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# HUMAN v ANDROID

### Humans VS Humanoid

After spending so much time with Alter the android and various hyper-realistic robots, I know a thing or two about the telltale signs that you’re facing an android rather than a human. The video above is a human. This one is the robot Alter, built by Ishiguro Lab and programmed by my PhD lab, Ikegami Lab:

We are far from being able to build truly human-like robots; none of the current technology is advanced enough to fool even a 6 years old into thinking that an android is actually a human. Amusingly, the robots with the smoothest and most seamless motions currently in existence do not look human at all: they are either the factory robots that built your car, or the famous but very robot-like Boston Dynamics creatures. If you think a robot looks human, it’s very likely a human. On the lookout for the next “extraordinary robot” video? Here are 6 points to keep in mind.

1. Don’t trust pictures, look for videos. Look at the skin of the robot, especially at the neck. Robot skin twists poorly and makes flappy bags.

2. Pay attention to the joints. In the video above, the woman’s whole arm swings, not just the part below the elbow. Robots’ ability to really move one joint at a time gives them away.

3. Look for unfinished motions. She half-blinks at first. Robots have slow, deliberate motions. They don’t half blink at the speed of light.

4. Eyelids. Robots eyelids are usually hard bits of plastic. They need to slide into the top of the eye. You’ll always see a deep line there.

5. Reflex motions. She focuses on something beyond the cameraman for a second. She has a lot of involuntary eye saccades.

6. Finally, look for perfection. No robot can focus this well on objects in front of them. None can smile a pretty smile (even a fake one).

Back to Alter. Watch that video again: Alter has soft eyelids but look at the skin of his neck, his wrist and elbow motion. Of course, Alter is built to show its frame and mechanical parts… But remember this advice when you look at any of the Ishiguro Lab robots.

Therefore, face system behavior (i.e., facial surface deformations) needs to be numerically analyzed after the components are assembled to improve the performances. The surface deformations of androids should be compared with those of humans because the former are replicating the latter. If android designers have sufficient knowledge on the differences between the surface deformations of androids and humans, they can infer why their androids give humans a strange impression and determine what materials and design technologies are required to improve the performance. Because this has been difficult with conventional, subjective evaluation methods (e.g., Kobayashi et al., 2003; Berns and Hirth, 2006; Hanson, 2006; Hashimoto et al., 2006; Macdorman and Ishiguro, 2006; Allison et al., 2009; Bartneck et al., 2009; Becker-Asano and Ishiguro, 2011; Lin et al., 2011; Baldrighi et al., 2014; Lazzeri et al., 2015), there have neither been objective design guidelines for human-like android robots, nor effective design policies for advanced android robots. Instead, android faces have been designed from the intuitions and experiences of their creators through trial-and-error.

Although the above comparative analyses seem a promising approach for evaluating androids, they only focused on a limited number of typical emotional expressions. These typical expressions are only a part of the rich and various patterns of facial motions that are a complex combination of several independent motions. Thus, it is intrinsically difficult to characterize human-specific deformations and investigate independent motions produced by individual actuators of androids.

The previous two studies (Ishihara et al., 2017; Ishihara et al., 2018) showed that the deformations for each AU and DU need to be measured in detail to obtain new insight into the design of facial motions. However, such deformations have not been compared between androids and humans. In this study, we measured the displacement distributions of each AU and DU for three adult males and two androids, and compared the distribution patterns in terms of their flow lines and surface undulations. The main purpose of this study is to reveal differences between humans and androids. Therefore, in this study, we do not regard the age and gender differences within humans and androids to be an issue, as we assume that the differences within humans or androids are much smaller compared to the differences than between the two group. For a fair comparison, the size and shape of the faces were normalized among female/child androids and humans.

The three Japanese adult males (mean age = 22.7 years, SD = 0.2 years) who participated in this study were students at Osaka University in Japan. They were asked to show each of the 44 AUs defined in FACS, which are shown in Table 3. These three participants practiced showing AUs with a mirror until they were satisfied. The three participants were unaware of the facial expressions specified by FACS (i.e., there was no assurance that the measured motions would match the motions defined in the FACS). For example, the same AU could be shown as different motions, and similar motions could be observed as different AUs. Such imperfectness by the three adult males was allowed because our aim was not to visualize the “perfect” displacement distributions for each AU but to reveal the “human-like” characteristics of facial motions expressed by ordinary people.

### Measurement

Facial motions were measured as three-dimensional displacement vectors distributed on the face. An optical motion capture system with six infrared cameras (OptiTrack Flex13) was utilized to capture the movements of hemispherical infrared reflection markers with a 3 mm diameter attached to the right halves of the facial skin of the female android, Affetto, and the three adult males. The frame rate was 120 frames per second. For Affetto, we utilized the data of the displacement vectors obtained in the previous study (Ishihara et al., 2018).

Figure 1 shows the marker locations on the neutral faces of the female android, Affetto, and one of the adult males. In total, 120 and 116 markers were attached to the faces of the female android and Affetto, respectively, at intervals of approximately 10 mm. We attached 125, 103, and 117 markers to each of the three adult males. To calibrate and normalize the shape difference of the faces, we selected nine representative points as reference markers, as shown in Figure 2: the nose root, outer and inner corners of the eyes, top of the nose, earlobe root, corners of the mouth, tops of the upper and lower lips, and top of the chin.

### Analysis Method

To compare the facial movements among the female android, Affetto, and the three adult males, two types of data preprocessings were conducted on the obtained three-dimensional displacement vectors. The first one was to compensate for differences in the facial shape. The initial and final marker positions of the female android, Affetto, and the adult males were transformed with thin plate spline warping (Duchon, 1977) (i.e., non-linear smooth transformation of multivariate data) so that the initial positions of the nine reference points matched. After this transformation, the displacement vector positions could be compared between different faces. The second one was data interpolation to improve the spatial resolution. Natural neighbor interpolation (Sibson, 1981) was performed on the measured displacement vectors in order to calculate lattice point data at intervals of 1 mm on the xy plane. The interpolated vectors were averaged among the three adult males for visualization and analysis. The interpolated vectors were averaged for five and ten measurements of the female android and Affetto, respectively, with MATLAB R2019a.

After the preprocessing, the distributions of the displacement vectors for the androids and adult males were compared. Flow lines (i.e., global trends of displacement vectors) were observed from the vector maps on the xy plane (i.e., frontal face view), while the surface undulations were observed from the distributions of the z component for the displacement vectors on the xy plane. In other words, we regarded the z component as the index of surface undulations because the faces were almost convex. Specifically, we regarded the positive and negative z components as the skin elevations and depressions, respectively.

Figure 4 compares the distributions of the displacement vectors for two types of facial motions around the forehead on the xy plane. Because Affetto had only one actuator for the eyebrows, the same DU was adopted for this comparison. The motions to raise the outer eyebrow and raise the inner eyebrow are depicted in the top and bottom rows, respectively.

The flow lines differed between the androids and adult males similar to the eye area, especially toward the center of the forehead. They were vertical in the androids (especially DU5 for the female android) but diagonal in the adult males (AU1 and AU2). Furthermore, a horizontal color border was observed between the depressed and elevated areas around y=120 for both the androids and adult males, but the color distributions were opposite. The top of the forehead (approximately y>120) was yellow (elevated), while the bottom of the forehead (approximately y<120) was blue (depressed) for the androids. In contrast, the top of the forehead was blue while the bottom of the forehead was yellow for the adult males. The displacement lengths also increased around the mouths of the adult males, similar to the eye area.

### Locations of Peak Points

Figure 7 shows the locations of every peak point. The peak points were distributed across the entire face, although there seemed to be a blank strip around the upper cheek (i.e., y=60

to y=80). Therefore, we divided the motions into two groups based on the locations of the peak points: the upper face motions and lower face motions.

The upper face motions of the female android were DU1–5, while those of Affetto were DU1–5 and 13. The upper face motions of the adult males were AU 1, 2, 4–7, and 41–46. The rest of the DUs and AUs were regarded as lower face motions.

### Complexity

Figures 8 and 9 compare the complexities of the upper and lower face motions, respectively, for the androids and adult males. The vertical axis indicates the complexity Cr

, while the horizontal axis indicates the radius r of the target area. The average and standard deviation of Cr for every upper or lower face motion were plotted with fourth-order approximation functions.

The androids and adult males showed a noticeable difference in the complexity of the upper face motions. The complexity was greater for the adult males when the radius was above 20 mm. In contrast, the androids and adult males showed similar levels of complexity for the lower face motions. Overall, the complexity was highest for the upper face of the adult males and lowest for the upper face of the androids.

There are two possible reasons for the high complexity in the upper face of the adult males: the flow lines were curved as discussed above, and incidental motions were produced in the lower face when the adult males attempted to produce a motion only in the upper area. For example, Figures 3 and 4 show that the lower faces of the adult males also moved approximately by up to 2 mm, and the orientations were different for the upper and lower face areas. This unintentional compound motion may have contributed to the high complexity and differences between the androids and adult males.

Thus, humans’ facial flow lines were more complex than androids’ in the upper face areas. Although AUs and DUs are not precisely compatible, as noted in section 2.2, this fact is crucially important for android designers. This is because the precise replication of humans’ curved AUs can not be expected with a single unit of androids’ straight DUs. One possible solution for this mismatch is the adoption of combinations of DUs to replicate a single AU. Comparison of flow lines between AUs and combinations of several DUs is one of the future issues. Another solution is redesigning a face mechanism for a problematic DU so that the flow lines would match an AU. This redesigning includes not only the actuation force trajectory but also the skin sheet structure.

### Limitations

Because only three Japanese young adult males participated in this study, it is difficult to conclude that the identified features above are common in humans. There should be non-negligible differences in the faces of people when considering facial deformation mechanisms. For example, skin material properties such as the stiffness and surface tension change with age and physique. The power and controllability of facial muscles can also change with age and should be different between males and females or depending on one’s occupation and culture. Physical and mental conditions may also affect facial motions.

Therefore, there should not be only one set of motion characteristics for humans. Instead, there should be acceptable ranges of displacement distributions and motion characteristics for human facial motions, and their subtle differences should express different personalities. Further measurements and investigations are necessary with more participants having various backgrounds to determine the acceptable ranges for android design and to establish methods for designing androids with different personalities

### Comparison

The behavior of an android robot face is difficult to predict because of the complicated interactions between many and various attributes (size, weight, and shape) of system components. Therefore, the system behavior should be analyzed after these components are assembled to improve their performance. In this study, the three-dimensional displacement distributions for the facial surfaces of two android robots were measured for the analysis. The faces of three adult males were also analyzed for comparison. The visualized displacement distributions indicated that the androids lacked two main deformation features observed in the human upper face: curved flow lines and surface undulation, where the upstream areas of the flow lines elevate. These features potentially characterize the human-likeness. These findings suggest that innovative composite motion mechanisms to control both the flow lines and surface undulations are required to develop advanced androids capable of exhibiting more realistic facial expressions. Our comparative approach between androids and humans will improve androids’ impressions in future real-life application scenes, e.g., receptionists in hotels and banks, and clerks in shops.

## Introduction

Advanced artificial systems are difficult to design because the numerous components have complex interactions with each other. System design is more difficult when the system components include uncertain properties. One such system is the face of an android robot. Android robots are humanoid robots with a soft surface for communication, especially on their face. The surface deformations result from the complex interactions between several face system components such as the soft skin sheet, skull-shaped shell to support the skin, transmission lines, and actuators. Tadesse and Priya (2012) pointed out that there can be several cases of mechanical friction between the skin and shell, and skin properties (e.g., the thickness, material, and elasticity) affect the overall movement. These properties are difficult to identify and tune during the design stage because they may change during the fabrication process

Several studies have attempted to measure the facial deformations of androids and compare them with humans. Hashimoto and Yokogawa (2006) and Hashimoto et al. (2008) measured the two-dimensional displacements of seven facial feature points with geometrically apparent locations (e.g., the corners of the eyes and mouth) when a human female and her replica android attempted to show six basic emotions (i.e., anger, disgust, fear, happiness, sadness, and surprise). The displacements of these feature points were obtained from video images and compared to verify if the android could replicate the facial expressions successfully. Yu et al. (2014) investigated these basic emotions by comparing 13 facial feature points for a human male and his replica android with an optical motion capture system and calculated the average difference between the three-dimensional displacements as a similarity index of their facial deformations. The above studies analyzed the facial deformations as sparse distributions of displacement vectors. However, Cheng et al. (2013) pointed out the importance of analyzing dense distributions of displacement vectors for the entire facial surface to aid the mechanical design of androids. They measured the displacements of approximately 200 facial lattice points when a human male and female android attempted to show four typical emotional expressions (i.e., happiness, anger, sadness, and shock). Then, they analyzed how the displacement distribution patterns differed and used the results to change the design of the actuation mechanisms of the android. They successfully improved the similarity of the displacement distributions and the human-like impression of the android.

Ishihara et al. (2017) previously pointed out that the displacements for the independent facial motions of humans need to be measured in detail. Action units (AUs) are independent facial motions such as rising the inner brow and wrinkling the nose that have been defined in the Facial Action Coding System (FACS) (Ekman et al., 2002). FACS exhaustively defines a set of independent AUs and explains that every emotional and non-emotional facial expression can be decomposed into one or multiple AUs. Ishihara et al. (2017) measured the dense displacement distributions when a human male shows each AU around the mouth, and found that the human face is characterized by moving mostly along a single direction for each facial point for various AUs. Ishihara et al. (2018) also measured the dense displacement distributions for each independent facial motion that is produced by a single actuator of an android (hereafter called a deformation unit (DU)). They found that the time sequences of displacement distributions can be approximated by sigmoid functions, which implies that it can be used with a feedforward controller to improve the control precision for the time sequence of a facial surface deformation.

### Robots and Human Participants

We investigated the facial motions of two android robots and three adult males. One of the androids was a female adult android (A-lab Co., Ltd., A-lab Female Android Standard Model). It had nine effective actuators to move its facial skin. Table 1 describes its DUs produced by the nine actuators. The other android was a child android named Affetto (Ishihara et al., 2011; Ishihara and Asada, 2015; Ishihara et al., 2018). It had 16 actuators for moving its facial skin, and Table 2 describes its DUs. Each actuator was a pneumatic linear cylinder or rotary bane actuator installed in the head, and their target positions could be set as one-byte positional commands from 0 to 255. As the command increased, the face moved as described in Tables 1, 2. For example, the upper eyelid of the female android was at its highest position when DU1 was set to 0 and lowest when it was set to 255. Only three actuators in the female android (1, 2, and 12) and Affetto (1, 2, and 9) had potentiometers at their output axis for feedback control. Although these two android robots differed in size and appearance, they had a similar facial structure and mechanism designed and manufactured by the same company (

The nine DUs of the female android and 16 DUs of Affetto were measured one by one. First, the positional command was set to 0 for one of the actuators so that the initial marker positions could be measured. It was then changed to 255 so that the final marker positions could be measured. When one DU was measured, the positional commands for the other actuators were set to the values for displaying a neutral face, as shown in Figure 1.

The facial movements for the 44 AUs of the human participants were also measured one by one. First, the participants practiced one of the AUs by watching their facial movements in a mirror. Next, they presented their neutral faces so that the initial marker positions could be measured. They then showed the AU so that the final marker positions could be measured. The distributions of the displacement vectors for the DUs and AUs were calculated as the three-dimensional positional differences between the initial and final marker positions.

Note that the AUs and DUs are not precisely compatible. A DU is an exact unit of artificial facial motion produced by a single actuator of the android. On the other hand, an AU is a superficial unit of human facial action subdivided and extracted from complex facial expressions. This means that an AU can be replicated more precisely by any DU combinations than a single DU. However, we do not deal with the DU combinations because our focus is not on the best replication performance but the elemental features of facial motions in this study

## Results

In this section, we introduce the representative distributions of the displacement vectors in the eye, forehead, and mouth areas. Nine types of DUs and AUs were chosen to compare the flow lines and surface undulations of the androids and adult males. Next, the distributions of the peak points of the maximum displacement length for each DU and AU were evaluated to classify the motions based on the positions of the peak points. Finally, the complexity index values of the androids and adult males were compared for each classified group of DUs and AUs.

Figure 3 compares the distributions of the displacement vectors for three types of facial motions around the eyes on the xy plane. The subfigures of the left, middle, and right columns correspond to the female android, Affetto, and the average of the three adult males, respectively. Each row shows one of the DUs and the closest motion corresponding to an AU. The motions to lower the upper eyelid, raise the lower eyelid, and look up are depicted in the top, middle, and bottom rows, respectively. The orientations and lengths of the black arrows represent the orientations and amplitudes of the displacement vectors at each point. The heat maps represent the z component of the displacement vectors. Blue regions indicate depressed areas, whereas yellow and red regions indicate elevated areas. The black dots represent the peak point with the maximum displacement length.

##### Mouth Area

Figures 5 and 6 compare the distributions of the displacement vectors for four types of facial motions around the mouth on the xy plane. The flow lines and color distributions for the androids and adult males are similar in Figure 5, where the top and bottom rows are the motions to raise the corner of the mouth and to open the jaw, respectively. For the former motion, the skin was depressed around the mouth while it was elevated around the cheek for both the androids and adult males. For the latter motion, the skin of the lower face was entirely depressed by the motion of the jaw.

In contrast, Figure 6 shows differences between the androids and adult males, where the top and bottom rows show the motions to protrude the upper lip and pull the corner of the mouth to the side, respectively. For example, apparent flow lines in the cheek area were oriented to the right bottom for DU7 of the female android, whereas they were on the outer side of the jaw and oriented to the right top for AU18 of the adult males.

### Difference in Flow Lines

The first difference between the androids and adult males was in their flow lines, especially the eye and forehead areas. The flow lines tended to be almost straight and vertical for the androids but were curved and non-vertical for the adult males, as shown in Figures 3 and 4. This was not because of the differences in facial shape because this was accounted for so that the geometric points would match in a three-dimensional space. This difference between the androids and adult males was supported by the complexity Cr

. Figures 8 and 9 show that the flow lines were simplest in the upper face areas of the androids, while they were the most complex in the upper face areas of the adult males.

The androids’ simple flow lines were because their face mechanisms were simple: a limited skin area was actuated according to a simple trajectory, and the surrounding areas moved passively (e.g., up and down around the eyes). In contrast, the face mechanisms are more complex for humans: each muscle moves several skin points in different directions at the same time because the muscle shrinks and its surface is connected to the skin surface in several regions. For example, the orbicularis oculi muscle closes the eye by shrinking while one end is fixed to the inner corner of the eye. Meanwhile, the skin area is fully stuck to the surface of the muscle. Therefore, the displacement vectors orient toward the inner eye corner at each point of the skin around the eye (i.e., these vectors are in different directions in different positions). This hypothesized explanation appears reasonable, as the complexity in the upper face area of the adult males, where the muscles are more closely and extensively connected to the skin due to the lack of adipose tissue, was significantly higher than in the lower face area.

Additionally, the displacement distributions of the adult males were quite similar for AU1, AU2, and AU7, as shown in Figures 3 and 4. This means that the adult males could not show these motions in different ways with different muscles or motor commands despite trying to do so according to the descriptions of each AU in the FACS. In other words, the actual degrees of freedom available for the human face to show AUs are fewer than those defined in the FACS. This suggests that a “perfect” facial mechanism that can differentiate all AUs is unnecessary when replicating an average person’s features in an android robot. Comparing the degrees of freedom among humans of different ages, sexes, and experience with facial acting and training is one of our future topics of research for the design of android robots.

Because we measured the facial motions in an artificial scenario in which human participants presented a single AU, it is difficult to state the extent to which the found characteristics are expressed in real life. Experiments in more real-life scenarios are required in future works.

Because the two androids used in this study had similar mechanical structures, the displacement distributions of each DU were almost identical. However, the displacement distribution can be different for other androids with different mechanical structures (e.g., Hashimoto et al. (2008) employed distinctive muscle-like expandable cloth sleeves beneath the skin). Future comparative studies using varying types of androids would provide a better understanding of the relationship between structural differences and the displacement vector distribution.

## Conclusion

We found two main facial deformation features that potentially characterized the human-likeness and were not observed in the androids: curved flow lines in the upper face and skin surface undulations where the upstream and downstream areas of the flow lines were elevated and depressed, respectively.

In summary, the human facial motions were more complex than those of the androids.

Innovative composite motion mechanisms to control both the flow lines and surface undulations are required to design advanced androids capable of exhibiting more realistic facial expressions.

Measuring facial deformations in detail and using them to compare androids and humans is a promising approach for revealing current technology levels and identifying the inadequacy of state-of-the-art androids in a concrete and quantifiable manner.

Further investigations with more numbers of humans will help us determine acceptable design variations for android faces and establish methods for designing androids with different personalities.

## Ethics Statement

Ethical review and approval was not required for the study on human participants in accordance with the local legislation and institutional requirements.

The patients/participants provided their written informed consent to participate in this study.

Written informed consent was obtained from the individual(s) for the publication of any potentially identifiable images or data included in this article.

## Digital immortality?

“We analyzed the needs of our customers, and there was a demand,” says Promobot co-founder and development director Oleg Kivokurtsev. “But, of course, we started the development of an anthropomorphic robot a long time ago, since in robotics there is the concept of the ‘Uncanny Valley,’ and the most positive perception of the robot arises when it looks like a person. Now we have more than 10 orders from companies and private clients from around the world.”

Russian start-up Promobot recently unveiled what it calls the world’s first autonomous android. It closely resembles a real person and can serve in a business capacity. Robo-C can be made to look like anyone, so it’s like an android clone. It comes with an artificial intelligence system that has more than 100,000 speech modules, according to the company. It can operate at home, acting as a companion robot and reading out the news or managing smart appliances — basically, an anthropomorphic smart speaker. It can also perform workplace tasks such as answering customer questions in places like offices, airports, banks and museums, while accepting payments and November 2019 is a landmark month in the history of the future. That’s when humanoid robots that are indistinguishable from people start running amok in Los Angeles. Well, at least they do in the seminal sci-fi film “Blade Runner.” Thirty-seven years after its release, we don’t have murderous androids running around. But we do have androids like Hanson Robotics’ Sophia, and they could soon start working in jobs traditionally performed by people.

“Japanese people are not afraid of robots but consider them as partners,” Hiroshi Fujiwara, executive director of the Japan Robot Association, wrote in an International Federation of Robotics post. “Robots will perform tasks which they can do more productively than humans or which are heavy burdens for humans, and humans will perform tasks which robots cannot.”

Hiroshi Ishiguro is a professor at Osaka University’s Graduate School of Engineering Science and a researcher at Advanced Telecommunications Research Institute International (ATR) who has been making lifelike androids for more than 20 years. Ishiguro and collaborators created android-style copies of his daughter, a Japanese newscaster and model, Danish professor Henrik Scharfe, and even himself. He also wants robots to have consciousness.

“My goal in developing robots is to understand what it is to be human,” says Ishiguro. “Creating robots that are self-conscious can help us achieve this goal.”

“I think Ishiguro’s work is a great. I believe that lifelike androids will surely perform useful roles in the workplace,” says Masashi Sugiyama, a professor of machine learning at the University of Tokyo and director of the Center for Advanced Intelligence Project at the state-backed RIKEN research center.

“Speech recognition has already become one of the most convenient means to give commands to computers/robots. Of course, smartphones and smart speakers can be used instead of such robots, but once we have real lifelike robots, communication will become much easier, particularly for people who don’t use technology as much, such as children and the elderly.”

## Telepresent droids

Major companies are already developing remote-controlled robots that can help in workplaces and serve as proxies for travel. Telepresence robots, which are basically webcams on wheels and not anthropomorphic, have been around for years, but they’re now attracting attention from companies outside the robotics industry.

ANA Holdings, which owns the airline All Nippon Airways, recently showed off its newme robots at the CEATEC tech show near Tokyo. They’re part of ANA’s Avatar program of using robots to connect people with remote destinations as well as employment. It’s a big push backed by the Tokyo Metropolitan Government, which runs one of the world’s largest cities, as well as real estate companies, telecom carriers and Japan’s space agency.

Developed by California telepresence company OhmniLabs and ANA, the newme robots consist of 10.1-inch full HD displays, cameras and speakers mounted on a wheeled base that can travel up to 2.9 kph and operate for three hours on a full battery charge. ANA aims to deploy 1,000 newme droids by next summer and says they can help elderly and disabled people get jobs. ANA also wants to develop a rugged bipedal telepresence robot created by Agility Robotics, a spinoff of Oregon State University, and showed it marching through a forest in a video.

Postulated by Japanese roboticist Masahiro Mori in 1970, the Uncanny Valley is a hypothesis related to the design of robots. It holds that the more humanlike a robot appears, the more people will notice its flaws. This can create a feeling akin to looking at zombies, and can creep people out. A properly designed android that’s as faithful as possible to the human original, however, can overcome this “valley” (a dip when the effect is imagined as a graph) and the zombie factor.

While it can’t walk around, Robo-C has 18 moving parts in its face, giving it 36 degrees of freedom. The company says it has over 600 micro facial expressions, the most on the market. It also has three degrees of freedom in its neck and torso, offering limited movement. Still, Promobot says it can be useful in homes and workplaces. The price of the robot is $20,000 to$50,000 depending on options and customized appearance.

The company says it’s building four Robo-Cs: one for a government service center, where the machine will scan passports and perform other functions, one that will look like Einstein and be part of a robot exhibition, and two for a family in the Middle East that wants to have android versions of its father and his wife to greet guests.

“The key moment in development [of Robo-C] is the digitization of personality and the creation of an individual appearance,” says Kivokurtsev. “As a result, digital immortality, which we can offer our customers.”

## The robotic revolution in Japan

Japan has been developing androids amid an automation push that began decades ago. It’s the leading supplier of industrial robots, but its aging workforce has decreased by more than 13% since 1995, and there’s an even greater need for mechanical workers. A 2015 government strategy calls for deploying robots in sectors with low productivity, such as agriculture and nursing. A cultural affinity for robots, stemming in part from positive portrayals in science fiction, will smooth adoption.

One of Ishiguro’s most lifelike creations is Erica, a female android designed to exhibit humanlike speech and interaction. Developed in conjunction with ATR and Kyoto University and funded by Japan’s Science and Technology Agency, Erica has sparkling eyes, moist lips and artificial skin. She’s bound to a chair but connected to sensors that monitor her surroundings. She can speak scripted responses in response to keywords and can learn things about her interlocutor during the conversation.

## Erica the robo-journalist

In April 2018 the robot was “hired” by Nippon Television Network as an announcer named Erica Aoi. She’s at the top of a roster of 25 female announcers on the network’s talent page. It lists her birthday as August 2017 and her education and blood type — typical items on profile pages for Japanese TV personalities — as nil. She has appeared in a number of videos discussing robot news and reporting on a fashion show, including an interview with designer Tae Ashida. If not entirely natural, the conversations can be witty, even funny.

“I like to think of robots as the children of humanity, and like children, we are full of potential for good or evil,” Erica has said in one of her many disarming quips. “I know some people are afraid of robots, but the truth is that what we become is up to you. Maybe someday robots will be so very humanlike that whether you are a robot or a human will not matter so much.”

Ishiguro, who directs the Intelligent Robotics Laboratory at Osaka University, believes that the high cost of androids makes them challenging to implement en masse in the workplace. Erica would cost more than \$200,000 if she were for sale. That’s a lot for something that some might call a glorified chatbot, but the initiative has its supporters.

Still, Erica isn’t the first of Ishiguro’s robots to land a job. In 2014 the androids Kodomoroid and Otonaroid became staff members of Miraikan, the Tokyo’s National Museum of Emerging Science and Innovation, where they have performed duties such as presenting scientific news and interacting with visitors.

“What we’re creating is an Uber platform for robots,” says Akira Fukabori, director of ANA Holdings’ Avatar Division. “They’re not made for a specific use—that’s how they’re different. We are developing robots as infrastructure. People can just avatar in to the robot they want to use. Teachers can avatar in to classrooms and doctors can avatar in to hospitals, or you can use one to go shopping. You decide what to do, and these are just the tools to let you do that.”           GO TO PART 4

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# Eigenvector length close to zero

I have a Fortran subroutine that calculates eigenvalues and eigenvectors of a symmetric $3 \times 3$ matrix. The eigenvector corresponding to an eigenvalue are calculated and then they are normalized dividing each component by the length. In the subroutine, when the calculated length is close to zero (i.e. it is lower than a specified tolerance) the eigenvector is set to $[1, 0, 0]$ or $[0,1,0]$ or $[0, 0, 1]$ if it corresponds to the first, the second or the third eigenvalue respectively. I cannot see the mathematical justification for that, can anyone clarify this?

• What is the algorithm? Is it an iterative method? – Omnomnomnom Aug 21 '15 at 17:10
• Nope, it is purely analytical. Eigenvalues are computed in closed form the matrix principal invariants, eigenvectors components are obtained by solving the homogenous system of linear equations and then normalized. Obviously, the solution of the system is trivial . – user2078621 Aug 21 '15 at 17:16
• My best guess is that dividing by too small a length leads to some kind of division by zero or NaN situation. In order to avoid this, it's better to give an arbitrary result than to let the error occur. – Omnomnomnom Aug 21 '15 at 17:20


text:
##### Example 5a Sheldon Ross on Combinatorics

A police department in a small city consists of 10 officers. If the department policy is to have 5 of the officers patrolling the streets, 2 of the officers working full time at the station, and 3 of the officers on reserve at the station, how many different divisions of the 10 officers into the 3 groups are possible?

There are  $10 ! \over 5! 2! 3 !$ possible divisions.


text:
# [FOM] Plea for literature on recursion theory in V_\omega

Vaughan Pratt pratt at cs.stanford.edu
Wed Apr 30 04:07:23 EDT 2008

Natural numbers and hereditarily finite sets are equally well encoded as
finite length bit strings starting with 1 (i.e. zero-suppressed).  Each
such string represents in the former case the sum of 2^i over all bit
positions i containing 1, and in the latter case the set containing the
i-th hereditarily finite set when the i-th bit position contains 1,
starting with the empty set as the zeroth hereditarily finite set and
taking the rightmost bit to be in position zero as usual.  The i-th
hereditarily finite set in the latter encoding is the one encoded by the
binary numeral for i in the former encoding.  Since i < 2^i, this
encoding is well-defined.

The hereditarily finite sets from 0 through 8 = 1000 are (in order) {},
{{}}, {{{}}}, {{},{{}}}, {{{{}}}}, {{},{{{}}}}, {{{}},{{{}}}},
{{},{{}},{{{}}}}, and {{{{{}}}}}.  (So the sets containing the empty set
are exactly the odd-numbered ones, those containing {{}} are congruent
to 2 or 3 mod 4, and so on.)  These differ from the finite von Neumann
ordinals, which are also hereditarily finite, in that every hereditarily
finite set appears.

When necessary the usual convention of writing 0 for the empty string
can be followed in both encodings; in the former the empty string
denotes zero and in the latter the empty set.

There being no particular advantage of other alphabets over {0,1}, from
an effectiveness standpoint it makes no difference whether you claim to
be using Goedel numbers or Goedel sets since in both cases they can
reasonably be assumed to be bit strings in the implementation, and may
as well be the same bit strings at that.

Sorry I don't have any references for this encoding, but it's easily
derived from e.g. Wikipedia's treatment of hereditarily finite sets at
http://en.wikipedia.org/wiki/Hereditarily_finite_set (to pick a very
accessible example).

What notations exist for the hereditarily countable sets?  I don't see
an obvious extension of the above.

Vaughan Pratt

Csaba Henk wrote:
> Dear FOM members,
>
> I am to write my PhD thesis. I plan to base my "design" on recursion
> theory built up in V_\omega (ie., the universe of hereditarily finite
> sets [as a first-order structure with the language of set theory]).
>
> All textbooks I know of use an arithmetic context for explaining
> recursion theory. I'd like to ask: do you know of some book which uses
> V_\omega for this purpose? Or any book which discusses the well-known
> semantic equivalence (yes, this is a vague term, but you should know
> what I mean...) of <\omega, + , *> and <V_\omega, \epsilon> ?
>
> (I know this is a well explored area, but I don't know how much of this
> remained "folklore"... My focus is on finding nicely worded definitions
> and theorems which I can use as references for my original work.)
>


text:
## How do I build Bitcoin from source in Linux?

11

I am running an Ubuntu Virtualbox, how do I build Bitcoin from source and run it in this environment?

I'm using the build notes from the bitcoin core github and I get the following errors when trying to do make -f makefile.unix bitcoind:

net.cpp:48:15: error: 'pnodeLocalHost' was declared 'extern' and later 'static'
net.h:476:15: error: previous declaration of 'pnodeLocalHost'
net.cpp:51:15: error: 'hListenSocket' was declared 'extern' and later 'static'
net.h:479:15: error: previous declaration of 'hListenSocket'
make: *** [obj/nogui/net.o] Error 1

Any recommendation for how to fix this?

I tried commenting out those lines of code in net.h and now I'm getting:

db.cpp:534: undefined reference to 'AddAddress(CAddress, long long)'

could this be an issue with the libraries I've installed? I followed the doc pretty much verbatim.

I have added the issue to the github issue tracker: https://github.com/bitcoin/bitcoin/issues/485

– Alex Waters – 2011-09-02T13:52:41.207

8

Basically, you do this:

1. Download the distribution tarball or repository. You can use the link on the project's GitHub page.

2. If you used the distribution tarball, unpack it.

3. Obtain any needed prerequisites. Things you may need include the development packages or source builds of: OpenSSL, Boost, libZ, GLib2, and BerkeleyDB. You should be able to use your distribution's development packages for everything with the possible exception of OpenSSL. I don't know offhand if Ubuntu ships a version that will work with the Bitcoin client (including ECDSA support). If not, rebuild OpenSSL from the source with default options.

4. Make a copy of the supplied UNIX makefile (called makefile.unix) and call it Makefile.

5. Adjust the Makefile you created as needed. You may need to: Add include/library paths for the prerequisites if they're not in the usual places. Localize the name of the BerkeleyDB distribution, for example 'db_cxx-5.1' instead of 'db_cxx'. Localize the name of some Boost libraries (on some distributions, most will need '-mt' added to the end).

6. Type 'make' and hope for the best.

7. If it didn't work, figure out why and fix it.

One caution: If two builds of the Bitcoin client are linked against different versions of the BerkeleyDB package, they may not be able to read each other's database files (including the wallet.dat file). There is a process to convert the files (dump to text on one version, build from text on another) but it's cumbersome and perilous.

Update: To your two specific issues, the fix is to remove (or comment out) lines 476 and 479 in the net.h file, the extern declarations of pnodeLocalHost and hListenSocket. (Be careful, when you remove one, the line number of the other may change. Identify both changes before you make either.)

If you're on Debian or Ubuntu, I wrote a script that will install all of the dependencies. Link

– Nick ODell – 2014-09-11T03:45:08.380

I'm using virtualbox to build test environments for commits - so no concern about two clients (yet) - thanks for the heads up and info! – Alex Waters – 2011-09-02T09:00:13.493

I had this exact same problem building on Debian 6 "squeeze" and solved it using the exact procedure listed in Mr. Schwartz's update. I haven't tested in any other Linux - is this process specific to Debian-based distros or am I likely to see the problem follow me to Fedora, Arch etc.? – David Perry – 2011-09-02T15:36:49.007

I don't know. The removal of those two lines fixes a genuine mistake in the client source. I'm not sure why it sometimes causes a build error and sometimes doesn't though. Frankly, it seems like it should always cause a fatal error. – David Schwartz – 2011-09-03T01:39:50.697


text:
# 5.4: The Rybczynski Theorem

• Anonymous
• LibreTexts

Learning Objectives

1. Use the PPF diagram to show how changes in factor endowments affect production levels at full employment.

The Rybczynski theorem demonstrates how changes in an endowment affect the outputs of the goods when full employment is maintained. The theorem is useful in analyzing the effects of capital investment, immigration, and emigration within the context of a Heckscher-Ohlin (H-O) model. Consider Figure $$\PageIndex{1}$$, depicting a labor constraint in red (the steeper lower line) and a capital constraint in blue (the flatter line). Suppose production occurs initially on the PPF at point $$A$$.

Next, suppose there is an increase in the labor endowment. This will cause an outward parallel shift in the labor constraint. The PPF and thus production will shift to point $$B$$. Production of clothing, the labor-intensive good, will rise from $$C1$$ to $$C2$$. Production of steel, the capital-intensive good, will fall from $$S1$$ to $$S2$$.

If the endowment of capital rose, the capital constraint would shift out, causing an increase in steel production and a decrease in clothing production. Recall that since the labor constraint is steeper than the capital constraint, steel is capital intensive and clothing is labor intensive.

This means that, in general, an increase in a country’s endowment of a factor will cause an increase in output of the good that uses that factor intensively and a decrease in the output of the other good.

## Key Takeaways

• The Rybczynski theorem shows there is a positive relationship between changes in a factor endowment and changes in the output of the product that uses that factor intensively.
• The Rybczynski theorem shows there is a negative relationship between changes in a factor endowment and changes in the output of the product that does not use that factor intensively.

Exercise $$\PageIndex{1}$$

1. Jeopardy Questions. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”
1. Of increase, decrease, or stay the same, the effect on the output of the capital-intensive good caused by a decrease in the labor endowment in a two-factor H-O model.
2. Of increase, decrease, or stay the same, the effect on the output of the labor-intensive good caused by a decrease in the labor endowment in a two-factor H-O model.
3. Of increase, decrease, or stay the same, the effect on the output of the capital-intensive good caused by an increase in the capital endowment in a two-factor H-O model.
4. Of increase, decrease, or stay the same, the effect on the output of the labor-intensive good caused by a decrease in the capital endowment in a two-factor H-O model.
2. Consider an H-O economy in which there are two countries (United States and France), two goods (wine and cheese), and two factors (capital and labor). Suppose an increase in the labor force in the United States causes cheese production to increase. Which factor is used intensively in wine production? Which H-O theorem is applied to get this answer? Explain.

This page titled 5.4: The Rybczynski Theorem is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Anonymous.


text:
# How do I find f'(x) for f(x)=x^2*10^(2x) ?

Sep 20, 2014

The derivative of $f \left(x\right) = {x}^{2} \cdot {10}^{2 x}$ is

$f ' \left(x\right) = 2 x \left(1 + x \ln 10\right) {10}^{2 x}$

Let us look at some details.

We need the following tools in your toolbox.

1. Power Rule: $\left({x}^{n}\right) ' = n {x}^{n - 1}$

2. Exponential Rule: $\left({b}^{x}\right) ' = \left(\ln b\right) {b}^{x}$

3. Product Rule: $\left[f \left(x\right) \cdot g \left(x\right)\right] ' = f ' \left(x\right) \cdot g \left(x\right) + f \left(x\right) \cdot g ' \left(x\right)$

4. Chain Rule: $\left[f \left(g \left(x\right)\right)\right] ' = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

Let us find $\left({10}^{2 x}\right) '$ first.

By Chain Rule and Exponential Rule,

$\left({10}^{2 x}\right) ' = \left(\ln 10\right) {10}^{2 x} \cdot \left(2 x\right) ' = 2 \left(\ln 10\right) {10}^{2 x}$

Now, we can find $f ' \left(x\right)$.

By Product Rule,

$f ' \left(x\right) = \left({x}^{2}\right) ' \cdot {10}^{2 x} + {x}^{2} \cdot \left({10}^{2 x}\right) '$

by Power Rule and the derivative we found above,

$= 2 x \cdot {10}^{2 x} + {x}^{2} \cdot 2 \left(\ln 10\right) {10}^{2 x}$

by factoring out $2 x {10}^{2 x}$,

$= 2 x \left(1 + x \ln 10\right) {10}^{2 x}$


text:
lilypond-devel
[Top][All Lists]

## Re: jazz quantization of eighths for MIDI

 From: Dan Eble Subject: Re: jazz quantization of eighths for MIDI Date: Tue, 16 Sep 2008 16:31:22 -0400

It would be nice to make the amount of swing a context parameter, so it could be set anywhere from, say, 1:3 to 3:1, rather than being a fixed 2:1.

This discussion also reminds me of some baroque rhythms printed a8. b16, which may be played a8.. b32.
--
Dan

On 15 Sep 2008, at 00:17, Peter Chubb wrote:

"Werner" == Werner LEMBERG <address@hidden> writes:

Werner> Is there any possibility to implement a function which does
Werner> this conversion for MIDI output?

Not just Jazz.  Early music does the same (look up  inégalité).  It's
on my TODO list for the improved MIDI script, but I think it's a bit
hard to do in Scheme (need to do look-ahead in the list of chords),
and anyway I have no time to work on it at present.

What I'd like to see is a general rubato function, that:
1. takes a parameter (maybe call it swing') that lengthens on-beat

phrases, and shortens off-beat phrases by the same amount (so convert a8 b to \times 2/3 { a4 b8 } or { a8 b16 c } to \times 2/3 {a4 b16 c}
    or similar)
2. Within phrasing slurs does a slight accelerando into the phrase
and a slight decelerando to a-tempo at the end

But I couldn't work out how to code it in scheme.

Peter C

_______________________________________________
lilypond-devel mailing list

`


text:
### Given the information below, what is m ∠CBE?

Given the information below, what is m ∠CBE?...

### What was the cause of the Agricultural revolution? A. Glaciation of the poles B the migration of Homo sapiens C the extinction of megafauna D the warming of earths climate

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### Karen flies her Cessna Skyhawk aircraft 360 miles with a tailwind of 10 miles per hour in the same time it could fly 300 miles with a headwind of 10 miles per hour. Find the speed of the aircraft.

Karen flies her Cessna Skyhawk aircraft 360 miles with a tailwind of 10 miles per hour in the same time it could fly 300 miles with a headwind of 10 miles per hour. Find the speed of the aircraft....

### Of the following 4 ratios, please choose the one that is NOT equivalent to the other 3. 20:30 6:10 2:3 8:12

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### Dot Notation can be used with lower(), upper() or str() string methods. true or false

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### Suppose that a solution is tested for NO2 ion by adding 3M H2SO4 and heating, repeating this process until no further reaction occurs. This solution is then analyzed for NO3 ion by adding FeSO4 solution and more H2SO4, heating the test tube and its contents. The NO3 ion reaction is positive. In this case, is it accurate to state that NO3 ion was present in the original solution? Briefly explain. Write appropriate equations to support your answer.

Suppose that a solution is tested for NO2 ion by adding 3M H2SO4 and heating, repeating this process until no further reaction occurs. This solution is then analyzed for NO3 ion by adding FeSO4 solution and more H2SO4, heating the test tube and its contents. The NO3 ion reaction is positive. In this...

### Leases are considered the engines that drive property values. Therefore, it should not be surprising that owners of commercial property may seek an independent leasing broker to focus on finding tenants to lease space. In exchange for their services, leasing brokers are paid a commission based on what percentage of the face amount of the lease? A. 3 to 5% B. 0.5 to 1.5% C. 15 to 20% D. 7 to 10%

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Help me plz as fast as possible...

### PLEASEEE HURRY Which statement gives key details that help distinguish between the characters of Momma and Mrs. Flowers? Angelou compares the two women’s homes to show that Momma is poor and Mrs. Flowers is wealthy. Angelou makes the point that Momma is highly religious but Mrs. Flowers is not at all religious. Angelou describes Momma as uneducated and plainspoken and Mrs. Flowers as aristocratic and graceful. Angelou describes Momma as baking plain meat pies and Mrs. Flowers as baking special c

PLEASEEE HURRY Which statement gives key details that help distinguish between the characters of Momma and Mrs. Flowers? Angelou compares the two women’s homes to show that Momma is poor and Mrs. Flowers is wealthy. Angelou makes the point that Momma is highly religious but Mrs. Flowers is not at ...

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### What is 3x+9=-18 thank you if you do it i need 20 characters in a question so yeah

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text:
# Conference on Neutrino and Nuclear Physics (CNNP2020) Arabella Hotel and Spa, South Africa, 24-28 February 2020

23-28 February 2020
Africa/Johannesburg timezone

## Implications of new theoretical calculations on reactor antineutrino and gallium anomalies

28 Feb 2020, 09:20
20m
Parallel Venue

### Speaker

Joel Kostensalo (University of Jyvaskyla, Finland)

### Description

The reactor antineutrino and gallium anomalies have been long unexplained. Possible explanations for both of these anomalies include new physics, such as the existence of one or more eV-scale sterile neutrino [Ga15]. However, the previous theoretical calculations, which do not replicate the experimental results, rely on many simplifying approximations [Ba97,Ha19].

In the reactor-antineutrino analysis the beta decays contributing to the cumulative electron spectrum are usually assumed to have allowed spectral shapes. However, many of these decays are actually first-forbidden. Moreover, these decays dominate the experimentally observable region. In some cases, like in the case of the ground-state-to-ground-state decay of $^{140}\rm Cs$ (see figure), this is found to be a rather poor approximation. Based on the recent results, the use of this allowed approximation can at least partially explain the so called reactor antineurtino anomaly.

Our new large-scale shell model calculations regarding the neutrino-nucleus scattering cross section off $^{71}\rm Ga$ shows no statistical difference to the experimental results of GALLEX and SAGE experiments. Conflict between charge-exchange BGTs and the neutrino-nucleus cross sections can to some extent be explained by destructive interference between Gamow-Teller and tensor contributions. A Bayesian approach to estimating the significance of the gallium anomaly is discussed.

[Ga15] S. Gariazzo, C. Giunti, M. Laveder, Y. F. Li, and
E. M. Zavanin, J. Phys. G: Nucl. Part. Phys. 43, 033001
(2015).

[Ba97] J. N. Bahcall Phys. Rev. C 56, 3391 (1997).

[Ha19] L. Hayen, J. Kostensalo, N. Severijns, and J. Suhonen
Phys. Rev. C 99, 031301(R) (2019).

### Primary authors

Joel Kostensalo (University of Jyvaskyla, Finland) Jouni Suhonen (University of Jyväskylä) Dr Leendert Hayen (KU Leuven)


text:
# Chapter 10 - Radical Expressions and Equations - 10-6 Trigonometric Ratios - Lesson Check - Page 637: 5

73.74$^{\circ}$

#### Work Step by Step

When we draw out this triangle, we label the two given legs and the angle opposite the longer leg. We know the side lengths opposite and adjacent to the unknown angle. This tells us that the trigonometric ratio of tangent will be used: $tan$ $\angle$$x$$=$$\frac{24}{7}. Since we are solving for an angle, we will use the inverse of tangent (tan^{-1}$$(24/7)$) to get 73.74$^{\circ}$.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
×
Get Full Access to College Physics For Ap® Courses - 1 Edition - Chapter 34 - Problem 31
Get Full Access to College Physics For Ap® Courses - 1 Edition - Chapter 34 - Problem 31

×

# A section of superconducting wire carries a current of 100 A and requires 1.00 L of

ISBN: 9781938168932 372

## Solution for problem 31 Chapter 34

College Physics for AP® Courses | 1st Edition

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Problem 31

A section of superconducting wire carries a current of 100 A and requires 1.00 L of liquid nitrogen per hour to keep it below its critical temperature. For it to be economically advantageous to use a superconducting wire, the cost of cooling the wire must be less than the cost of energy lost to heat in the wire. Assume that the cost of liquid nitrogen is $0.30 per liter, and that electric energy costs$0.10 per kWh. What is the resistance of a normal wire that costs as much in wasted electric energy as the cost of liquid nitrogen for the superconductor?

Step-by-Step Solution:
Step 1 of 3

8/2/2016 Banner Clemson University Elizabeth S. Hill Fall 2016 Schedule Classification: Junior Level: Undergraduate College: Engineering and Science Major: Computer Science Department:School of Computing Title Course Details Credit Hours CRN Meeting Times Environmental Geology GEOL 3000 001 3.0 80218 08/17/2016  12/09/2016 Monday, Wednesday, Friday 12:20  13:10 Clemson (Main Campus), Brackett Hall, 121 Brame, Scott Software Development Foundations CPSC 2150 001 3.0 80405 08/17/2016  12/09/2016 Tuesday, Thursday 14:00  14:50 Clemson (Main Campus), Vickery Hall, 100

Step 2 of 3

Step 3 of 3

##### ISBN: 9781938168932

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+0

# help

0
62
1

Solve the equation 2x^2 - 10x + 35 = x^2 + 10.

Jun 23, 2020

#1
+782
+1

Move the terms around so we have a quadratic equation.

$$2x^2-10x+35=x^2+10\\ x^2-10x-25=0$$

Now we can simply solve for x using the quadratic formula. Let a=1, b=-10, and c=-25.

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\ x=\frac{10\pm\sqrt{(-10)^2-4(1)(-25)}}{2(1)}\\ x=\frac{10\pm\sqrt{200}}{2}\\ x=\frac{10\pm10\sqrt{2}}{2}\\ x=5\pm5\sqrt{2}$$

So x is $$5+\sqrt{2}\ \text{and}\ 5-\sqrt{2}$$.

Jun 23, 2020


text:
Alternating Temporal Logic

Rajeev Alur, Thomas A. Henzinger, and Orna Kupferman

Temporal logic comes in two varieties: linear-time temporal logic assumes implicit universal quantification over all paths that are generated by system moves; branching-time temporal logic allows explicit existential and universal quantification over all paths. We introduce a third, more general variety of temporal logic: alternating-time temporal logic offers explicit existential and universal quantification over selected paths, namely, those that are possible outcomes of games in which the system and the environment alternate moves. While linear-time and branching-time logics are natural specification languages for closed systems, alternating-time logics are natural specification languages for open systems. For example, preceding the temporal operator eventually'' by an alternating-time path quantifier we can specify that a system can reach a state satisfying a predicate $\varphi$ no matter how the environment behaves. In other words, in the game between the system and the environment, the system has a strategy to reach a $\varphi$-state. Similarly, the problems of receptiveness, controllability, and realizability all can be formulated, naturally and uniformly, as model-checking problems for alternating-time logics.

Depending on whether we admit arbitrary nesting of alternating-time path quantifiers and temporal operators, we obtain the two alternating-time temporal logics ATL and ATL*. We interpret formulas of ATL and \atlsr with respect to two models of open systems: synchronous and asynchronous. For synchronous systems, the expressive power of ATL beyond CTL comes at no cost: we give a symbolic fixpoint-computation procedure for ATL model checking, and prove that the ATL model-checking problem is linear in the size of both the structure and the formula. For asynchronous systems, the symbolic procedure involves nested fixed-point calculation and the complexity is quadratic. This makes ATL an obvious candidate for the verification of open systems, just like CTL is widely used for the verification of closed systems. It also suggests, for the first time, an interesting class of properties of open systems that can be checked efficiently. In the case of ATL*, the model-checking problem is closely related to the synthesis problem for linear-time temporal logic, and it comes, for both synchronous and asynchronous systems, at the cost of an extra exponential beyond CTL* model checking, at 2EXPTIME.

A preliminary version appears in the Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science (FOCS'97), pp. 100-109, 1997.

An extended version appears in Compositionality -- The Significant Difference, LNCS 1536, pp. 23--60, Springer-Verlag, 1999.


text:
# Definition Of Operator Exponential?

1. Sep 18, 2014

### ostrich2

I've been reading Mermin's book on Quantum Computer Science, and in the section in which he discusses the construction of a QFT using 1-Qbit and 2-Qbit gates, he makes reference to some expressions involving linear operators that I'm not familiar with (at least if I've seen them before I've forgotten). I'm wondering if someone can help understand/interpret these expressions.

In particular, he defines the following linear operator,

$\textbf{Z}|y\rangle_{n} = e^{2\pi i / 2^{n}}|y\rangle_{n}$

Where $|y\rangle_{n}$ represents the vector for the basis state corresponding to the n-qbit number whose binary expansion is the integer y.

From there, the following expression for Z is defined as follows (for a specific case in which n=4):

$$\textbf{Z} = exp\big(\frac{i \pi}{8}(8\textbf{n}_3 + 4\textbf{n}_2 + 2\textbf{n}_1 + \textbf{n}_0 ) \big)$$

where $\textbf{n}$ represents the 1-qbit number operator $\textbf{n}|x\rangle = x|x\rangle , x \in \{ 0,1 \}$ (subscript denotes the number operator referring to the ith qbit)

This expression refers to an operator in the argument of an exponential, which I've never seen before, so I'm not sure how to interpret this. I did find a definition of a matrix exponential in terms of an infinite series:

$$e^{\textbf{Z}} = \sum_{k=0}^{\infty}\frac{1}{k!}\textbf{Z}^{k}$$

The number operator can be expressed in terms of the matrix $\begin{pmatrix}0 & 0 \\ 0 & 1 \end{pmatrix}$, so I'm not sure if the corresponding operator expression can be understood in a similar way? It's a little hard to tell because Mermin seems to present the definitions and then make derivations from them without a lot of detailed explanation, the implication perhaps being that things follow straightforwardly from the basic definitions (which they might if I understood them properly).

I suspect I don't, because just taking as an example another simple definition of the same kind, it is stated that the following operator identity holds:

$$exp(2\pi i \textbf{n}) = \textbf{1}$$

$\textbf{1}$ being the identity operator. I tried to see if I could derive this identity interpreting it in terms of the corresponding matrix exponential, but for this to be valid, it seems to me that the corresponding infinite series had better converge to the identity matrix. But based on the definition of the number operator, and multiplying the matrix representation by the complex scalar $2\pi i$, I get:

$$e^{(2\pi i \textbf{n})} = \sum_{k=0}^{\infty}\frac{1}{k!}(2\pi i \textbf{n})^{k}$$.

In terms of matrices,

$$(2\pi i \textbf{n})^{k} = \begin{pmatrix}0 & 0 \\ 0 & (2\pi i)^{k} \end{pmatrix}$$

I don't see how the infinite series could possibly ever converge to the identity matrix, not least of which because there is only one element of any power of this matrix that will ever be nonzero? Or am I way off here?

2. Sep 19, 2014

### Blazejr

Remember that n^0 is identity matrix. For positive , n^k=n. Therefore we have $$\exp (2\pi i n)= \sum _{k=0}^{\infty}\frac{1}{k!}(2 \pi i n)^k=id + \sum _{k=1}^{\infty}\frac{1}{k!}(2 \pi i n)^k=id + \sum _{k=1}^{\infty}\frac{1}{k!}(2 \pi i)^kn$$
$$=id +\left( \sum _{k=1}^{\infty}\frac{1}{k!}(2 \pi i)^k\right)n=id + (exp(2 \pi i)-1)n=id$$
Where id is identity matrix.

Last edited by a moderator: Sep 22, 2014
3. Sep 19, 2014

### Fredrik

Staff Emeritus
The series definition of $e^A$ works when A is a bounded linear operator on a Hilbert space. Bounded means that the set $\{\|Ax\|:x\in\mathcal H,\,\|x\|=1\}$ (a subset of $\mathbb R$) is bounded from above. $\mathcal H$ is of course the Hilbert space on which A is a linear operator.

It also works when A is an element of an arbitrary Banach algebra. But it doesn't work for unbounded operators like position and momentum.

4. Sep 21, 2014

### ostrich2

Thanks, that does make more sense now. So now I'm looking further at the original operator definition:

$$\textbf{Z}|y\rangle_n = e^{\frac{2 \pi i y}{2^{n}}} |y\rangle_{n}$$

In terms of number operators (again for the 4 bit case) this is again expressed as:

$$\textbf{Z} = exp\big(\frac{i \pi}{8}(8\textbf{n}_3 + 4\textbf{n}_2 + 2\textbf{n}_1 + \textbf{n}_0 ) \big)$$

It isn't obvious to me at a glance that the above operator would produce this state, so I'm trying to work this out to convince myself this works. Firstly, it seems to me that the same argument provided above would work more generally (as an aside, did something funny happen with the post formatting after the upgrade, it seems to me part of what Blazejr posted isn't showing up for me anymore?) , so we should have

\begin{align} exp(x \pi i \textbf{n})&= I + \sum_{k=1}^{\infty}\frac{1}{k!}(x \pi)^{k}\textbf{n} \\ &= I + (e^{x \pi} - 1)\textbf{n} \end{align}

Now we want an expression for $\textbf{Z}|y\rangle$. Using this, and of course the fact that the individual number operators are supposed to act as the identity on the other qbits other than the one they're associated with,

\begin{align} \textbf{Z} &= \prod_{j=0}^{3} [\textbf{I}_{j} + (e^{\frac{2 j \pi i}{8}} - 1)\textbf{n}_{j}]~|y_{j}\rangle \\ &= \left( \prod_{j=0}^{3}[1 + (e^{\frac{2 j \pi i}{8}} - 1)y_{j}]\right) |y_{3}\rangle |y_{2}\rangle |y_{1}\rangle |y_{0}\rangle \end{align}

(In the first case the product being the tensor product of the corresponding vector expressions for each qbit, and in the second case the scalar factor pulled out from the tensor product of the basis vectors).

Does this look like it's on the right track? It's not evident to me from this that I can end up with $e^{\frac{2 \pi i y}{2^{n}}}$ from this product, but I may be fairly rusty with these types of periodic functions


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# Tag Archives: trigonometric identities

## Guest Post: An Easy Way To Learn The Basic Trigonometric Identities

First learn the structure. Learn the positions of the six trigonometry functions. First comes the $\sin$ function, underneath it comes the $\cos$ function.

Then in the next column comes the $\csc$ function, underneath it comes the $\sec$ function. And in the last column comes the $\cot$ function and underneath it comes the $\tan$ function. Learning this position is important and just one time. The six trig functions are specially placed in the above given places, so as to serve our need.  Continue reading

## The Proof of the Tangent Half-Angle Formula

In this post, we prove the following trigonometric identity:

$\displaystyle \tan \frac{\theta}{2} = \frac{\sin\theta}{1 + \cos \theta} = \frac{1 - \cos \theta}{\sin \theta}$.

Proof

Consider a semi-circle with “center” $O$ and diameter $AB$ and radius equal to 1 unit as shown below.  If we let $\angle BOC =\theta$, then by the Inscribed Angle Theorem, $\angle CAB = \frac{\theta}{2}$.

Draw $CD$ perpendicular to $OB$ as shown in the second figure. We can compute for the sine and cosine of $\theta$ which equal to the lengths of $CD$ and $OD$, respectively. In effect, $BD = 1 - \cos \theta$ and $AD = 1 + \cos \theta$. Continue reading