Electrophysiological Evidence of the Amodal Representation of Symmetry in Extrastriate Areas

Extrastriate visual areas are strongly activated by image symmetry. Less is known about symmetry representation at object-, rather than image-, level. Here we investigated electrophysiological responses to symmetry, generated by amodal completion of partially-occluded polygon shapes. We used a similar paradigm in four experiments (N=112). A fully-visible abstract shape (either symmetric or asymmetric) was presented for 250ms (t0). A large rectangle covered it entirely for 250ms (t1) and then moved to one side to reveal half of the shape hidden behind (t2, 1000ms). Note that at t2 no symmetry could be inferred from retinal image information. In half of the trials the shape was the same as previously presented, in the other trials it was replaced by a novel shape. Participants matched shapes similarity (Exp. 1 and Exp. 2), or their colour (Exp. 3) or the orientation of a triangle superimposed to the shapes (Exp. 4). The fully-visible shapes (t0-t1) elicited automatic symmetry-specific ERP responses in all experiments. Importantly, there was an exposure-dependent symmetry-response to the occluded shapes that were recognised as previously seen (t2). Exp. 2 and Exp.4 confirmed this second ERP (t2) did not reflect a reinforcement of a residual carry-over response from t0. We conclude that the extrastriate symmetry-network can achieve amodal representation of symmetry from occluded objects that have been previously experienced as wholes.

Divide and conquer, just like Riemann sums in calculus

“Divide and conquer, just like Riemann sums in calculus” offers a basic introduction for how to estimate—with any desired non-zero margin of error—the area of an irregular shape. For an initial underestimate, readers are encouraged to draw a large rectangle that fits inside the irregular shape and then use the grade school formula to calculate the rectangle’s area: area equals length times width. To refine this underestimate, readers learn to draw multiple smaller rectangles inside the shape whose areas they also sum. An analogous method is provided for overestimates. The discussion concerning how to obtain underestimates or overestimates with any desired margin of error is illustrated with numerous hand-drawn sketches. Mathematics students and enthusiasts are encouraged to consider dividing and conquering in all challenges they face in mathematics or life. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Exact fit problem generator for cutting and packing, revisiting of the upper deck placement algorithm

Problem generators are practical solutions for generating a set of inputs to specific problems. These inputs are widely used for testing, comparing and optimizing placement algorithms. The problem generator presented in this paper fills the gap in the area of 2D Cutting & Packing as the sum of the area of the small objects is equal to the area of the Large Object and has at least one perfect solution. In this paper, the already proposed Upper Deck algorithm is revisited and used to test the proposed generator outputs. This algorithm bypasses the dead area problem that occurs in most of all well-known strategies of the 2D Single Knapsack Problem where we have a single large rectangle to cover with small, heterogeneous rectangle shapes, whom total area exceeds the large object’s area. The idea of placing the small shapes in a free corner simplifies and speeds the placement algorithm as only the available angles are checked for possible placements, and collision detection only requires the checking of corners and edges of the placed shape. Since the proposed generator output has at least one exact solution, a series of optimization performed on the algorithm is also presented.

Application of Asymmetric Eigenvalue Method for Dynamic Analysis of Aqueducts

Taking into consideration of the boundary conditions in fluid-solid interactions, the author built the FSI large rectangle aqueduct model of dynamic analysis according to the equation dynamic performance of the fluid-solid coupling system. Large aqueduct of the dynamic properties were analyzed with asymmetric eigenvalue method, dynamic properties rules were calculated by changing the depth of the water in the aqueduct. Aqueduct, also called elevated canal, is usually built over valley, lower land or river for conducting water from a distance or even for shipping.The aqueduct is one of the most important hydraulic structures in Yindaruqin Irrigation Project. After aqueduct is built, all kinds of reasons get its degree of safety descent so as to affect its ordinary running with the time going. Based on fluid-solid coupling system, a FSI analysis model of the aqueduct structure is established.

Structure Analysis of Large Rectangle Aqueduct during Construction and Operation Periods

Using large structure analysis software, the stress and displacement are calculated for Shuangji River rectangle aqueduct structure during construction and operation periods. The 19 different load combination cases are considered. The variation rule of mechanics behavior is obtained for concrete material during different load combination period. The simulation results show that the conventional method simulating prestress can lead to stress concentration in the point the force is acted on. It is a relatively good simulation method to add a steel plate to the area where the prestressed strands are acted on. In this way, the local stress concentration can be reduced. When use pseudo-static method for calculation of the seismic load, the load case considering the seismic load is a control load case. The Shuangji River aqueduct can meet the requirement of intensity and rigidity during construction and operation periods.

Some Plethysm Results related to Foulkes' Conjecture

We provide several classes of examples to show that Stanley's plethysm conjecture and a reformulation by Pylyavskyy, both concerning the ranks of certain matrices $K^{\lambda}$ associated with Young diagrams $\lambda$, are in general false. We also provide bounds on the rank of $K^{\lambda}$ by which it may be possible to show that the approach of Black and List to Foulkes' conjecture does not work in general. Finally, since Black and List's work concerns $K^{\lambda}$ for rectangular shapes $\lambda$, we suggest a constructive way to prove that $K^{\lambda}$ does not have full rank when $\lambda$ is a large rectangle.