IWARP

iWARP is a computer networking protocol that implements remote direct memory access (RDMA) for efficient data transfer over Internet Protocol networks. Contrary to some accounts, iWARP is not an acronym. Because iWARP is layered on Internet Engineering Task Force (IETF)-standard congestion-aware protocols such as Transmission Control Protocol (TCP) and Stream Control Transmission Protocol (SCTP), it makes few requirements on the network, and can be successfully deployed in a broad range of environments. == History == In 2007, the IETF published five Request for Comments (RFCs) that define iWARP: RFC 5040 A Remote Direct Memory Access Protocol Specification is layered over Direct Data Placement Protocol (DDP). It defines how RDMA Send, Read, and Write operations are encoded using DDP into headers on the network. RFC 5041 Direct Data Placement over Reliable Transports is layered over MPA/TCP or SCTP. It defines how received data can be directly placed into an upper layer protocols receive buffer without intermediate buffers. RFC 5042 Direct Data Placement Protocol (DDP) / Remote Direct Memory Access Protocol (RDMAP) Security analyzes security issues related to iWARP DDP and RDMAP protocol layers. RFC 5043 Stream Control Transmission Protocol (SCTP) Direct Data Placement (DDP) Adaptation defines an adaptation layer that enables DDP over SCTP. RFC 5044 Marker PDU Aligned Framing for TCP Specification defines an adaptation layer that enables preservation of DDP-level protocol record boundaries layered over the TCP reliable connected byte stream. These RFCs are based on the RDMA Consortium's specifications for RDMA over TCP. The RDMA Consortium's specifications are influenced by earlier RDMA standards, including Virtual Interface Architecture (VIA) and InfiniBand (IB). Since 2007, the IETF has published three additional RFCs that maintain and extend iWARP: RFC 6580 IANA Registries for the Remote Direct Data Placement (RDDP) Protocols published in 2012 defines IANA registries for Remote Direct Data Placement (RDDP) error codes, operation codes, and function codes. RFC 6581 Enhanced Remote Direct Memory Access (RDMA) Connection Establishment published in 2011 fixes shortcomings with iWARP connection setup. RFC 7306 Remote Direct Memory Access (RDMA) Protocol Extensions published in 2014 extends RFC 5040 with atomic operations and RDMA Write with Immediate Data. == Protocol == The main component in the iWARP protocol is the Direct Data Placement Protocol (DDP), which permits the actual zero-copy transmission. DDP itself does not perform the transmission; the underlying protocol (TCP or SCTP) does. However, TCP does not respect message boundaries; it sends data as a sequence of bytes without regard to protocol data units (PDU). In this regard, DDP itself may be better suited for SCTP, and indeed the IETF proposed a standard RDMA over SCTP. To run DDP over TCP requires a tweak known as marker PDU aligned (MPA) framing to guarantee boundaries of messages. Furthermore, DDP is not intended to be accessed directly. Instead, a separate RDMA protocol (RDMAP) provides the services to read and write data. Therefore, the entire RDMA over TCP specification is really RDMAP over DDP over either MPA/TCP or SCTP. All of these protocols can be implemented in hardware. Unlike IB, iWARP only has reliable connected communication, as this is the only service that TCP and SCTP provide. The iWARP specification omits other features of IB, such as Send with Immediate Data operations. With RFC 7306, the IETF is working to reduce these omissions. == Implementation == Because a kernel implementation of the TCP stack can be seen as a bottleneck, the protocol is typically implemented in hardware RDMA network interface controllers (rNICs). As simple data losses are rare in tightly coupled network environments, the error-correction mechanisms of TCP may be performed by software while the more frequently performed communications are handled strictly by logic embedded on the rNIC. Similarly, connections are often established entirely by software and then handed off to the hardware. Furthermore, the handling of iWARP specific protocol details is typically isolated from the TCP implementation, allowing rNICs to be used for both as RDMA offload and TCP offload (in support of traditional sockets based TCP/IP applications). The portion of the hardware implementation used for implementing the TCP protocol is known as the TCP Offload Engine (TOE). TOE itself does not prevent copying on the reception side, and must be combined with RDMA hardware for zero-copy results. The RDMA / TCP specification is a set of different wire protocols intended to be implemented in hardware (though it seems feasible to emulate it in software for compatibility but without the performance benefits). == Interfaces == iWARP is a protocol, not an implementation, but defines protocol behavior in terms of the operations that are legal for the protocol, known as Verbs. As such, iWARP does not have any single standard programming interface. However, programming interfaces tend to very closely correspond to the Verbs. Several programmatic interfaces have been proposed, including OpenFabrics Verbs, Network Direct, uDAPL, kDAPL, IT-API, and RNICPI. Implementations of some of these interfaces are available for different platforms, including Windows and Linux. == Services available == Networking services implemented over iWARP include those offered in the OpenFabrics Enterprise Distribution (OFED) by the OpenFabrics Alliance for Linux operating systems, and by Microsoft Windows via Network Direct. NVMe over Fabrics (NVMEoF) iSCSI Extensions for RDMA (iSER) Server Message Block Direct (SMB Direct) Sockets Direct Protocol (SDP) SCSI RDMA Protocol (SRP) Network File System over RDMA (NFS over RDMA) GPUDirect

Buckeye Corpus

The Buckeye Corpus of conversational speech is a speech corpus created by a team of linguists and psychologists at Ohio State University led by Prof. Mark Pitt. It contains high-quality recordings from 40 speakers in Columbus, Ohio conversing freely with an interviewer. The interviewer's voice is heard only faintly in the background of these recordings. The sessions were conducted as Sociolinguistics interviews, and are essentially monologues. The speech has been orthographically transcribed and phonetically labeled. The audio and text files, together with time-aligned phonetic labels, are stored in a format for use with speech analysis software (Xwaves and Wavesurfer). Software for searching the transcription files is also available at the project web site. The corpus is available to researchers in academia and industry. The project was funded by the National Institute on Deafness and Other Communication Disorders and the Office of Research at Ohio State University.

Katia Sycara

Ekaterini Panagiotou Sycara (Greek: Κάτια Συκαρά) is a Greek computer scientist. She is an Edward Fredkin Research Professor of Robotics in the Robotics Institute, School of Computer Science at Carnegie Mellon University internationally known for her research in artificial intelligence, particularly in the fields of negotiation, autonomous agents and multi-agent systems. She directs the Advanced Agent-Robotics Technology Lab at Robotics Institute, Carnegie Mellon University. She also serves as academic advisor for PhD students at both Robotics Institute and Tepper School of Business. == Education and early life == Born in Greece, she went to the United States to pursue advanced education through various scholarships, including a Fulbright (1965-1969). She received a B.S. in applied mathematics from Brown University, M.S. in electrical engineering from the University of Wisconsin–Milwaukee, and PhD in computer science from Georgia Institute of Technology. == Research and career == Sycara is a pioneer in the field of semantic web, case-based reasoning, autonomous agents and multi-agent systems. She has authored or co-authored more than 700 technical papers dealing with multi-agent systems, software agents, web services, semantic web, human–computer interaction, human-robot interaction, negotiation, case-based reasoning and the application of these techniques to crisis action planning, scheduling, manufacturing, healthcare management, financial planning and e-commerce.[1] She has led multimillion-dollar research effort funded by DARPA, NASA, AFOSR, ONR, AFRL, NSF and industry. Through an ONR MURI program and though the COABS DARPA program, Prof. Sycara's group has developed the RETSINA multiagent infrastructure, a toolkit that enables the development of heterogeneous software agents that can dynamically coordinate in open information environments (e.g. the Internet). RETSINA has been used in multiple applications including supporting human joint mission teams for crisis response; creating autonomous agents for situation awareness and information fusion; financial portfolio management, negotiations and coalition formation for e-commerce, and coordinating robots for Urban Search and Rescue. Sycara is one of the contributors to the development of OWL-S, the Darpa-sponsored language for Semantic Web services, as well as matchmaking and brokering software for agent discovery, service integration and semantic interoperation. === Academic service === Sycara is the founding Editor-in-Chief of the journal Autonomous Agents and Multi-Agent Systems; Editor-in-Chief, of the Springer Series on Agents; and Area Editor of AI and Management Science, the journal "Group Decision and Negotiation." She is a member of the Editorial Board, the Kluwer book series on "Multiagent Systems, Artificial Societies and Simulated Organizations"; member of the editorial board, the journals "Agent Oriented Software Engineering", "Web Intelligence and Agent Technologies", "Journal of Infonomics", "Fundamenda Informaticae", and "Concurrent Engineering: Research and Applications"; and member of the editorial board of the "ETAI journal on the Semantic Web" (1998–2001). She was on the Editorial Board of "IEEE Intelligent Systems and their Applications" (1992–1996), and "AI in Engineering" (1990–1996). She is a member of the Scientific Advisory Board of France Telecom, 2003-2009; member of the Scientific Advisory Board of the Institute of Informatics and Telecommunications of the Greek National Research Center Demokritos, 2004-2012; member of the AAAI Executive Council (1996–99); member of the OASIS Technical committee on the development of UDDI (Universal Description and Discovery for Interoperability) software which is an industry standard; and an invited expert for W3C (the World Wide Web Consortium) Working Group on Web Services Architecture. She was a founding member of the Board of Directors of the International Foundation of Multiagent Systems (IFMAS), and founding member of the Semantic Web Science Association. Sycara served as the program chair of the Second International Semantic Web Conference (ISWC 2003); general chair, of the Second International Conference on Autonomous Agents (Agents 98); chair of the Steering Committee of the Agents Conference (1999–2001); scholarship chair of AAAI (1993–1999); and the US co-chair for the US-Europe Semantic Web Services Initiative. === Awards and honors === Sycara is a Fellow of Institute of Electrical and Electronics Engineers (IEEE), and a Fellow of American Association for Artificial Intelligence (AAAI). Sycara is the recipient of the 2002 ACM/SIGART Agents Research Award. She is also the recipient of the 2015 Group Decision and Negotiation (GDN) Award of the Institute for Operations Research and the Management Sciences (INFORMS) GDN Section for her outstanding contributions to the field of group decision and negotiation. According to the citation of the award: Katia Sycara is widely acknowledged as one of the leading researchers in the field of autonomous software agents and in particular on problems related to joint decision making and negotiations of such agents. Her work is characterized by a unique combination of methods from Artificial Intelligence and research on human negotiations, and thus has contributed to significant advances in both fields. Sycara's robot teams have won multiple international awards. In the 2005 Robocup Urban Search and Rescue (US Open) held in Atlanta, her team won the First-in-Class Award for Autonomy, and the First-in-Class Award for Mobility. Two years later, again in Atlanta, she led another team that became a world champions in the 2007 International Robocup Search and Rescue Simulation League Competition. In 2008, her robotic team placed third in the Worldwide Robocup Championship Competition in the Urban Search and Rescue Virtual robots League held in Beijing, China. In 2005, she received the Outstanding Alumnus Award from the University of Wisconsin–Milwaukee. She was awarded an Honorary Doctorate from the University of the Aegean in 2004.

Ω-automaton

In automata theory, a branch of theoretical computer science, an ω-automaton (or stream automaton) is a variation of a finite automaton that runs on infinite, rather than finite, strings as input. Since ω-automata do not stop, they have a variety of acceptance conditions rather than simply a set of accepting states. ω-automata are useful for specifying behavior of systems that are not expected to terminate, such as hardware, operating systems and control systems. For such systems, one may want to specify a property such as "for every request, an acknowledge eventually follows", or its negation "there is a request that is not followed by an acknowledge". The former is a property of infinite words: one cannot say of a finite sequence that it satisfies this property. Classes of ω-automata include the Büchi automata, Rabin automata, Streett automata, parity automata and Muller automata, each deterministic or non-deterministic. These classes of ω-automata differ only in terms of acceptance condition. They all recognize precisely the regular ω-languages except for the deterministic Büchi automata, which is strictly weaker than all the others. Although all these types of automata recognize the same set of ω-languages, they nonetheless differ in succinctness of representation for a given ω-language. == Deterministic ω-automata == Formally, a deterministic ω-automaton is a tuple A = ( Q , Σ , δ , q 0 , A a c c ) {\textstyle A=(Q,\Sigma ,\delta ,q_{0},A_{acc})} , that consists of the following components: Q {\textstyle Q} , is a finite set. The elements of Q {\textstyle Q} are called the states of A {\textstyle A} . Σ {\textstyle \Sigma } , is a finite set called the alphabet of A {\textstyle A} . δ : Q × Σ → Q {\textstyle \delta \colon Q\times \Sigma \rightarrow Q} is a function, called the transition function of A {\textstyle A} . Q 0 {\textstyle Q_{0}} is an element of Q {\textstyle Q} , called the initial state. A a c c {\textstyle A_{acc}} is a set of accepting states of A {\textstyle A} , formally a subset of Q ω {\textstyle Q^{\omega }} . An input for A {\textstyle A} is an infinite string over the alphabet Σ {\textstyle \Sigma } , i.e. it is an infinite sequence α = ( a 1 , a 2 , a 3 , … ) {\textstyle \alpha =(a_{1},a_{2},a_{3},\ldots )} . The run of A {\textstyle A} on such an input is an infinite sequence ρ = ( r 0 , r 1 , r 2 , … ) {\textstyle \rho =(r_{0},r_{1},r_{2},\ldots )} of states, defined as follows: r 0 = q 0 {\textstyle r_{0}=q_{0}} . r 1 = δ ( r 0 , a 1 ) {\textstyle r_{1}=\delta (r_{0},a_{1})} . r 2 = δ ( r 1 , a 2 ) {\textstyle r_{2}=\delta (r_{1},a_{2})} . ... that is, for every i {\textstyle i} : r i = δ ( r i − 1 , a i ) {\textstyle r_{i}=\delta (r_{i-1},a_{i})} . The main purpose of an ω-automaton is to define a subset of the set of all inputs: The set of accepted inputs. Whereas in the case of an ordinary finite automaton every run ends with a state r n {\textstyle r_{n}} and the input is accepted if and only if r n {\textstyle r_{n}} is an accepting state, the definition of the set of accepted inputs is more complicated for ω-automata. Here we must look at the entire run ρ {\textstyle \rho } . The input is accepted if the corresponding run is in Acc {\textstyle {\text{Acc}}} . The set of accepted input ω-words is called the recognized ω-language by the automaton, which is denoted as L ( A ) {\textstyle L(A)} . The definition of Acc {\textstyle {\text{Acc}}} as a subset of Q ω {\textstyle Q^{\omega }} is purely formal and not suitable for practice because normally such sets are infinite. The difference between various types of ω-automata (Büchi, Rabin etc.) consists in how they encode certain subsets Acc {\textstyle {\text{Acc}}} of Q ω {\textstyle Q^{\omega }} as finite sets, and therefore in which such subsets they can encode. == Nondeterministic ω-automata == Formally, a nondeterministic ω-automaton is a tuple A = ( Q , Σ , Δ , Q 0 , Acc ) {\textstyle A=(Q,\Sigma ,\Delta ,Q_{0},{\text{Acc}})} that consists of the following components: Q {\textstyle Q} is a finite set. The elements of Q {\textstyle Q} are called the states of A {\textstyle A} . Σ {\textstyle \Sigma } is a finite set called the alphabet of A {\textstyle A} . Δ {\textstyle \Delta } is a subset of Q × Σ × Q {\textstyle Q\times \Sigma \times Q} and is called the transition relation of A {\textstyle A} . Q 0 {\textstyle Q_{0}} is a subset of Q {\textstyle Q} , called the initial set of states. Acc {\textstyle {\text{Acc}}} is the acceptance condition, a subset of Q ω {\textstyle Q^{\omega }} . Unlike a deterministic ω-automaton, which has a transition function δ {\textstyle \delta } , the non-deterministic version has a transition relation Δ {\textstyle \Delta } . Note that Δ {\textstyle \Delta } can be regarded as a function Q × Σ → P ( Q ) {\textstyle Q\times \Sigma \rightarrow {\mathcal {P}}(Q)} from Q × Σ {\textstyle Q\times \Sigma } to the power set P ( Q ) {\textstyle {\mathcal {P}}(Q)} . Thus, given a state q n {\textstyle q_{n}} and a symbol a n {\textstyle a_{n}} , the next state q n + 1 {\textstyle q_{n+1}} is not necessarily determined uniquely, rather there is a set of possible next states. A run of A {\textstyle A} on the input α = ( a 1 , a 2 , a 3 , … ) {\textstyle \alpha =(a_{1},a_{2},a_{3},\ldots )} is any infinite sequence ρ = ( r 0 , r 1 , r 2 , … ) {\textstyle \rho =(r_{0},r_{1},r_{2},\ldots )} of states that satisfies the following conditions: r 0 {\textstyle r_{0}} is an element of Q 0 {\textstyle Q_{0}} . r 1 {\textstyle r_{1}} is an element of Δ ( r 0 , a 1 ) {\textstyle \Delta (r_{0},a_{1})} . r 2 {\textstyle r_{2}} is an element of Δ ( r 1 , a 2 ) {\textstyle \Delta (r_{1},a_{2})} . ... that is, for every i {\textstyle i} : r i {\textstyle r_{i}} is an element of Δ ( r i − 1 , a i ) {\textstyle \Delta (r_{i-1},a_{i})} . A nondeterministic ω-automaton may admit many different runs on any given input, or none at all. The input is accepted if at least one of the possible runs is accepting. Whether a run is accepting depends only on Acc {\textstyle {\text{Acc}}} , as for deterministic ω-automata. Every deterministic ω-automaton can be regarded as a nondeterministic ω-automaton by taking Δ {\textstyle \Delta } to be the graph of δ {\textstyle \delta } . The definitions of runs and acceptance for deterministic ω-automata are then special cases of the nondeterministic cases. == Acceptance conditions == Acceptance conditions may be infinite sets of ω-words. However, people mostly study acceptance conditions that are finitely representable. The following lists a variety of popular acceptance conditions. Before discussing the list, let's make the following observation. In the case of infinitely running systems, one is often interested in whether certain behavior is repeated infinitely often. For example, if a network card receives infinitely many ping requests, then it may fail to respond to some of the requests but should respond to an infinite subset of received ping requests. This motivates the following definition: For any run ρ {\textstyle \rho } , let Inf ( ρ ) {\textstyle {\text{Inf}}(\rho )} be the set of states that occur infinitely often in ρ {\textstyle \rho } . This notion of certain states being visited infinitely often will be helpful in defining the following acceptance conditions. A Büchi automaton is an ω-automaton A {\textstyle A} that uses the following acceptance condition, for some subset F {\textstyle F} of Q {\textstyle Q} : Büchi condition A {\textstyle A} accepts exactly those runs ρ {\textstyle \rho } for which Inf ( ρ ) ∩ F ≠ ∅ {\textstyle {\text{Inf}}(\rho )\cap F\neq \emptyset } , i.e. there is an accepting state that occurs infinitely often in ρ {\textstyle \rho } . A Rabin automaton is an ω-automaton A {\textstyle A} that uses the following acceptance condition, for some set Ω {\textstyle \Omega } of pairs ( B i , G i ) {\textstyle (B_{i},G_{i})} of sets of states: Rabin condition A {\textstyle A} accepts exactly those runs ρ {\textstyle \rho } for which there exists a pair ( B i , G i ) {\textstyle (B_{i},G_{i})} in Ω {\textstyle \Omega } such that B i ∩ Inf ( ρ ) = ∅ {\textstyle B_{i}\cap {\text{Inf}}(\rho )=\emptyset } and G i ∩ Inf ( ρ ) ≠ ∅ {\textstyle G_{i}\cap {\text{Inf}}(\rho )\neq \emptyset } . A Streett automaton is an ω-automaton A {\textstyle A} that uses the following acceptance condition, for some set Ω {\textstyle \Omega } of pairs ( B i , G i ) {\textstyle (B_{i},G_{i})} of sets of states: Streett condition A {\textstyle A} accepts exactly those runs ρ {\textstyle \rho } such that for all pairs ( B i , G i ) {\textstyle (B_{i},G_{i})} in Ω {\textstyle \Omega } , B i ∩ Inf ( ρ ) ≠ ∅ {\textstyle B_{i}\cap {\text{Inf}}(\rho )\neq \emptyset } or G i ∩ Inf ( ρ ) = ∅ {\textstyle G_{i}\cap {\text{Inf}}(\rho )=\emptyset } . A parity automaton is an automaton A {\textstyle A} whose set of states is Q = { 0 , 1 , 2 , … , k } {\textstyle Q=\{0,1,2,\ldots ,k\}} for some natural number k {\textst

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And–or tree

An and–or tree is a graphical representation of the reduction of problems (or goals) to conjunctions and disjunctions of subproblems (or subgoals). == Example == The and–or tree: represents the search space for solving the problem P, using the goal-reduction methods: P if Q and R P if S Q if T Q if U == Definitions == Given an initial problem P0 and set of problem solving methods of the form: P if P1 and … and Pn the associated and–or tree is a set of labelled nodes such that: The root of the tree is a node labelled by P0. For every node N labelled by a problem or sub-problem P and for every method of the form P if P1 and ... and Pn, there exists a set of children nodes N1, ..., Nn of the node N, such that each node Ni is labelled by Pi. The nodes are conjoined by an arc, to distinguish them from children of N that might be associated with other methods. A node N, labelled by a problem P, is a success node if there is a method of the form P if nothing (i.e., P is a "fact"). The node is a failure node if there is no method for solving P. If all of the children of a node N, conjoined by the same arc, are success nodes, then the node N is also a success node. Otherwise the node is a failure node. == Search strategies == An and–or tree specifies only the search space for solving a problem. Different search strategies for searching the space are possible. These include searching the tree depth-first, breadth-first, or best-first using some measure of desirability of solutions. The search strategy can be sequential, searching or generating one node at a time, or parallel, searching or generating several nodes in parallel. == Relationship with logic programming == The methods used for generating and–or trees are propositional logic programs (without variables). In the case of logic programs containing variables, the solutions of conjoint sub-problems must be compatible. Subject to this complication, sequential and parallel search strategies for and–or trees provide a computational model for executing logic programs. == Relationship with two-player games == And–or trees can also be used to represent the search spaces for two-person games. The root node of such a tree represents the problem of one of the players winning the game, starting from the initial state of the game. Given a node N, labelled by the problem P of the player winning the game from a particular state of play, there exists a single set of conjoint children nodes, corresponding to all of the opponents responding moves. For each of these children nodes, there exists a set of non-conjoint children nodes, corresponding to all of the player's defending moves. For solving game trees with proof-number search family of algorithms, game trees are to be mapped to and–or trees. MAX-nodes (i.e. maximizing player to move) are represented as OR nodes, MIN-nodes map to AND nodes. The mapping is possible, when the search is done with only a binary goal, which usually is "player to move wins the game".

Andrei Knyazev (mathematician)

Andrew Knyazev is an American mathematician. He graduated from the Faculty of Computational Mathematics and Cybernetics of Moscow State University under the supervision of Evgenii Georgievich D'yakonov (Russian: Евгений Георгиевич Дьяконов) in 1981 and obtained his PhD in Numerical Mathematics at the Russian Academy of Sciences under the supervision of Vyacheslav Ivanovich Lebedev (Russian: Вячеслав Иванович Лебедев) in 1985. He worked at the Kurchatov Institute between 1981–1983, and then to 1992 at the Marchuk Institute of Numerical Mathematics (Russian: ru:Институт вычислительной математики имени Г. И. Марчука РАН) of the Russian Academy of Sciences, headed by Gury Marchuk (Russian: Гурий Иванович Марчук). From 1993–1994, Knyazev held a visiting position at the Courant Institute of Mathematical Sciences of New York University, collaborating with Olof B. Widlund. From 1994 until retirement in 2014, he was a Professor of Mathematics at the University of Colorado Denver, supported by the National Science Foundation and United States Department of Energy grants. He was a recipient of the 2008 Excellence in Research Award, the 2000 college Teaching Excellence Award, and a finalist of the CU President's Faculty Excellence Award for Advancing Teaching and Learning through Technology in 1999. He was awarded the title of Professor Emeritus at the University of Colorado Denver and named the SIAM Fellow Class of 2016 and AMS Fellow Class of 2019. From 2012–2018, Knyazev worked at the Mitsubishi Electric Research Laboratories on algorithms for image and video processing, data sciences, optimal control, and material sciences, resulting in dozens of publications and 13 patent applications. Since 2018, he contributed to numerical techniques in quantum computing at Zapata Computing, real-time embedded anomaly detection in automotive data, and algorithms for silicon photonics-based hardware. Knyazev is mostly known for his work in numerical solution of large sparse eigenvalue problems, particularly preconditioning and the iterative method LOBPCG. Knyazev's implementation of LOBPCG is available in many open source software packages, e.g., BLOPEX, SciPy, and ABINIT. Knyazev collaborated with John Osborn on the theory of the Ritz method in the finite element method context and with Nikolai Sergeevich Bakhvalov (Russian: Николай Серге́евич Бахвалов) (Erdős number 3 via Leonid Kantorovich) on numerical solution of elliptic partial differential equations with large jumps in the main coefficients. Jointly with his Ph.D. students, Knyazev pioneered using majorization for bounds in the Rayleigh–Ritz method (see and references there) and contributed to the theory of angles between flats.