On the van der Waerden numbers w(2;3,t)

We present results and conjectures on the van der Waerden numbers w(2;3,t) and on the new palindromic van der Waerden numbers pdw(2;3,t). We have computed the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39, where for t <= 30 we conjecture these lower bounds to be exact. The lower bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we present an improved conjecture. We also investigate regularities in the good partitions (certificates) to better understand the lower bounds. Motivated by such reglarities, we introduce *palindromic van der Waerden numbers* pdw(k; t_0,...,t_{k-1}), defined as ordinary van der Waerden numbers w(k; t_0,...,t_{k-1}), however only allowing palindromic solutions (good partitions), defined as reading the same from both ends. Different from the situation for ordinary van der Waerden numbers, these "numbers" need actually to be pairs of numbers. We compute pdw(2;3,t) for 3 <= t <= 27, and we provide lower bounds, which we conjecture to be exact, for t <= 35. All computations are based on SAT solving, and we discuss the various relations between SAT solving and Ramsey theory. Especially we introduce a novel (open-source) SAT solver, the tawSolver, which performs best on the SAT instances studied here, and which is actually the original DLL-solver, but with an efficient implementation and a modern heuristic typical for look-ahead solvers (applying the theory developed in the SAT handbook article of the second author).Comment: Second version 25 pages, updates of numerical data, improved formulations, and extended discussions on SAT. Third version 42 pages, with SAT solver data (especially for new SAT solver) and improved representation. Fourth version 47 pages, with updates and added explanation

On the asymptotic minimum number of monochromatic 3-term arithmetic progressions

Let V(n) be the minimum number of monochromatic 3-term arithmetic progressions in any 2-coloring of {1,2,...,n}. We show that (1675/32768) n^2 (1+o(1)) <= V(n) <= (117/2192) n^2(1+o(1)). As a consequence, we find that V(n) is strictly greater than the corresponding number for Schur triples (which is (1/22) n^2 (1+o(1)). Additionally, we disprove the conjecture that V(n) = (1/16) n^2(1+o(1)), as well as a more general conjecture.Comment: 9 pages. Revised version fixes formatting errors (same text

ParaFPGA15 : exploring threads and trends in programmable hardware

The symposium ParaFPGA focuses on parallel techniques using FPGAs as accelerator in high performance computing. The green computing aspects of low power consumption at high performance were somewhat tempered by long design cycles and hard programmability issues. However, in recent years FPGAs have become new contenders as versatile compute accelerators because of a growing market interest, extended application domains and maturing high-level synthesis tools. The keynote paper highlights the historical and modern approaches to high-level FPGA and the contributions cover applications such as NP-complete satisfiability problems and convex hull image processing as well as performance evaluation, partial reconfiguration and systematic design exploration

Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer

We solved a long-outstanding open problem in Ramsey theory, using SAT solving

Key Agreement Protocol (KAP) Based on Matrix Power Function

* Work is partially supported by the Lithuanian State Science and Studies Foundation.The key agreement protocol (KAP) is constructed using matrix power functions. These functions are based on matrix ring action on some matrix set. Matrix power functions have some indications as being a one- way function since they are linked with certain generalized satisfiability problems which are potentially NP- Complete. A working example of KAP with guaranteed brute force attack prevention is presented for certain algebraic structures. The main advantage of proposed KAP is considerable fast computations and avoidance of arithmetic operations with long integers