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Articles 1 - 11 of 11
Full-Text Articles in Mathematics
Motivic Classes Of Degeneracy Loci And Pointed Brill-Noether Varieties, D. Anderson, Linda Chen, N. Tarasca
Motivic Classes Of Degeneracy Loci And Pointed Brill-Noether Varieties, D. Anderson, Linda Chen, N. Tarasca
Mathematics & Statistics Faculty Works
Motivic Chern and Hirzebruch classes are polynomials with K-theory and homology classes as coefficients, which specialize to Chern–Schwartz–MacPherson classes, K-theory classes, and Cappell–Shaneson L-classes. We provide formulas to compute the motivic Chern and Hirzebruch classes of Grassmannian and vexillary degeneracy loci. We apply our results to obtain the Hirzebruch χy-genus of classical and one-pointed Brill–Noether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature.
Amphichiral Knots With Large 4-Genus, Allison N. Miller
Amphichiral Knots With Large 4-Genus, Allison N. Miller
Mathematics & Statistics Faculty Works
For each we give g > 0 infinitely many knots that are strongly negative amphichiral, hence rationally slice and representing 2-torsion in the smooth concordance group, yet which do not bound any locally flatly embedded surface in the 4-ball with genus less than or equal to g. Our examples also allow us to answer a question about the four-dimensional clasp number of knots.
Spectra Of Variants Of Distance Matrices Of Graphs And Digraphs: A Survey, L. Hogben, Carolyn Reinhart
Spectra Of Variants Of Distance Matrices Of Graphs And Digraphs: A Survey, L. Hogben, Carolyn Reinhart
Mathematics & Statistics Faculty Works
Distance matrices of graphs were introduced by Graham and Pollack in 1971 to study a problem in communications. Since then, there has been extensive research on the distance matrices of graphs—a 2014 survey by Aouchiche and Hansen on spectra of distance matrices of graphs lists more than 150 references. In the last 10 years, variants such as the distance Laplacian, the distance signless Laplacian, and the normalized distance Laplacian matrix of a graph have been studied. After a brief description of the early history of the distance matrix and its motivating problem, this survey focuses on comparing and contrasting techniques …
Functional Implications Of Dale's Law In Balanced Neuronal Network Dynamics And Decision Making, Victor J. Barranca, Asha Bhuiyan , '23, Max Sundgren , '22, Fangzhou Xing , '22
Functional Implications Of Dale's Law In Balanced Neuronal Network Dynamics And Decision Making, Victor J. Barranca, Asha Bhuiyan , '23, Max Sundgren , '22, Fangzhou Xing , '22
Mathematics & Statistics Faculty Works
The notion that a neuron transmits the same set of neurotransmitters at all of its post-synaptic connections, typically known as Dale's law, is well supported throughout the majority of the brain and is assumed in almost all theoretical studies investigating the mechanisms for computation in neuronal networks. Dale's law has numerous functional implications in fundamental sensory processing and decision-making tasks, and it plays a key role in the current understanding of the structure-function relationship in the brain. However, since exceptions to Dale's law have been discovered for certain neurons and because other biological systems with complex network structure incorporate individual …
An Affine Approach To Peterson Comparison, Linda Chen, E. Milićević, J. Morse
An Affine Approach To Peterson Comparison, Linda Chen, E. Milićević, J. Morse
Mathematics & Statistics Faculty Works
The Peterson comparison formula proved by Woodward relates the three-pointed Gromov-Witten invariants for the quantum cohomology of partial flag varieties to those for the complete flag. Another such comparison can be obtained by composing a combinatorial version of the Peterson isomorphism with a result of Lapointe and Morse relating quantum Littlewood-Richardson coefficients for the Grassmannian to k-Schur analogs in the homology of the affine Grassmannian obtained by adding rim hooks. We show that these comparisons on quantum cohomology are equivalent, up to Postnikov’s strange duality isomorphism.
Topological Speedups Of ℤD-Actions, Aimee S.A. Johnson, D. M. Mcclendon
Topological Speedups Of ℤD-Actions, Aimee S.A. Johnson, D. M. Mcclendon
Mathematics & Statistics Faculty Works
We study minimal ℤd-Cantor systems and the relationship between their speedups, their collections of invariant Borel measures, their associated unital dimension groups, and their orbit equivalence classes. In the particular case of minimal ℤd-odometers, we show that their bounded speedups must again be odometers but, contrary to the 1-dimensional case, they need not be conjugate, or even isomorphic, to the original. Furthermore, we give examples of speedups of ℤd-odometers which show the significant role played by a choice of ‘cone’ associated to the speedup.
Maximal And Minimal Weak Solutions For Elliptic Problems With Nonlinearity On The Boundary, S. Bandyopadhyay, M. Chhetri, B. B. Delgado, Nsoki Mavinga, R. Pardo
Maximal And Minimal Weak Solutions For Elliptic Problems With Nonlinearity On The Boundary, S. Bandyopadhyay, M. Chhetri, B. B. Delgado, Nsoki Mavinga, R. Pardo
Mathematics & Statistics Faculty Works
This paper deals with the existence of weak solutions for semilinear elliptic equation with nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution for both monotone and nonmonotone nonlinearities. We use iteration argument when the nonlinearity is monotone. For the nonmonotone case, we utilize the surjectivity of a pseudomonotone and coercive operator, Zorn's lemma and a version of Kato's inequality.
Reconstruction Of Sparse Recurrent Connectivity And Inputs From The Nonlinear Dynamics Of Neuronal Networks, Victor J. Barranca
Reconstruction Of Sparse Recurrent Connectivity And Inputs From The Nonlinear Dynamics Of Neuronal Networks, Victor J. Barranca
Mathematics & Statistics Faculty Works
Reconstructing the recurrent structural connectivity of neuronal networks is a challenge crucial to address in characterizing neuronal computations. While directly measuring the detailed connectivity structure is generally prohibitive for large networks, we develop a novel framework for reverse-engineering large-scale recurrent network connectivity matrices from neuronal dynamics by utilizing the widespread sparsity of neuronal connections. We derive a linear input-output mapping that underlies the irregular dynamics of a model network composed of both excitatory and inhibitory integrate-and-fire neurons with pulse coupling, thereby relating network inputs to evoked neuronal activity. Using this embedded mapping and experimentally feasible measurements of the firing rate …
On The Finiteness Of Quantum K-Theory Of A Homogeneous Space, D. Anderson, Linda Chen, H.-H. Tseng
On The Finiteness Of Quantum K-Theory Of A Homogeneous Space, D. Anderson, Linda Chen, H.-H. Tseng
Mathematics & Statistics Faculty Works
We show that the product in the quantum K-ring of a generalized flag manifold G/P involves only finitely many powers of the Novikov variables. In contrast to previous approaches to this finiteness question, we exploit the finite difference module structure of quantum K-theory. At the core of the proof is a bound on the asymptotic growth of the J-function, which in turn comes from an analysis of the singularities of the zastava spaces studied in geometric representation theory. An appendix by H. Iritani establishes the equivalence between finiteness and a quadratic growth condition on certain shift operators.
The Complexity Threshold For The Emergence Of Kakutani Inequivalence, V. Cyr, Aimee S.A. Johnson, B. Kra, A. Şahi̇n
The Complexity Threshold For The Emergence Of Kakutani Inequivalence, V. Cyr, Aimee S.A. Johnson, B. Kra, A. Şahi̇n
Mathematics & Statistics Faculty Works
We show that linear complexity is the threshold for the emergence of Kakutani inequivalence for measurable systems supported on a minimal subshift. In particular, we show that there are minimal subshifts of arbitrarily low superlinear complexity that admit both loosely Bernoulli and non-loosely Bernoulli ergodic measures and that no minimal subshift with linear complexity can admit inequivalent measures.
The Topological Slice Genus Of Satellite Knots, P. Feller, Allison N. Miller, J. Pinzón-Caicedo
The Topological Slice Genus Of Satellite Knots, P. Feller, Allison N. Miller, J. Pinzón-Caicedo
Mathematics & Statistics Faculty Works
We present evidence supporting the conjecture that, in the topological category, the slice genus of a satellite knot P(K) is bounded above by the sum of the slice genera of K and P(U). Our main result establishes this conjecture for a variant of the topological slice genus, the ℤ–slice genus. Notably, the conjectured upper bound does not involve the algebraic winding number of the pattern P. This stands in stark contrast with the smooth category, where, for example, there are many genus 1 knots whose (n,1)–cables have arbitrarily large smooth 4–genera. …