| 1. |
- Baird, Julia, et al.
(author)
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Introducing Resilience Practice to Watershed Groups : What Are the Learning Effects?
- 2016
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In: Society & Natural Resources. - : Informa UK Limited. - 0894-1920 .- 1521-0723. ; 29:10, s. 1214-1229
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Journal article (peer-reviewed)abstract
- Resilience as an organizing framework for addressing dynamics of social-ecological systems has experienced strong uptake; however, its application is nascent. This research study aimed to address the gap between resilience thinking and practice by focusing on learning, a key aspect of resilience. Two Canadian watershed groups were led in 2-day workshops focused on resilience. Learning effects were measured using a survey administered both before and after the workshop, and a qualitative survey was administered 6 months later to understand longer term effects. Short-term learning effects were similar between the two case studies, with strong cognitive and relational learning and less normative learning. Longer term effects showed enduring cognitive and normative learning in both cases; however, relational learning persisted only in the watershed where a resilience practice approach to watershed planning had been incorporated. Future research directions include refinements to the learning measurement methodology and continuing to build resilience practice literature.
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| 2. |
- Brandes, Julia, 1986
(author)
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Linear spaces on hypersurfaces over number fields
- 2017
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In: Michigan Mathematical Journal. - : Michigan Mathematical Journal. - 1945-2365 .- 0026-2285. ; 66:4, s. 769-784
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Journal article (peer-reviewed)abstract
- We establish an analytic Hasse principle for linear spaces of affine dimension m on a complete intersection over an algebraic field extension K of Q. The number of variables required to do this is no larger than what is known for the analogous problem over ?. As an application, we show that any smooth hypersurface over K whose dimension is large enough in terms of the degree is K-unirational, provided that either the degree is odd or K is totally imaginary.
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| 3. |
- Brandes, Julia, et al.
(author)
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On generating functions in additive number theory, II: lower-order terms and applications to PDEs
- 2021
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In: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 379, s. 347-76
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Journal article (peer-reviewed)abstract
- We obtain asymptotics for sums of the form Sigma(p)(n=1) e(alpha(k) n(k) + alpha(1)n), involving lower order main terms. As an application, we show that for almost all alpha(2) is an element of [0, 1) one has sup(alpha 1 is an element of[0,1)) | Sigma(1 <= n <= P) e(alpha(1)(n(3) + n) + alpha(2)n(3))| << P3/4+epsilon, and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrodinger and Airy equations.
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| 4. |
- Brandes, Julia, 1986, et al.
(author)
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ON THE INHOMOGENEOUS VINOGRADOV SYSTEM
- 2022
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In: Bulletin of the Australian Mathematical Society. - 1755-1633 .- 0004-9727. ; 106:3, s. 396-403
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Journal article (peer-reviewed)abstract
- We show that the system of equations Sigma(s)(i=1)(x(i)(j) - y(i)(j)) = a(j) (1 <= j <= k) has appreciably fewer solutions in the subcritical range s < 1/2k(k + 1) than its homogeneous counterpart, provided that al not equal 0 for some l <= k - 1. Our methods use Vinogradov's mean value theorem in combination with a shifting argument.
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| 5. |
- Brandes, Julia, et al.
(author)
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ON THE INHOMOGENEOUS VINOGRADOV SYSTEM
- 2022
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In: Bulletin of the Australian Mathematical Society. - : Cambridge University Press (CUP). - 0004-9727 .- 1755-1633. ; 106:3, s. 396-403
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Journal article (peer-reviewed)abstract
- We show that the system of equations Sigma(s)(i=1)(x(i)(j) - y(i)(j)) = a(j) (1 <= j <= k) has appreciably fewer solutions in the subcritical range s < 1/2k(k + 1) than its homogeneous counterpart, provided that al not equal 0 for some l <= k - 1. Our methods use Vinogradov's mean value theorem in combination with a shifting argument.
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| 6. |
- Brandes, Julia, 1986
(author)
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On the number of linear spaces on hypersurfaces with a prescribed discriminant
- 2018
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In: Mathematische Zeitschrift. - : Springer Science and Business Media LLC. - 1432-1823 .- 0025-5874. ; 289:3-4, s. 803-827
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Journal article (peer-reviewed)abstract
- For a given form we apply the circle method in order to give an asymptotic estimate of the number of m-tuples spanning a linear space on the hypersurface with the property that . This allows us in some measure to count rational linear spaces on hypersurfaces whose underlying integer lattice is primitive.
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| 7. |
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| 8. |
- Brandes, Julia, 1986, et al.
(author)
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Rational lines on cubic hypersurfaces
- 2021
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In: Mathematical Proceedings of the Cambridge Philosophical Society. - 0305-0041 .- 1469-8064. ; 171:1, s. 99-112
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Journal article (peer-reviewed)abstract
- We show that any smooth projective cubic hypersurface of dimension at least 29 over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous results due to the second author and Wooley.We include an appendix in which we highlight some slight modifications to a recent result of Papanikolopoulos and Siksek. It follows that the set of rational points on smooth projective cubic hypersurfaces of dimension at least 29 is generated via secant and tangent constructions from just a single point.
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| 9. |
- Brandes, Julia, 1986, et al.
(author)
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Simultaneous Additive Equations: Repeated and Differing Degrees
- 2017
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In: Canadian Journal of Mathematics. - 1496-4279 .- 0008-414X. ; 69:2, s. 258-283
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Journal article (peer-reviewed)abstract
- We obtain bounds for the number of variables required to establish Hasse principles, both for the existence of solutions and for asymptotic formula, for systems of additive equations containing forms of differing degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning-Heath-Brown, which give weak results when specialized to the diagonal situation, this is the first result on such "hybrid" systems. We also obtain specialized results for systems of quadratic and cubic forms, where we are able to take advantage of some of the stronger methods available in that setting. In particular, we achieve essentially square root cancellation for systems consisting of one cubic and r quadratic equations.
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| 10. |
- Brandes, Julia, 1986
(author)
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The density of rational lines on hypersurfaces: a bihomogeneous perspective
- 2021
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In: Monatshefte für Mathematik. - : Springer Science and Business Media LLC. - 1436-5081 .- 0026-9255. ; 195:2, s. 191-231
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Journal article (peer-reviewed)abstract
- Let F be a non-singular homogeneous polynomial of degree d in n variables. We give an asymptotic formula of the pairs of integer points (x, y) with | x| ⩽ X and | y| ⩽ Y which generate a line lying in the hypersurface defined by F, provided that n> 2 d-1d4(d+ 1) (d+ 2). In particular, by restricting to Zariski-open subsets we are able to avoid imposing any conditions on the relative sizes of X and Y.
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