Special curves and postcritically-finite polynomials

We study the postcritically-finite (PCF) maps in the moduli space of complex polynomials $\mathrm{MP}_d$. For a certain class of rational curves $C$ in $\mathrm{MP}_d$, we characterize the condition that $C$ contains infinitely many PCF maps. In particular, we show that if $C$ is parameterized by polynomials, then there are infinitely many PCF maps in $C$ if and only if there is exactly one active critical point along $C$, up to symmetries; we provide the critical orbit relation satisfied by any pair of active critical points. For the curves $\mathrm{Per}_1(\lambda)$ in the space of cubic polynomials, introduced by Milnor (1992), we show that $\mathrm{Per}_1(\lambda)$ contains infinitely many PCF maps if and only if $\lambda=0$. The proofs involve a combination of number-theoretic methods (specifically, arithmetic equidistribution) and complex-analytic techniques (specifically, univalent function theory). We provide a conjecture about Zariski density of PCF maps in subvarieties of the space of rational maps, in analogy with the Andr\'e-Oort Conjecture from arithmetic geometry.Comment: Final version, appeared in Forum of Math. P

Eye muscle proprioception is represented bilaterally in the sensorimotor cortex

The cortical representation of eye position is still uncertain. In the monkey a proprioceptive representation of the extraocular muscles (EOM) of an eye were recently found within the contralateral central sulcus. In humans, we have previously shown a change in the perceived position of the right eye after a virtual lesion with rTMS over the left somatosensory area. However, it is possible that the proprioceptive representation of the EOM extends to other brain sites, which were not examined in these previous studies. The aim of this fMRI study was to sample the whole brain to identify the proprioceptive representation for the left and the right eye separately. Data were acquired while passive eye movement was used to stimulate EOM proprioceptors in the absence of a motor command. We also controlled for the tactile stimulation of the eyelid by removing from the analysis voxels activated by eyelid touch alone. For either eye, the brain area commonly activated by passive and active eye movement was located bilaterally in the somatosensory area extending into the motor and premotor cytoarchitectonic areas. We suggest this is where EOM proprioception is processed. The bilateral representation for either eye contrasts with the contralateral representation of hand proprioception. We suggest that the proprioceptive representation of the two eyes next to each other in either somatosensory cortex and extending into the premotor cortex reflects the integrative nature of the eye position sense, which combines proprioceptive information across the two eyes with the efference copy of the oculomotor comman

Solving exponential diophantine equations using lattice basis reduction algorithms

Let S be the set of all positive integers with prime divisors from a fixed finite set of primes. Algorithms are given for solving the diophantine inequality 0< x − y < yδ in x, y S for fixed δ (0, 1), and for the diophantine equation x + Y = z in x, y, z S. The method is based on multi-dimensional diophantine approximation, in the real and p-adic case, respectively. The main computational tool is the L3-Basis Reduction Algorithm. Elaborate examples are presented

Effective String Theory of Vortices and Regge Trajectories

Starting from a field theory containing classical vortex solutions, we obtain an effective string theory of these vortices as a path integral over the two transverse degrees of freedom of the string. We carry out a semiclassical expansion of this effective theory, and use it to obtain corrections to Regge trajectories due to string fluctuations.Comment: 27 pages, revtex, 3 figures, corrected an error with the cutoff in appendix E (was previously D), added more discussion of Fig. 3, moved some material in section 9 to a new appendi

Understanding Confinement From Deconfinement

We use effective magnetic SU(N) pure gauge theory with cutoff M and fixed gauge coupling g_m to calculate non-perturbative magnetic properties of the deconfined phase of SU(N) Yang-Mills theory. We obtain the response to an external closed loop of electric current by reinterpreting and regulating the calculation of the one loop effective potential in Yang-Mills theory. This effective potential gives rise to a color magnetic charge density, the counterpart in the deconfined phase of color magnetic currents introduced in effective dual superconductor theories of the confined phase via magnetically charged Higgs fields. The resulting spatial Wilson loop has area law behavior. Using values of M and g_m determined in the confined phase, we find SU(3) spatial string tensions compatible with lattice simulations in the temperature interval 1.5T_c < T < 2.5T_c. Use of the effective theory to analyze experiments on heavy ion collisions will provide applications and further tests of these ideas.Comment: 18 pages, 5 figures, v2: fixed archive title (only

Kinetic cross coupling between non-conserved and conserved fields in phase field models

We present a phase field model for isothermal transformations of two component alloys that includes Onsager kinetic cross coupling between the non-conserved phase field and the conserved concentration field. We also provide the reduction of the phase field model to the corresponding macroscopic description of the free boundary problem. The reduction is given in a general form. Additionally we use an explicit example of a phase field model and check that the reduced macroscopic description, in the range of its applicability, is in excellent agreement with direct phase field simulations. The relevance of the newly introduced terms to solute trapping is also discussed