Problem Statement A random walk on the 2-dimensional integer lattice begins at the origin. At each step, the walker moves one unit either left, right, or up, each with probability $\frac13$. (No downward steps ever.) A walk is a success if it reaches the point (1, 1). What is the probability of success? Note: One can vary the problem by varying the target point. Eg., use (1, 0) or (0, 1) instead. Perhaps there is a good method to resolve the general case of target (a, b). Source: Bruce Torrence, Randolph-Macon College
Objective $$ (\lambda, ) = f(\lambda) + ^n h_i(\lambda, \phi_i). $$ Define λ-parameter function, $$ (\lambda) = f(\lambda) + \sum_i^n h_i(\lambda, \phi_i) $$ Let ϕi* = minϕihi(λ, ϕi). The gradient of L has the following form $$ \del _i(\lambda) = \del f(\lambda) + n [\del h_i(\lambda, \phi_i^*)] $$ The SGD is reduced down to $$ \lambda \gets (1 - \rho)\lambda + \rho[n \phi_i] $$
1. _Describe the genus Australopithecus, including the range of dates it appears in the fossil record, major physical features including brain size, tooth complex (teeth and muscle for chewing) and what those features suggest for the Australopithecine diet, and any evidence for bipedalism. Why is this genus considered a hominid?_ Australopithecus lived from 3.9 mya to 1 mya. They all had small brains. They had huge chewing complex like large molars and bigger chewing muscles evidenced by huge saggital crests and zygomatics; this all suggests that their diets consisted of vegetation and tree barks: things that require heavy gnawing. They also had close-to humanlike pelvic structures, occipital crests. They are considered in the hominid family because they share a number of traits such as full bipedalism. 2. _Why is Aegyptopithecus an important primate fossil?_ The Aegyptopithecus is sometimes considered the very first Ape, due to it having many ape-like features such as being large, smaller eyes, active during daylight, and had low, rounded tooth clasps. Due to the fact that Evolution is a more continuous process, it is often difficult to track down the first known ape that diverged from its old-world monkey ancestors. However, the increase in traits observed in this species as well as the presence of old-world monkey-like traits suggest that Aegyptopithecus occured during a turbulant time where primates were beginning to appear. As such, Aegyptopithecus is important because 1) it may have been the first ape, but also 2) because even if it isn’t the first ape, its records give us a good indication of when our line first began, and therefore give us context as to why primates began to thrive. 3. _Give two possible reasons for the selection of human bipedalism._ One possible reason for the selection of human bipedalism comes from the hypothesis that humans diverged from their ancestors and preferred instead to cover large tracks of territory and hence walking on two legs was a trait that was selected because it is ENERGETICALLY EFFICIENT to cover more territory walking on two rather than four legs. Another reason was that bipedalism allowed us to free our arms from use during walking. Therefore, this allowed us to throw and catch and carry things. 4. _Describe the species Homo erectus. Include in your answer the range of dates it appears in the fossil record, geographic location, and the physical features of brain size, shape of the skull, tooth complex, and post-crania. Describe any evidence of culture for this species._ They span from 1.5mya to 150,000 years ago; it has been found in diverse areas such as Indonesia, China, Russia, North/East Africa, and Europe; they had large brains up to 1200 ccs, thick cranial bones, no post-orbital constriction, filled out instead by parietal bones, low, flat cranial vault, and a pointed skull in the back. They have smaller teeth which meant a diverse set of diets. There are also various evidences of social behaviour: the spread of tools across the world; but more importantly, evidence of hunting suggests that they are living groups, and must have shared and cooperated with each other. There are also evidence of stable home bases for groups, which suggest the development of culture for these species. 5. _Define the Multiregional Hypothesis and the Out of Africa Hypothesis for the appearance of fully modern humans._ The Multiregional Hypothesis, now largely considered dated, claims that modern humans evolved from Homo Erectus in SEPARATE geographic locations, with enough mixing that we all still remain one species. In contrast, the Out of Africa Hypothesis claims that we all evolved in Africa, and then dispersed out all over the world, letting selection take its place and allowed us to replaced whatever other hominids there were in the other worlds. 6. _What are the differences between Ramapiths and Dryopiths and why is this information important to the human fossil record?_ The Ramapiths and Dryopiths are the two types of the Myocene apes with Dryopiths predating Ramapiths. Dryopiths are Quadrapaths and are like tree-dwellers. However, their teeth patterns were similar to that of the Ramapiths, which are the more modern homonoids. Therefore, the fossils of these two types are important because they give insight and record to the split between apes-like (Ramapiths) and human-like homonoids.
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