In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials over the field F p {\displaystyle \mathbb {F} _{p}} with p {\displaystyle p} elements. The method was discovered by Elwyn Berlekamp in 1970 as an auxiliary to the algorithm for polynomial factorization over finite fields. The algorithm was later modified by Rabin for arbitrary finite fields in 1979. The method was also independently discovered before Berlekamp by other researchers. == History == The method was proposed by Elwyn Berlekamp in his 1970 work on polynomial factorization over finite fields. His original work lacked a formal correctness proof and was later refined and modified for arbitrary finite fields by Michael Rabin. In 1986 René Peralta proposed a similar algorithm for finding square roots in F p {\displaystyle \mathbb {F} _{p}} . In 2000 Peralta's method was generalized for cubic equations. == Statement of problem == Let p {\displaystyle p} be an odd prime number. Consider the polynomial f ( x ) = a 0 + a 1 x + ⋯ + a n x n {\textstyle f(x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n}} over the field F p ≃ Z / p Z {\displaystyle \mathbb {F} _{p}\simeq \mathbb {Z} /p\mathbb {Z} } of remainders modulo p {\displaystyle p} . The algorithm should find all λ {\displaystyle \lambda } in F p {\displaystyle \mathbb {F} _{p}} such that f ( λ ) = 0 {\textstyle f(\lambda )=0} in F p {\displaystyle \mathbb {F} _{p}} . == Algorithm == === Randomization === Let f ( x ) = ( x − λ 1 ) ( x − λ 2 ) ⋯ ( x − λ n ) {\textstyle f(x)=(x-\lambda _{1})(x-\lambda _{2})\cdots (x-\lambda _{n})} . Finding all roots of this polynomial is equivalent to finding its factorization into linear factors. To find such factorization it is sufficient to split the polynomial into any two non-trivial divisors and factorize them recursively. To do this, consider the polynomial f z ( x ) = f ( x − z ) = ( x − λ 1 − z ) ( x − λ 2 − z ) ⋯ ( x − λ n − z ) {\textstyle f_{z}(x)=f(x-z)=(x-\lambda _{1}-z)(x-\lambda _{2}-z)\cdots (x-\lambda _{n}-z)} where z {\displaystyle z} is some element of F p {\displaystyle \mathbb {F} _{p}} . If one can represent this polynomial as the product f z ( x ) = p 0 ( x ) p 1 ( x ) {\displaystyle f_{z}(x)=p_{0}(x)p_{1}(x)} then in terms of the initial polynomial it means that f ( x ) = p 0 ( x + z ) p 1 ( x + z ) {\displaystyle f(x)=p_{0}(x+z)p_{1}(x+z)} , which provides needed factorization of f ( x ) {\displaystyle f(x)} . === Classification of === F p {\displaystyle \mathbb {F} _{p}} elements Due to Euler's criterion, for every monomial ( x − λ ) {\displaystyle (x-\lambda )} exactly one of following properties holds: The monomial is equal to x {\displaystyle x} if λ = 0 {\displaystyle \lambda =0} , The monomial divides g 0 ( x ) = ( x ( p − 1 ) / 2 − 1 ) {\textstyle g_{0}(x)=(x^{(p-1)/2}-1)} if λ {\displaystyle \lambda } is quadratic residue modulo p {\displaystyle p} , The monomial divides g 1 ( x ) = ( x ( p − 1 ) / 2 + 1 ) {\textstyle g_{1}(x)=(x^{(p-1)/2}+1)} if λ {\displaystyle \lambda } is quadratic non-residual modulo p {\displaystyle p} . Thus if f z ( x ) {\displaystyle f_{z}(x)} is not divisible by x {\displaystyle x} , which may be checked separately, then f z ( x ) {\displaystyle f_{z}(x)} is equal to the product of greatest common divisors gcd ( f z ( x ) ; g 0 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{0}(x))} and gcd ( f z ( x ) ; g 1 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{1}(x))} . === Berlekamp's method === The property above leads to the following algorithm: Explicitly calculate coefficients of f z ( x ) = f ( x − z ) {\displaystyle f_{z}(x)=f(x-z)} , Calculate remainders of x , x 2 , x 2 2 , x 2 3 , x 2 4 , … , x 2 ⌊ log 2 p ⌋ {\textstyle x,x^{2},x^{2^{2}},x^{2^{3}},x^{2^{4}},\ldots ,x^{2^{\lfloor \log _{2}p\rfloor }}} modulo f z ( x ) {\displaystyle f_{z}(x)} by squaring the current polynomial and taking remainder modulo f z ( x ) {\displaystyle f_{z}(x)} , Using exponentiation by squaring and polynomials calculated on the previous steps calculate the remainder of x ( p − 1 ) / 2 {\textstyle x^{(p-1)/2}} modulo f z ( x ) {\textstyle f_{z}(x)} , If x ( p − 1 ) / 2 ≢ ± 1 ( mod f z ( x ) ) {\textstyle x^{(p-1)/2}\not \equiv \pm 1{\pmod {f_{z}(x)}}} then gcd {\displaystyle \gcd } mentioned below provide a non-trivial factorization of f z ( x ) {\displaystyle f_{z}(x)} , Otherwise all roots of f z ( x ) {\displaystyle f_{z}(x)} are either residues or non-residues simultaneously and one has to choose another z {\displaystyle z} . If f ( x ) {\displaystyle f(x)} is divisible by some non-linear primitive polynomial g ( x ) {\displaystyle g(x)} over F p {\displaystyle \mathbb {F} _{p}} then when calculating gcd {\displaystyle \gcd } with g 0 ( x ) {\displaystyle g_{0}(x)} and g 1 ( x ) {\displaystyle g_{1}(x)} one will obtain a non-trivial factorization of f z ( x ) / g z ( x ) {\displaystyle f_{z}(x)/g_{z}(x)} , thus algorithm allows to find all roots of arbitrary polynomials over F p {\displaystyle \mathbb {F} _{p}} . === Modular square root === Consider equation x 2 ≡ a ( mod p ) {\textstyle x^{2}\equiv a{\pmod {p}}} having elements β {\displaystyle \beta } and − β {\displaystyle -\beta } as its roots. Solution of this equation is equivalent to factorization of polynomial f ( x ) = x 2 − a = ( x − β ) ( x + β ) {\textstyle f(x)=x^{2}-a=(x-\beta )(x+\beta )} over F p {\displaystyle \mathbb {F} _{p}} . In this particular case problem it is sufficient to calculate only gcd ( f z ( x ) ; g 0 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{0}(x))} . For this polynomial exactly one of the following properties will hold: GCD is equal to 1 {\displaystyle 1} which means that z + β {\displaystyle z+\beta } and z − β {\displaystyle z-\beta } are both quadratic non-residues, GCD is equal to f z ( x ) {\displaystyle f_{z}(x)} which means that both numbers are quadratic residues, GCD is equal to ( x − t ) {\displaystyle (x-t)} which means that exactly one of these numbers is quadratic residue. In the third case GCD is equal to either ( x − z − β ) {\displaystyle (x-z-\beta )} or ( x − z + β ) {\displaystyle (x-z+\beta )} . It allows to write the solution as β = ( t − z ) ( mod p ) {\textstyle \beta =(t-z){\pmod {p}}} . === Example === Assume we need to solve the equation x 2 ≡ 5 ( mod 11 ) {\textstyle x^{2}\equiv 5{\pmod {11}}} . For this we need to factorize f ( x ) = x 2 − 5 = ( x − β ) ( x + β ) {\displaystyle f(x)=x^{2}-5=(x-\beta )(x+\beta )} . Consider some possible values of z {\displaystyle z} : Let z = 3 {\displaystyle z=3} . Then f z ( x ) = ( x − 3 ) 2 − 5 = x 2 − 6 x + 4 {\displaystyle f_{z}(x)=(x-3)^{2}-5=x^{2}-6x+4} , thus gcd ( x 2 − 6 x + 4 ; x 5 − 1 ) = 1 {\displaystyle \gcd(x^{2}-6x+4;x^{5}-1)=1} . Both numbers 3 ± β {\displaystyle 3\pm \beta } are quadratic non-residues, so we need to take some other z {\displaystyle z} . Let z = 2 {\displaystyle z=2} . Then f z ( x ) = ( x − 2 ) 2 − 5 = x 2 − 4 x − 1 {\displaystyle f_{z}(x)=(x-2)^{2}-5=x^{2}-4x-1} , thus gcd ( x 2 − 4 x − 1 ; x 5 − 1 ) ≡ x − 9 ( mod 11 ) {\textstyle \gcd(x^{2}-4x-1;x^{5}-1)\equiv x-9{\pmod {11}}} . From this follows x − 9 = x − 2 − β {\textstyle x-9=x-2-\beta } , so β ≡ 7 ( mod 11 ) {\displaystyle \beta \equiv 7{\pmod {11}}} and − β ≡ − 7 ≡ 4 ( mod 11 ) {\textstyle -\beta \equiv -7\equiv 4{\pmod {11}}} . A manual check shows that, indeed, 7 2 ≡ 49 ≡ 5 ( mod 11 ) {\textstyle 7^{2}\equiv 49\equiv 5{\pmod {11}}} and 4 2 ≡ 16 ≡ 5 ( mod 11 ) {\textstyle 4^{2}\equiv 16\equiv 5{\pmod {11}}} . == Correctness proof == The algorithm finds factorization of f z ( x ) {\displaystyle f_{z}(x)} in all cases except for ones when all numbers z + λ 1 , z + λ 2 , … , z + λ n {\displaystyle z+\lambda _{1},z+\lambda _{2},\ldots ,z+\lambda _{n}} are quadratic residues or non-residues simultaneously. According to theory of cyclotomy, the probability of such an event for the case when λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} are all residues or non-residues simultaneously (that is, when z = 0 {\displaystyle z=0} would fail) may be estimated as 2 − k {\displaystyle 2^{-k}} where k {\displaystyle k} is the number of distinct values in λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} . In this way even for the worst case of k = 1 {\displaystyle k=1} and f ( x ) = ( x − λ ) n {\displaystyle f(x)=(x-\lambda )^{n}} , the probability of error may be estimated as 1 / 2 {\displaystyle 1/2} and for modular square root case error probability is at most 1 / 4 {\displaystyle 1/4} . == Complexity == Let a polynomial have degree n {\displaystyle n} . We derive the algorithm's complexity as follows: Due to the binomial theorem ( x − z ) k = ∑ i = 0 k ( k i ) ( − z ) k − i x i {\textstyle (x-z)^{k}=\sum \limits _{i=0}^{k}{\binom {k}{i}}(-z)^{k-i}x^{i}} , we may transition from f ( x ) {\displaystyle f(x)} to f ( x − z ) {\displaystyle f(x-z)} in O ( n 2 ) {\displaystyle O(n^{2})} time. Polynomial multiplication a
Elowan
Elowan is a plant-robot cyborg. Using its own internal bioelectrical signals, The plant has a robotic extension that makes it move towards light sources. Electrodes are inserted into the leaves, stem, and ground to detect the faint bioelectrical signals the plant produces. Then they are amplified so the robot can read them. So when the plant "wants" to go to light, the cyborg automatically goes to the nearest light source. Future extensions of the robot could provide: Protection, growth frameworks, and nutrients. Other factors that could make the cyborg move are temperature, soil, and gravity conditions Elowan is one in a series of plant-electronic hybrid experiments.
Jiaya Jia
Jiaya Jia (Chinese: 贾佳亚) is a Chair Professor of the Department of Computer Science and Engineering at The Hong Kong University of Science and Technology (HKUST). He is an IEEE Fellow, the associate editor-in-chief of one of IEEE’s flagship and premier journals- Transactions on Pattern Analysis and Machine Intelligence (TPAMI), as well as on the editorial board of International Journal of Computer Vision (IJCV). == Early life and education == Jiaya Jia joined CUHK in 2004 as an assistant professor, and was promoted to full professor in 2015. He obtained his PhD degree in computer science jointly from Hong Kong University of Science and Technology and Microsoft Research in 2004. From March 2003 to August 2004, he was a visiting scholar at Microsoft. He conducted collaborative research at Adobe Research in 2007. == Career == Jiaya Jia is a distinguished scientist in the fields of computer vision and artificial intelligence. His research team at HKUST, DV Lab, is one of the largest vision AI research teams in the world and has been making significant contribution to advanced development of computer vision algorithms and technologies with focuses on image/video understanding, detection and segmentation, multi-modal AI, computational imaging, practical optimization, and advanced learning for visual content since 2000. Jiaya Jia has published 200+ top papers and was cited 80,000+ times on Google Scholar with H-Index 110+. 40+ PhDs and fellows from this group are now active in academia and industry, and have become prominent AI tech leaders as professors, directors in major research labs, and founders of several successful startups. Jiaya Jia assumes the position of associate editor-in-chief of IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI) since 2021. He is also on the editorial board of International Journal of Computer Vision (IJCV). Jiaya Jia has served as the area chair of ICCV, CVPR, AAAI, ECCV, and several other premium international AI conferences for years. He was on program committees of major conferences in graphics and computational imaging, including ICCP, SIGGRAPH, and SIGGRAPH Asia. == Research == The research areas of Jiaya Jia are computer vision, large X models, and deep learning. Jiaya Jia has made outstanding contributions to computer vision technology, algorithms and engineering, and is among the world's leading experts in the field. His research partners include numerous renowned multinational technology companies, such as Microsoft, Qualcomm, Adobe, Intel, NVIDIA, Amazon, and Lenovo. Jia has cultivated a number of outstanding talents with Master's and PhDs who continue to engage in scientific research and development in computer vision. Many technologies in image analysis and processing developed by Jiaya Jia are still leading in the field worldwide. Wherein, his achievements in image deblurring, filtering, image sparse processing, multi-band image signal fusion and enhancement, large range motion estimation, texture and structure-based layering, etc. have been published in the industry's most influential conferences and publications, and implemented in the real-world applications. These achievements have demonstrated outstanding performance in established systems, and most of which are open source so as to enable wider applications across industries such as aviation, medical imaging, safety management, robotic design, meteorological analysis and many more. == Selected publications == In his over 20 years of research experience, Jiaya Jia has published 200+ top papers that have been cited more than 80,000 times. According to HKUST Website in August 2024, Jiaya Jia has accumulatively published over 200 scientific papers in books, journals and conferences, such as IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI), International Journal of Computer Vision (IJCV) "Computer Vision and Pattern Recognition (CVPR)", and "International Conference on Computer Vision (ICCV)". Representative papers include: Jiaya Jia: Mathematical Models and Practical Solvers for Uniform Motion Deblurring (in Motion Deblurring: Algorithms and Systems), Cambridge University Press, ISBN 9781107044364, 2014; Jiaya Jia: “Matte Extraction” Book: Computer Vision - A Reference Guide, Springer, ISBN 9780387307718 Editor-in-chief: Ikeuchi, Katsushi; Jiaya Jia, Chi-Keung Tang：Image Stitching Using Structure Deformation，IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI), Vol. 30, No. 4, 2008; Jiaya Jia, Jian Sun, Chi-Keung Tang, Heung-Yeung Shum：Drag-and-Drop Pasting，ACM Transactions on Graphics (also in SIGGRAPH 2006), Vol. 25, No. 3, 2006. Xiaojuan Qi, Zheng zhe Liu, Renjie Liao, Philip HS Torr, Raquel Urtasun, Jiaya Jia：GeoNet++: Iterative Geometric Neural Network with Edge-Aware Refinement for Joint Depth and Surface Normal Estimation，IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI). Accepted. == Selected honors and awards == ACM Fellow. 1st Place of WAD Drivable Area Segmentation Challenge 2018; 1st Place of LSUN'17 Instance and Semantic Segmentation Challenges; 1st Place of COCO Instance Segmentation Challenge 2017; 2nd Place in COCO Detection Challenge 2017; 1st Place of ImageNet Scene Parsing Challenge 2016 with the paper PSPNet presented in CVPR 2017.
Noémie Elhadad
Noémie Elhadad is an American data scientist who is an associate professor of biomedical informatics at the Columbia University Vagelos College of Physicians and Surgeons. As of 2022, she serves as the chair of the Department of Biomedical Informatics. Her research considers machine learning in bioinformatics, natural language processing and medicine. == Early life and education == Elhadad studied computer software engineering at École nationale supérieure d'électronique, informatique, télécommunications, mathématique et mécanique de Bordeaux (ENSEIRB). She completed her doctoral research at Columbia University. She was based in the Department of Computer Science, where she developed patient-focused text summaries of clinical literature. == Research and career == Elhadad joined the faculty at the City College of New York. In 2007 she joined the Department of Biomedical Informatics at Columbia University. She was made Chair of the Health Analytics Center at the Columbia Data Science Institute in 2013. Her research considers how clinical data, electronic health records and patient-generated data can enhance access to information for researchers, patients and physicians. She developed an artificial intelligence tool that supported patients in the NewYork-Presbyterian Hospital. Elhadad is interested in using data to advance women's health. She led the Citizen Endo Project that looks to comprehensively describe how patients experience endometriosis. It was built using principles of citizen science, using patient testimonials from focus groups in New York City and data aggregation. She created the app, Phendo, which asks patients about their experience of the disease. The name Phendo is a portmanteau of phenotyping endometriosis. Elhadad was announced as chair of the Department of Biomedical Informatics in December 2022. == Selected publications == Caruana, Rich; Lou, Yin; Gehrke, Johannes; Koch, Paul; Sturm, Marc; Elhadad, Noemie (August 10, 2015). "Intelligible Models for HealthCare". Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. New York, NY, USA: ACM. pp. 1721–1730. doi:10.1145/2783258.2788613. ISBN 9781450336642. S2CID 14190268. Chaitanya Shivade; Preethi Raghavan; Eric Fosler-Lussier; Peter J Embi; Noemie Elhadad; Stephen B Johnson; Albert M Lai (November 7, 2013). "A review of approaches to identifying patient phenotype cohorts using electronic health records". Journal of the American Medical Informatics Association. 21 (2): 221–230. doi:10.1136/AMIAJNL-2013-001935. ISSN 1067-5027. PMC 3932460. PMID 24201027. Wikidata Q37598951. Shivade, Chaitanya; Raghavan, Preethi; Fosler-Lussier, Eric; Embi, Peter J; Elhadad, Noemie; Johnson, Stephen B; Lai, Albert M (March 2014). "A review of approaches to identifying patient phenotype cohorts using electronic health records". Journal of the American Medical Informatics Association. 21 (2): 221–230. doi:10.1136/amiajnl-2013-001935. ISSN 1067-5027. PMC 3932460. PMID 24201027. == Personal life == Elhadad suffers from endometriosis.
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Artificial psychology
Artificial psychology (AP) has had multiple meanings dating back to 19th century, with recent usage related to artificial intelligence (AI).Artificial psychology is a theoretical field related to artificial intelligence, cognitive science, and psychology, which explores how advanced AI systems may develop human-like decision-making processes. In 1999, Zhiliang Wang and Lun Xie presented a theory of artificial psychology based on artificial intelligence. They analyze human psychology using information science research methods and artificial intelligence research to probe deeper into the human mind. == Main Theory == Dan Curtis (b. 1963) proposed AP is a theoretical discipline. The theory considers the situation when an artificial intelligence approaches the level of complexity where the intelligence meets two conditions: Condition I A: Makes all of its decisions autonomously B: Is capable of making decisions based on information that is New Abstract Incomplete C: The artificial intelligence is capable of reprogramming itself based on the new data, allowing it to evolve. D: And is capable of resolving its own programming conflicts, even in the presence of incomplete data. This means that the intelligence autonomously makes value-based decisions, referring to values that the intelligence has created for itself. Condition II All four criteria are met in situations that are not part of the original operating program When both conditions are met, then, according to this theory, the possibility exists that the intelligence will reach irrational conclusions based on real or created information. At this point, the criteria are met for intervention which will not necessarily be resolved by simple re-coding of processes due to extraordinarily complex nature of the codebase itself; but rather a discussion with the intelligence in a format which more closely resembles classical (human) psychology. If the intelligence cannot be reprogrammed by directly inputting new code, but requires the intelligence to reprogram itself through a process of analysis and decision based on information provided by a human, in order for it to overcome behavior which is inconsistent with the machines purpose or ability to function normally, then artificial psychology is by definition, what is required. The level of complexity that is required before these thresholds are met is currently a subject of extensive debate. The theory of artificial psychology does not address the specifics of what those levels may be, but only that the level is sufficiently complex that the intelligence cannot simply be recoded by a software developer, and therefore dysfunctionality must be addressed through the same processes that humans must go through to address their own dysfunctionalities. Along the same lines, artificial psychology does not address the question of whether or not the intelligence is conscious. As of 2022, the level of artificial intelligence does not approach any threshold where any of the theories or principles of artificial psychology can even be tested, and therefore, artificial psychology remains a largely theoretical discipline. Even at a theoretical level, artificial psychology remains an advanced stage of artificial intelligence.
Is an AI Blog Writer Worth It in 2026?
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