady state (QSS) before the Hopf bifurcation remain near that state for long times after the instantaneous Hopf bifurcation and the QSS has become repelling. In the complex time plane, the phase function of the linearized PDE has a saddle point, and the Stokes and anti-Stokes lines are central to the asymptotics. The non-linear terms are treated by applying an iterative method to the mild form of the PDE given by perturbations about the linear particular solution. This tracks the closeness of solutions near the attracting and repelling QSS in the full, non-linear PDE. Next, we show that beyond a key Stokes line through the saddle there is a curve in the space-time plane along which the particular solution of the linear PDE ceases to be exponentially small, causing the solution of the non-linear PDE to diverge from the repelling QSS and exhibit large-amplitude oscillations. This curve is called the space–time buffer curve. The homogeneous solution also stops being exponentially small in a spatially dependent manner, as determined also by the initial data and time. Hence, a competition arises between these two solutions, as to which one ceases to be exponentially small first, and this competition governs spatial dependence of the DHB. We find four different cases of DHB, depending on the outcomes of the competition, and we quantify to leading order how these depend on the main system parameters, including the Hopf frequency, initial time, initial data, source terms, and diffusivity. Examples are presented for each case, with source terms that are a uni-modal function, a smooth step function, a spatially periodic function and an algebraically growing function. Also, rich spatio-temporal dynamics are observed in the post-DHB oscillations. Finally, it is shown that large-amplitude source terms can be designed so that solutions spend substantially longer times near the repelling QSS, and hence, region-specific control over the delayed onset of oscillations can be achieved.
more »
« less
Classification of Frobenius Forms in Five Variables
Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the 19-dimensional space of all cubics, and that up to projective equivalence, there are finitely many non-Frobenius split cubic surfaces. We explicitly describe defining equations for each and characterize them as extremal in terms of configurations of lines on them. In particular, a (possibly singular) cubic surface in characteristic two fails to be Frobenius split if and only if no
three lines on it form a “triangle”.
more »
« less
- Award ID(s):
- 1945611
- NSF-PAR ID:
- 10326299
- Editor(s):
- Miller, Claudia; Striuli, Janet; Witt, Emily E.
- Date Published:
- Journal Name:
- Women in Commutative Algebra: Proceedings of the 2019 WICA Workshop
- Volume:
- 29
- Page Range / eLocation ID:
- 353-367
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
more » « less
Abstract We study the symbolic powers of square-free monomial ideals via symbolic Rees algebras and methods in prime characteristic. In particular, we prove that the symbolic Rees algebra and the symbolic associated graded algebra are split with respect to a morphism that resembles the Frobenius map and that exists in all characteristics. Using these methods, we recover a result by Hoa and Trung that states that the normalized $a$-invariants and the Castelnuovo–Mumford regularity of the symbolic powers converge. In addition, we give a sufficient condition for the equality of the ordinary and symbolic powers of this family of ideals and relate it to Conforti–Cornuéjols conjecture. Finally, we interpret this condition in the context of linear optimization.
-
null (Ed.)Abstract Let $K$ be an algebraically closed field of prime characteristic $p$ , let $X$ be a semiabelian variety defined over a finite subfield of $K$ , let $\unicode[STIX]{x1D6F7}:X\longrightarrow X$ be a regular self-map defined over $K$ , let $V\subset X$ be a subvariety defined over $K$ , and let $\unicode[STIX]{x1D6FC}\in X(K)$ . The dynamical Mordell–Lang conjecture in characteristic $p$ predicts that the set $S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$ is a union of finitely many arithmetic progressions, along with finitely many $p$ -sets, which are sets of the form $\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$ for some $m\in \mathbb{N}$ , some rational numbers $c_{i}$ and some non-negative integers $k_{i}$ . We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case $X$ is an algebraic torus, we can prove the conjecture in two cases: either when $\dim (V)\leqslant 2$ , or when no iterate of $\unicode[STIX]{x1D6F7}$ is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of $X$ . We end by proving that Vojta’s conjecture implies the dynamical Mordell–Lang conjecture for tori with no restriction.more » « less
-
International Ocean Discovery Program Expedition 397T sought to address the shortage of drilling time caused by COVID-19 mitigation during Expedition 391 (Walvis Ridge Hotspot) by drilling at two sites omitted from the earlier cruise. A week of coring time was added to a transit of JOIDES Resolution from Cape Town to Lisbon, which would cross Walvis Ridge on its way north. These two sites were located on two of the three seamount trails that emerge from the split in Walvis Ridge morphology into several seamount chains at 2°E. Site U1584 (proposed Site GT-6A) sampled the Gough track on the southeast side of the hotspot track, and Site U1585 (proposed Site TT-4A) sampled the Tristan track on the northwest side. Together with Site U1578, drilled on the Center track during Expedition 391, they form a transect across the northern Walvis Ridge Guyot Province. The goal was to core seamount basalts and associated volcanic material for geochemical and isotopic, geochronologic, paleomagnetic, and volcanological study. Scientifically, one emphasis was to better understand the split in isotopic signatures that occurs at the morphologic split. Geochronology would add to the established age progression but also give another dimension to understanding Walvis Ridge seamount formation by giving multiple ages at the same sites. The paleomagnetic study seeks to establish paleolatitudes for Walvis Ridge sites for comparison with those published from hotspot seamount chains in the Pacific, in particular to test whether a component of true polar wander affects hotspot paleolatitude. Hole U1584A cored a 66.4 m thick sedimentary and volcaniclastic section with two lithostratigraphic units. Unit I is a 23 m thick sequence of bioturbated clay and nannofossil chalk with increasing volcaniclastic content downhole. Unit II is a >43 m thick sequence of lapillistone with basalt fragments. Because the seismic section crossing the site shows no evidence as to the depth of the volcaniclastic cover, coring was terminated early. Because there were no other shallow sites nearby with different characteristics on existing seismic lines, the unused operations time from Site U1584 was shifted to the next site. The seismic reflector interpreted as the top of igneous rock at Site U1585 once again resulted from volcaniclastic deposits. Hole U1585A coring began at 144.1 mbsf and penetrated a 273.5 m thick sedimentary and volcaniclastic section atop a 81.2 m thick series of massive basalt flows. The hole was terminated at 498.8 mbsf because allotted operational time expired. The sedimentary section contains four main lithostratigraphic units. Unit I (144.1–157.02 mbsf) is a bioturbated nannofossil chalk with foraminifera, similar to the shallowest sediments recovered at Site U1584. Unit II (157.02–249.20 mbsf), which is divided into two subunits, is a 92.2 m thick succession of massive and bedded pumice and scoria lapillistone with increased reworking, clast alteration, and tuffaceous chalk intercalations downhole. Unit III (249.20–397.76 mbsf) is 148.6 m thick and consists of a complex succession of pink to greenish gray tuffaceous chalk containing multiple thin, graded ash turbidites and tuffaceous ash layers; intercalated tuffaceous chalk slumps; and several thick coarse lapilli and block-dominated volcaniclastic layers. Befitting its complexity, this unit is divided into eight subunits (IIIA–IIIH). Three of these subunits (IIIA, IIID, and IIIG) are mainly basalt breccias. Unit IV (397.76–417.60 mbsf) is a volcanic breccia, 19.8 m thick, containing mostly juvenile volcaniclasts. The igneous section, Unit V (417.60–498.80 mbsf) is composed of a small number of massive basaltic lava flows. It is divided into three igneous lithologic units, with Unit 2 represented by a single 3 cm piece of quenched basalt with olivine phenocrysts in a microcrystalline groundmass. This piece may represent a poorly recovered set of pillow lavas. Unit 1 is sparsely to highly olivine-clinopyroxene ± plagioclase phyric massive basalt and is divided into Subunits 1a and 1b based on textural and mineralogical differences, which suggests that they are two different flows. Unit 3 also consists of two massive lava flows with no clear boundary features. Subunit 3a is a 10.3 m thick highly clinopyroxene-plagioclase phyric massive basalt flow with a fine-grained groundmass. Subunit 3b is a featureless massive basalt flow that is moderately to highly clinopyroxene-olivine-plagioclase phyric and >43.7 m thick. Alteration of the lava flows is patchy and moderate to low in grade, with two stages, one at a higher temperature and one at a low temperature, both focused around fractures. The Site U1585 chronological succession from basalt flows to pelagic sediment indicates volcanic construction and subsidence. Lava eruptions were followed by inundation and shallow-water volcaniclastic sediment deposition, which deepened over time to deepwater conditions. Although the massive flows were probably erupted in a short time and have little variability, volcaniclasts in the sediments may provide geochemical and geochronologic data from a range of time and sources. Chemical analyses indicate that Site U1585 basalt samples are mostly alkalic basalt, with a few trachybasalt flow and clast samples and one basaltic trachyandesite clast. Ti/V values lie mostly within the oceanic island basalt (OIB) field but overlap the mid-ocean-ridge basalt (MORB) field. Only a handful of clasts from Site U1584 were analyzed, but geochemical data are similar. Paleomagnetic data from Site U1585 indicate that the sediments and basalt units are strongly magnetic and mostly give coherent inclination data, which indicates that the basaltic section and ~133 m of overlying volcaniclastic sediment is reversely polarized and that this reversal is preserved in a core. Above this, the rest of the sediment section records two normal and two reversed zones. Although there are not enough basalt flows to give a reliable paleolatitude, it may be possible to attain such a result from the sediments.more » « less
-
International Ocean Discovery Program Expedition 397T sought to address the shortage of drilling time caused by COVID-19 mitigation during Expedition 391 (Walvis Ridge Hotspot) by drilling at two sites omitted from the earlier cruise. A week of coring time was added to a transit of JOIDES Resolution from Cape Town to Lisbon, which would cross Walvis Ridge on its way north. These two sites were located on two of the three seamount trails that emerge from the split in Walvis Ridge morphology into several seamount chains at 2°E. Site U1584 (proposed Site GT-6A) sampled the Gough track on the east, and Site U1585 (proposed Site TT-4A) sampled the Tristan track on the west. Together with Site U1578, drilled on the Center track during Expedition 391, they form a transect across the northern Walvis Ridge Guyot Province. The goal was to core seamount basalts and associated volcanic material for geochemical and isotopic, geochronologic, paleomagnetic, and volcanologic study. Scientifically, one emphasis was to better understand the split in geochemical and isotopic signatures that occurs at the morphologic split. Geochronology would add to the established age progression but also give another dimension to understanding Walvis Ridge seamount formation by giving multiple ages at the same sites. The paleomagnetic study seeks to establish paleolatitudes for Walvis Ridge sites for comparison with those published from hotspot seamount chains in the Pacific, in particular to test whether a component of true polar wander affects hotspot paleolatitude. Hole U1584A cored a 66.4 m thick sedimentary and volcaniclastic section with two lithostratigraphic units. Unit I is a 23 m thick sequence of bioturbated clay and nannofossil chalk with increasing volcaniclastic content downhole. Unit II is a >43 m thick sequence of lapillistone with basalt fragments. Because the seismic section crossing the site shows no evidence as to the depth of the volcaniclastic cover, coring was terminated early. Because there were no other shallow nearby sites with different character on existing seismic lines, the unused operations time from Site U1584 was shifted to the next site. The seismic reflector interpreted as the top of igneous rock at Site U1585 once again resulted from volcaniclastic deposits. Hole U1585A coring began at 144.1 mbsf and penetrated a 273.5 m thick sedimentary and volcaniclastic section atop a 81.2 m thick series of massive basalt flows. The hole was terminated at 498.8 mbsf because allotted operational time expired. The sedimentary section contains four main units. Unit I (144.1–157.02 mbsf) is a bioturbated nannofossil chalk with foraminifera, similar to the shallowest sediments recovered at Site U1584. Unit II (157.02–249.20 mbsf), which is divided into two subunits, is a 92.2 m thick succession of massive and bedded pumice and scoria lapillistone with increased reworking, clast alteration, and tuffaceous chalk intercalations downhole. Unit III (249.20–397.76 mbsf) is 148.6 m thick and consists of a complex succession of pink to greenish gray tuffaceous chalk containing multiple thin, graded ash turbidites and tuffaceous ash layers; intercalated tuffaceous chalk slumps; and several thick coarse lapilli and block-dominated volcaniclastic layers. Befitting the complexity, it is divided into eight subunits (IIIA–IIIH). Three of these subunits (IIIA, IIID, and IIIG) are mainly basalt breccias. Unit IV (397.76–417.60 mbsf) is a volcanic breccia, 19.8 m thick, containing mostly juvenile volcaniclasts. The igneous section, Unit V (417.60–498.80 mbsf) is composed of a small number of massive basaltic lava flows. It is divided into three lithologic units, with Unit 2 represented by a single 3 cm piece of quenched basalt with olivine phenocrysts in a microcrystalline groundmass. This piece may represent a poorly recovered set of pillow lavas. Unit 1 is sparsely to highly olivine-clinopyroxene ± plagioclase phyric massive basalt and is divided into Subunits 1a and 1b based on textural and mineralogical differences, which suggests that they are two different flows. Unit 3 also consists of two massive lava flows with no clear boundary features. Subunit 3a is a 10.3 m thick highly clinopyroxene-plagioclase phyric massive basalt flow with a fine-grained groundmass. Subunit 3b is a featureless massive basalt flow that is moderately to highly clinopyroxene-olivine-plagioclase phyric and >43.7 m thick. Alteration of the lava flows is patchy and moderate to low in grade, with two stages, one at a higher temperature and one at a low temperature, both focused around fractures. The Site U1585 chronologic succession from basalt flows to pelagic sediment indicates volcanic construction and subsidence. Lava eruptions were followed by inundation and shallow-water volcaniclastic sediment deposition, which deepened over time to deepwater conditions. Although the massive flows were probably erupted in a short time and have little variability, volcaniclasts in the sediments may provide geochemical and geochronologic data from a range of time and sources. Chemical analyses indicate that Site U1585 basalt samples are mostly alkalic basalt, with a few trachybasalt flow and clast samples and one basaltic trachyandesite clast. Ti/V ratios lie mostly within the oceanic island basalt (OIB) field but overlap the mid-ocean-ridge basalt (MORB) field. Only a handful of clasts from Site U1584 were analyzed, but geochemical data are similar. Paleomagnetic data from Site U1585 indicate that the sediments and basalt units are strongly magnetic and mostly give coherent inclination data, which indicates that the basaltic section and ~133 m of overlying volcaniclastic sediment is reversely polarized and that this reversal is preserved in a core. Above this, the rest of the sediment section records two normal and two reversed zones. Although there are not enough basalt flows to give a reliable paleolatitude, it may be possible to attain such a result from the sediments.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-030-91986-3_15
