SINGULAR CONTINUOUS SPECTRUM OF HALF
... The main achievement of this construction is the stability of the singular continuous spectrum under “small” variations of the potential V in (1.1) and arbitrary self-adjoint variations of the boundary condition at the origin. This situation is non-typical for other known examples with singular cont ...
... The main achievement of this construction is the stability of the singular continuous spectrum under “small” variations of the potential V in (1.1) and arbitrary self-adjoint variations of the boundary condition at the origin. This situation is non-typical for other known examples with singular cont ...
The Probability that a Random - American Mathematical Society
... wishes to choose an odd prime, the trials n may be restricted to odd numbers. The expected number of trials is then about ¿ log x. There are many algorithms which can be used to decide if n is prime or composite. However, using the Fermât congruence is a very cheap test that is usually recommended a ...
... wishes to choose an odd prime, the trials n may be restricted to odd numbers. The expected number of trials is then about ¿ log x. There are many algorithms which can be used to decide if n is prime or composite. However, using the Fermât congruence is a very cheap test that is usually recommended a ...
Group knowledge is not always distributed (neither is it always implicit)
... Interestingly, we are able to prove a property like that in Lemma 3.1 even when the G operator is involved. Theorem 3.3. Let X and Y range over hK1 ,K2 , . . . ,Km ,Gj. Then: £Xw ⇔ £Yw. Theorem 3.3 has, for both the reading as group knowledge as well as that of a receiving agent for G, some remarkab ...
... Interestingly, we are able to prove a property like that in Lemma 3.1 even when the G operator is involved. Theorem 3.3. Let X and Y range over hK1 ,K2 , . . . ,Km ,Gj. Then: £Xw ⇔ £Yw. Theorem 3.3 has, for both the reading as group knowledge as well as that of a receiving agent for G, some remarkab ...
Mathematical writing - QMplus - Queen Mary University of London
... functions are introduced: ordering, symmetry, boundedness, continuity. Mathematical arguments are studied in detail in the second part of the book. Chapters 6 and 7 are devoted to basic proof techniques, while chapter 8 deals with existence statements and definitions. Some chapters are dedicated exp ...
... functions are introduced: ordering, symmetry, boundedness, continuity. Mathematical arguments are studied in detail in the second part of the book. Chapters 6 and 7 are devoted to basic proof techniques, while chapter 8 deals with existence statements and definitions. Some chapters are dedicated exp ...
Ans - Logic Matters
... it of the form e1 = e2 or e1 < e2 for some e1 and e2 ? It is a mechanical business to test if so. If ‘no’, then e isn’t an atomic wff. If ‘yes’, then e is an atomic wff iff both e1 and e2 are terms, and that’s effectively testable given answer (b). (e) You will want something like this: 1. If ϕ is a ...
... it of the form e1 = e2 or e1 < e2 for some e1 and e2 ? It is a mechanical business to test if so. If ‘no’, then e isn’t an atomic wff. If ‘yes’, then e is an atomic wff iff both e1 and e2 are terms, and that’s effectively testable given answer (b). (e) You will want something like this: 1. If ϕ is a ...
Interpreting Lattice-Valued Set Theory in Fuzzy Set Theory
... This paper presents a comparison of two axiomatic set theories over two non-classical logics. In particular, it suggests an interpretation of lattice-valued set theory as defined in [16] by S. Titani in fuzzy set theory as defined in [11] by authors of this paper. There are many different conception ...
... This paper presents a comparison of two axiomatic set theories over two non-classical logics. In particular, it suggests an interpretation of lattice-valued set theory as defined in [16] by S. Titani in fuzzy set theory as defined in [11] by authors of this paper. There are many different conception ...
Can Modalities Save Naive Set Theory?
... Section 3 considers (Comp2) in the strong modal logic S5, showing that (Comp2) is consistent in S5 and so a fortiorti in all weaker modal logics. Unfortunately, while the principle is consistent in these modal logics, the set theory it gives rise to is very weak. Indeed, we show that in S5, the non- ...
... Section 3 considers (Comp2) in the strong modal logic S5, showing that (Comp2) is consistent in S5 and so a fortiorti in all weaker modal logics. Unfortunately, while the principle is consistent in these modal logics, the set theory it gives rise to is very weak. Indeed, we show that in S5, the non- ...
Full text
... Several Interconnection networks have been proposed in literature for interconnecting computing elements. The interconnection network usually forms a regular pattern, which is exploited by the algorithms running on the network. Some of the commercially available networks are the hypercube, mesh, etc ...
... Several Interconnection networks have been proposed in literature for interconnecting computing elements. The interconnection network usually forms a regular pattern, which is exploited by the algorithms running on the network. Some of the commercially available networks are the hypercube, mesh, etc ...
Argumentative Approaches to Reasoning with Maximal Consistency Ofer Arieli Christian Straßer
... Dung’s semantics for abstract argumentation frameworks. Given a framework AF (Definition 1), a key issue in its understanding is the question what combinations of arguments (called extensions) can collectively be accepted from AF. According to Dung (1995), this is determined as follows: Definition 6 L ...
... Dung’s semantics for abstract argumentation frameworks. Given a framework AF (Definition 1), a key issue in its understanding is the question what combinations of arguments (called extensions) can collectively be accepted from AF. According to Dung (1995), this is determined as follows: Definition 6 L ...
neighborhood semantics for basic and intuitionistic logic
... neighborhood), and a modal formula ϕ is true at a world w, if the set of all states in which ϕ is true is a neighborhood of w. See [2] for more details on neighborhood semantics for modal logic. An interesting question is whether one can define similar neighborhood semantics for Intuitionistic Prop ...
... neighborhood), and a modal formula ϕ is true at a world w, if the set of all states in which ϕ is true is a neighborhood of w. See [2] for more details on neighborhood semantics for modal logic. An interesting question is whether one can define similar neighborhood semantics for Intuitionistic Prop ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.