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### On the Complexity of Finding Narrow Proofs [article]

Christoph Berkholz
2012 arXiv   pre-print
This lower bound is unconditional and does not rely on any unproven complexity theoretic assumptions. The lower bound is matched by a trivial upper bound of n^O(k).  ...  This confirms a conjecture by Vardi, who has first raised the question for the complexity of the resolution width problem.  ...  Most state-of-the-art SAT solvers try to find resolution refutations. One natural complexity measure for resolution is the length of a refutation.  ...

### On the Complexity of Finding Narrow Proofs

Christoph Berkholz
2012 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
the Boolean ex.  ...  (k + 1)-pebble game on Γ ⇐⇒ Γ has a width-k resolution refutation (Atserias, Dalmau '08) Γ = {D ∨ E, C ∨ D ∨ E, A ∨ B ∨ C, A ∨ D ∨ E, B ∨ C ∨ D, D} Spoiler wins the Boolean ex.  ...  Player 1 wins on I ⇐⇒ Spoiler wins on Γ(I) The reduction We reduce the k-pebble KAI-game on I to the Boolean existential (k+2)-pebble game on Γ(I).  ...

### The Skitovich–Darmois and Heyde theorems for complex and quaternion random variables [article]

G. M. Feldman
2020 arXiv   pre-print
We also study an analogue of the Heyde theorem for complex random variables.  ...  We prove the following analogue of the classical Skitovich–Darmois theorem for complex random variables. Let α=a+ib be a nonzero complex number. Then the following statements hold. 1.  ...  The proof of Theorem A, given in [14] , is based on the Polya characterization theorem for complex Gaussian random variables in the narrow sense (see [11] ).  ...

### Monte-Carlo Proof-Number Search for Computer Go [chapter]

Jahn-Takeshi Saito, Guillaume Chaslot, Jos W. H. M. Uiterwijk, H. Jaap van den Herik
2007 Lecture Notes in Computer Science
It enhances proof-number search by adding the flexible Monte-Carlo evaluation. We present the new algorithm and evaluate it on a sub-problem of Go, the Life-and-Death problem.  ...  In order to improve the efficiency of proof-number search, we introduce a new algorithm, Monte-Carlo Proof-Number search.  ...  Especially, we would like to thank one of them for his constructive criticism which lead to crucial improvements on the implementational work.  ...

### Conditional equational theories and complete sets of transformations

Steffen Hölldobler
1990 Theoretical Computer Science
The completeness proof is based on the observation that each refutation with respect to linear paramodulation and reflection can be modelled by the transformations.  ...  As certam conditions imposed on an equational theory restrict the search space generated by paramoduiation and reflection we can easily refine our transfiiinations -due to the completeness proof-if the  ...  The proof is by transfinite induction on the complexity of the refutation of EPu{CF} with respect to (+ip, +r}. We assume that the result holds for all M < &I'.  ...

### Polya's Characterization Theorem for Complex Random Variables

Nicholas N. Vakhania
1997 Journal of Complexity
Moreover, it is proved that this property characterizes the class of complex Gaussian random variables in the narrow sense; no other complex random variable enjoys it.  ...  A complex random variable can be Gaussian in either the narrow or the wide sense.  ...  If we want to maintain this property for the complex case we have to adopt the narrow definition. The proof of this fact is easy and we give it here for completeness.  ...

### Many Maybes Mean (Mostly) the Same Thing [chapter]

Tim Menzies, Harhsinder Singh
2004 Studies in Fuzziness and Soft Computing
Based on a literature review and a newly developed mathematics of "reachability", it is argued that searches through a space containing uncertainties, most of the reachable conclusions will be reached  ...  via a small number of "master variables" in a "narrow funnel".  ...  Acknowledgements The arguments in this chapter benefitted greatly from numerous lively discussions with Helen Burgess and Steve Easterbrook.  ...

### The Ackermann Award 2009 [chapter]

Johann A. Makowsky, Alexander Razborov
2009 Lecture Notes in Computer Science
The proof-search heuristic in this algorithm is represented by the choice of the branching literal, and, abstracting from the issue of finding the best heuristics, we arrive at the proof system that is  ...  On the theoretical side, propositional resolution had been considered in Proof Complexity (Tseitin 1968) well before the formal inauguration of the subject (Cook 1975) .  ...

### The asymptotics of optimal (equiripple) filters

Jianhong Shen, G. Strang
1999 IEEE Transactions on Signal Processing
In the transition band, the filter response is discovered to be asymptotically close to a scaled error function. The main tools are potential theory in the complex plane and asymptotic analysis.  ...  For equiripple filters, the relation among the filter length N + 1, the transition bandwidth 1!, and the optimal passband and stopband errors p and s has been a secret for more than 20 years.  ...  ACKNOWLEDGMENT The authors would like to thank J. Kaiser and A. Oppenheim for their interest in this problem.  ...

### The Interaction Between Public and Private Enforcement of EU Competition Law: a Case Study of the Swedish Booking Cases

Katharina Voss, Swedish Competition Authority
2020 Yearbook of Antitrust and Regulatory Studies
The focus of this article is placed on the assessment carried out by the Swedish courts to determine whether the MFN clauses were restrictive of competition by effect and on the standard of proof attached  ...  Booking is viewed as an illustration of the increased complexity of competition policy, in particular were novel practices are at issue  ...  With regard to the standard of proof required by Swedish courts, there is on the one hand a tradition of defining specific standards of proof.  ...

### On the numerical radius of operators in Lebesgue spaces [article]

Miguel Martin, Mikhail Popov
2010 arXiv   pre-print
We also give lower bounds for the best constant of equivalence between the numerical radius and the operator norm in L_p(μ) for atomless μ when restricting to rank-one operators or narrow operators.  ...  In other words, we prove that {∫ |x|^p-1|Tx| dμ : x∈ L_p(μ), x_p=1} ≥ p^-1/p q^-1/q T for every T∈L(L_p(μ)) and that this inequality is the best possible when the dimension of L_p(μ) is greater than one  ...  Acknowledgments: The authors would like to thank Rafael Payá and Beata Randrianantoanina for fruitful conversations concerning the matter of this paper.  ...

### A note on propositional proof complexity of some Ramsey-type statements

Jan Krajíček
2010 Archive for Mathematical Logic
As a consequence of Pudlák's work in bounded arithmetic [19] it is known that there are quasi-polynomial size constant depth Frege proofs of RAM(4 k , k), but the proof complexity of these formulas in  ...  In this paper we continue the study of propositional complexity of Ramsey theorem, motivated by the following problem: *  ...  Acknowledgements: I am indebted to Stefan Dantchev (Durham) and Neil Thapen (Prague) for comments on the draft of the paper.  ...

### Encoding and decoding of BCH codes using light and short codewords

R.M. Roth, G. Seroussi
1988 IEEE Transactions on Information Theory
For high-rate primitive BCH codes this is faster than the commonly used algorithms, which are nonlinear in n when run on sequential machines.  ...  It is shown that every q-ary primitive BCH code of designed distance δ and sufficiently large length n contains a codeword c 0 of weight w = O(δ ), and degree deg(c 0 ) = o(n).  ...  The technique used is similar to one proposed by Kaminski [3] for finding the remainder of a long polynomial (or integer) divided by a short one.  ...