Arithmetical Congruence Preservation: from Finite to Infinite

Various problems on integers lead to the class of congruence preserving functions on rings, i.e. functions verifying $a-b$ divides $f(a)-f(b)$ for all $a,b$. We characterized these classes of functions in terms of sums of rational polynomials (taking only integral values) and the function giving the least common multiple of $1,2,\ldots,k$. The tool used to obtain these characterizations is "lifting": if $\pi\colon X\to Y$ is a surjective morphism, and $f$ a function on $Y$ a lifting of $f$ is a function $F$ on $X$ such that $\pi\circ F=f\circ\pi$. In this paper we relate the finite and infinite notions by proving that the finite case can be lifted to the infinite one. For $p$-adic and profinite integers we get similar characterizations via lifting. We also prove that lattices of recognizable subsets of $Z$ are stable under inverse image by congruence preserving functions

Integral Difference Ratio Functions on Integers

number theoryInternational audienceTo Jozef, on his 80th birthday, with our gratitude for sharing with us his prophetic vision of Informatique Abstract. Various problems lead to the same class of functions from integers to integers: functions having integral difference ratio, i.e. verifying f (a) − f (b) ≡ 0 (mod (a − b)) for all a > b. In this paper we characterize this class of functions from Z to Z via their a la Newton series expansions on a suitably chosen basis of polynomials (with rational coefficients). We also exhibit an example of such a function which is not polynomial but Bessel like

Compressed Subsequence Matching and Packed Tree Coloring

We present a new algorithm for subsequence matching in grammar compressed strings. Given a grammar of size $n$ compressing a string of size $N$ and a pattern string of size $m$ over an alphabet of size $\sigma$, our algorithm uses $O(n+\frac{n\sigma}{w})$ space and $O(n+\frac{n\sigma}{w}+m\log N\log w\cdot occ)$ or $O(n+\frac{n\sigma}{w}\log w+m\log N\cdot occ)$ time. Here $w$ is the word size and $occ$ is the number of occurrences of the pattern. Our algorithm uses less space than previous algorithms and is also faster for $occ=o(\frac{n}{\log N})$ occurrences. The algorithm uses a new data structure that allows us to efficiently find the next occurrence of a given character after a given position in a compressed string. This data structure in turn is based on a new data structure for the tree color problem, where the node colors are packed in bit strings.Comment: To appear at CPM '1

Faster subsequence recognition in compressed strings

Computation on compressed strings is one of the key approaches to processing massive data sets. We consider local subsequence recognition problems on strings compressed by straight-line programs (SLP), which is closely related to Lempel--Ziv compression. For an SLP-compressed text of length $\bar m$, and an uncompressed pattern of length $n$, C{\'e}gielski et al. gave an algorithm for local subsequence recognition running in time $O(\bar mn^2 \log n)$. We improve the running time to $O(\bar mn^{1.5})$. Our algorithm can also be used to compute the longest common subsequence between a compressed text and an uncompressed pattern in time $O(\bar mn^{1.5})$; the same problem with a compressed pattern is known to be NP-hard

Pairs, sets and sequences in first-order theories

Asuransi sebagai aktivitas bisnis diharuskan memenuhi prinsip-prinsip hukum asuransi. Salah satu prinsip yang harus dipegang teguh adalah principle of  utmost good faith, di samping prinsip yang lain. Prinsip ini berbunyi bahwa seorang tertanggung wajib memberi informasi secara jujur terhadap apa yang dipertanggungkan kepada penanggung. Dalam bisnis Islam, kejujuran merupakan prinsip yang harus dijunjung tinggi. Secara hukum, prinsip ini diatur dalam KUH Dagang. Persoalannya adalah apakah prinsip ini dianggap cukup dari sudut pandang hukum perjanjian syariah. Secara sekilas bahwa prinsip iktikad baik sempurna ini telah memenuhi asas perjanjian syariah, namun demikian tidak memiliki kriteria maksimal kejujuran. Ketiadaan kejujuran dalam bisnis asuransi akan berdampak pada batalnya perjanjian asuransi karena ada unsur cacat kehendak (‘uyub ar-ridla). Insurance as a business activity must fulfill principles of insurance law. One of the principles that must be hold on is the principle of  utmost good faith. The principle says that an endured person must honestly give information of  what should be given responsibility to the guarantor. In Islamic business, honesty is a principle that should be respected. From point of  view of  law, the principle is settled in commerce law. The problem is that whether the principle is represenative enough if it is viewed from law of  syariah agreement. At glance, the principle has fulfilled the basic of syariah agreement, however, it does not have maximum criteria of  honesty. Unavailability of honesty in insurance business will give effect of  invalidate of  insurance agreement, for there is a deformity of desire (‘uyub ar-ridla).</p

Congruence preserving functions on free monoids

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