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### How long is a Proof? - A short note [article]

A. G. Yaneff
2003 arXiv   pre-print
In practical terms W (work left to do in proof) is W ≤ ⌈ S − S 0 d k ↑ ⌉ ≤ n⌈ S /n − w 0 d k ↑ ⌉ ≤ |K ↑ solve | The question of what K ↑ consists of, is outside the Length of this note.  ...  Keeping in mind that (i) an infinite proof is no proof; and (ii) a circular proof is an infinite proof; under the condition |K ↑ max | = 1, i.e. one move can complete (at least) one game ('of proof' here  ...

### Shorter Notes: A Short Proof of the Birkhoff-Smale Theorem

T. Mrowka
1985 Proceedings of the American Mathematical Society
A short proof of the Birkhoff-Smale theorem on homoclinic points of diffeomorphisms is given. The proof is elementary in that only the local properties of a hyperbolic fixed point are assumed.  ...  The following theorem is fundamental in understanding how homoclinic points complicate the dynamics of a diffeomorphism.  ...  A SHORT PROOF OF THE BIRKHOFF-SMALE THEOREM T. MROWKA Abstract. A short proof of the Birkhoff-Smale theorem on homoclinic points of diffeomorphisms is given.  ...

### A Verifiable Random Function with Short Proofs and Keys [chapter]

Yevgeniy Dodis, Aleksandr Yampolskiy
2005 Lecture Notes in Computer Science
We give a simple and efficient construction of a verifiable random function (VRF) on bilinear groups. Our construction is direct.  ...  Our proofs of security are based on a decisional bilinear Diffie-Hellman inversion assumption, which seems reasonable given current state of knowledge.  ...  The VRF value is the label of the leaf, and the proof of correctness is a tuple of VRF proofs-one proof per each node on the path traced by x. 6 We also note that both of the aforementioned techniques  ...

### Short Locally Testable Codes and Proofs: A Survey in Two Parts [chapter]

Oded Goldreich
2010 Lecture Notes in Computer Science
., be a codeword or a valid proof). We also review the related concept of local decodable codes.  ...  We survey known results regarding locally testable codes and locally testable proofs (known as PCPs), with emphasis on the length of these constructs.  ...  From this perspective, the question of how much redundancy is introduced by such a transformation is a fundamental one.  ...

### A short and flexible proof of strong normalization for the calculus of constructions [chapter]

Herman Geuvers
1995 Lecture Notes in Computer Science
short and flexible proof of Strong Normalization for the Calculus of Constructions.  ...  The approach that we use here is based on saturated sets. It yields a (relatively short) direct proof of SN for CC using two different interpretations, [-~< and a-Dp.  ...  IS. The performance of single-keyword and rnultiplekeyword pattern matching algorithms, p. 46.  ...

### How Long is a Piece of Cable? Equilibrium Determination of Local Telephone Areas

Gianni De Fraja, Fabio M. Manenti
2003 Journal of Economics
In many telephone systems, calls between users in the same calling area are local; calls to outside the calling area are long distance and priced differently.  ...  We end the paper by considering the case of a competitive market.  ...  The intuition for this result is that, when the community of interest effect is strong, few long distance calls are placed for every call to nearby locations; as a consequence, short distance calls are  ...

### A note on short cycles in diagraphs

C.T Hoàng, B Reed
1987 Discrete Mathematics
This conjecture is known to be true for k ~< 3. In this paper, we present a proof of the conjecture for the cases k = 4 and k = 5.  ...  In 1977, Caceetta and Haggkvist conjectured that if G is a directed graph with n vertices and minimal outdegree k, then G contains a directed cycle of length at most In~k].  ...  First note that the conjecture holds trivially for t ~< 2. When t ~> 3, our arguments shall rely on the arguments used in the proof of Theorem 1.  ...

### A long note on Mulders' short product

G. Hanrot, P. Zimmermann
2004 Journal of symbolic computation
The short product of two power series is the meaningful part of the product of these objects, i.e., i+ j <n a i b j x i+ j .  ...  This algorithm works by selecting a cutoff point k and performing a full k × k product and two (n − k) × (n − k) short products recursively.  ...  Acknowledgement The authors wish to thank Michel Quercia for valuable discussions, a useful collaboration on related subjects, and for suggesting Theorem 4 to them.  ...

### A note on short cycles in a hypercube

Maria Axenovich, Ryan Martin
2006 Discrete Mathematics
How many edges can a quadrilateral-free subgraph of a hypercube have? This question was raised by Paul Erdős about 27 years ago.  ...  His conjecture that such a subgraph asymptotically has at most half the edges of a hypercube is still unresolved.  ...  There is no monochromatic C 6 in Q n under coloring c. Proof. Let C = x 1 y 1 x 2 y 2 x 3 y 3 x 1 be a monochromatic C 6 .  ...

### A short note on graphs with long Thomason's chains [article]

2020 arXiv   pre-print
The base of the exponent is approximately 1.1812..., which exceeds previous results in the area.  ...  We present a family of 3-connected cubic planar Hamiltonian graphs with an exponential number of steps required by Thomason's algorithm.  ...  So in the case of a rightmost path, they differ only in how the path traverses the pac, and there is a lollipop operation that maps between two possibilities.  ...

### A short note on the Stanley-Wilf Conjecture for permutations on multisets [article]

Marie-Louise Bruner
2013 arXiv   pre-print
In this note we present a direct proof of the already known fact that the well-known Stanley-Wilf Conjecture, stating that the number of permutations avoiding a given pattern does not grow faster than  ...  The concept of pattern avoidance respectively containment in permutations can be extended to permutations on multisets in a straightforward way.  ...  This present note is devoted to providing a simple and direct proof of the Stanley-Wilf Conjecture for permutations on multisets.  ...

### A note on character sums over short moving intervals [article]

2022 arXiv   pre-print
Tools used in the proofs include the existence of characters with large partial sums on short initial segments, and moment estimates for trigonometric polynomials with random multiplicative coefficients  ...  On the other hand, we show it is true for "almost all" characters on the range q^1-o(1)≤ H = o(q).  ...  Soundararajan and Max Xu for sharing a draft of their forthcoming paper [21] , and Max Xu for some helpful comments.  ...

### A short note on type-inhabitation: Formula-trees vs. game semantics

S. Alves, S. Broda
2015 Information Processing Letters
In 2000 a new proof method has been presented, evidencing the close relationship between the structure of types and their inhabitants.  ...  Type-inhabitation is a topic of major importance, due to its close relationship to provability in logical systems and has been studied from di↵erent perspectives over the years.  ...  We now describe how, given a proof-tree of a type , we compute the corresponding finite set of long normal inhabitants of . Definition 15 Consider a proof-tree t for a type .  ...

### A short note on a Bernstein-Bezier basis for the pyramid [article]

Jesse Chan, T. Warburton
2015 arXiv   pre-print
We introduce a Bernstein-Bezier basis for the pyramid, whose restriction to the face reduces to the Bernstein-Bezier basis on the triangle or quadrilateral.  ...  The Bernstein pyramid space is identical to the pyramid space of Bergot, Cohen, Durufle [5] . Proof.  ...  Proof. Since each component of the Bernstein pyramid basis is pointwise positive on the cube, the basis is pointwise positive over the pyramid.  ...

### A Short Note on Soft-max and Policy Gradients in Bandits Problems [article]

Neil Walton
2020 arXiv   pre-print
This is a short communication on a Lyapunov function argument for softmax in bandit problems.  ...  We give a short argument that gives a regret bound for the soft-max ordinary differential equation for bandit problems.  ...  The author is grateful to Csaba Szepesvari for suggesting to make this note available and to Tor Lattimore for first suggesting Soft-Max as an alternative to the SAMBA o.d.e.. Appendix.  ...
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