Denominator-preserving maps

Let F be a continuous injective map from an open subset of R^n to R^n. Assume that, for infinitely many k>1, F induces a bijection between the rational points of denominator k in the domain and those in the image (the denominator of (a_1/b_1,...,a_n/b_n) being the l.c.m. of b_1,...,b_n). Then F preserves the Lebesgue measure.Comment: 11 pages, 1 figur

Multidimensional continued fractions and a Minkowski function

The Minkowski Question Mark function can be characterized as the unique homeomorphism of the real unit interval that conjugates the Farey map with the tent map. We construct an n-dimensional analogue of the Minkowski function as the only homeomorphism of an n-simplex that conjugates the piecewise-fractional map associated to the Monkemeyer continued fraction algorithm with an appropriate tent map.Comment: 17 pages, 3 figures. Revised version according to the referee's suggestions. Proof of Lemma 2.3 more detailed, other minor modifications. To appear in Monatshefte fur Mathemati

Generic substitutions

Up to equivalence, a substitution in propositional logic is an endomorphism of its free algebra. On the dual space, this results in a continuous function, and whenever the space carries a natural measure one may ask about the stochastic properties of the action. In classical logic there is a strong dichotomy: while over finitely many propositional variables everything is trivial, the study of the continuous transformations of the Cantor space is the subject of an extensive literature, and is far from being a completed task. In many-valued logic this dichotomy disappears: already in the finite-variable case many interesting phenomena occur, and the present paper aims at displaying some of these.Comment: 22 pages, 2 figures. Revised version according to the referee's suggestions. To appear in the J. of Symbolic Logi

Budaya Organisasi dan Kode Etik Pustakawan dalam Implementasinya

Librarians actually played a major role in contributing to the development of a civilized. but its role is not seen by most people . The problem is when people seeing the library, as if the librarian is blocked by a row of collection that is increasingly to be older and increasingly away from the elements to present . That said, when mentioning the word library, people thinks refer to a medium of human civilization, namely books. For a long time, the book became the primary knowledge resource, compiled by the library. This happens because the position of the library was considered only as a storage place only, and turns up the assumption that the modern age so shall still can not be eliminated. Judging by the cases me "minor" the role of librarians as noted above, provide opportunities for us to further tracing the pattern of the existing organizational cultur

Measures induced by units

The half-open real unit interval (0,1] is closed under the ordinary multiplication and its residuum. The corresponding infinite-valued propositional logic has as its equivalent algebraic semantics the equational class of cancellative hoops. Fixing a strong unit in a cancellative hoop -equivalently, in the enveloping lattice-ordered abelian group- amounts to fixing a gauge scale for falsity. In this paper we show that any strong unit in a finitely presented cancellative hoop H induces naturally (i.e., in a representation-independent way) an automorphism-invariant positive normalized linear functional on H. Since H is representable as a uniformly dense set of continuous functions on its maximal spectrum, such functionals -in this context usually called states- amount to automorphism-invariant finite Borel measures on the spectrum. Different choices for the unit may be algebraically unrelated (e.g., they may lie in different orbits under the automorphism group of H), but our second main result shows that the corresponding measures are always absolutely continuous w.r.t. each other, and provides an explicit expression for the reciprocal density.Comment: 24 pages, 1 figure. Revised version according to the referee's suggestions. Examples added, proof of Lemma 2.6 simplified, Section 7 expanded. To appear in the Journal of Symbolic Logi

Decreasing height along continued fractions

The fact that the euclidean algorithm eventually terminates is pervasive in mathematics. In the language of continued fractions, it can be stated by saying that the orbits of rational points under the Gauss map x-->1/x eventually reach zero. Analogues of this fact for Gauss maps defined over quadratic number fields have relevance in the theory of flows on translation surfaces, and have been established via powerful machinery, ultimately relying on the Veech dichotomy. In this paper, for each commensurability class of noncocompact triangle groups of quadratic invariant trace field, we construct a Gauss map whose defining matrices generate a group in the class; we then provide a direct and self-contained proof of termination. As a byproduct, we provide a new proof of the fact that noncocompact triangle groups of quadratic invariant trace field have the projective line over that field as the set of cross-ratios of cusps. Our proof is based on an analysis of the action of nonnegative matrices with quadratic integer entries on the Weil height of points. As a consequence of the analysis, we show that long symbolic sequences in the alphabet of our maps can be effectively split into blocks of predetermined shape having the property that the height of points which obey the sequence and belong to the base field decreases strictly at each block end. Since the height cannot decrease infinitely, the termination property follows

Slow continued fractions, transducers, and the Serret theorem

A basic result in the elementary theory of continued fractions says that two real numbers share the same tail in their continued fraction expansions iff they belong to the same orbit under the projective action of PGL(2,Z). This result was first formulated in Serret's Cours d'alg`ebre sup'erieure, so we'll refer to it as to the Serret theorem. Notwithstanding the abundance of continued fraction algorithms in the literature, a uniform treatment of the Serret result seems missing. In this paper we show that there are finitely many possibilities for the subgroups Sigma of PGL(2,Z) generated by the branches of the Gauss maps in a large family of algorithms, and that each Sigma-equivalence class of reals is partitioned in finitely many tail-equivalence classes, whose number we bound. Our approach is through the finite-state transducers that relate Gauss maps to each other. They constitute opfibrations of the Schreier graphs of the groups, and their synchronizability ---which may or may not hold--- assures the a.e. validity of the Serret theorem

Penerapan Metode Experiential Learning Untuk Meningkatkan Prestasi Bahasa Indonesia Materi Apresiasi Puisi Pada Siswa Kelas VIII-B SMP Negeri 1 Ngawi

Berdasarkan hasil pembelajaran puisi sebelumnya dapat diambil kesimpulan bahwa kemampuan siswa dalam mengapresiasi puisi ternyata masih belum banyak siswa yang memiliki ketuntasan belajar yang rendah. Oleh karena itu penelitian ini dilaksanakan dengan tujuan menerapkan metode experiential learning untuk meningkatkan prestasi bahasa Indonesia materi apresiasi puisi pada siswa kelas VIII-B SMP Negeri 1 Ngawi. Penelitian tindakan kelas ini dilaksanakan di SMP Negeri 1 Ngawi Penelitian ini dilaksanakan di SMP Negeri 1 Ngawi. Penelitian tindakan kelas ini akan dilaksanakan selama 2 bulan yakni bulan Januari sampai dengan Februari 2020. Subjek penelitian ini adalah siswa kelas VIII-B SMP Negeri 1 Ngawi. Jumlah siswa dalam kelas tersebut sebanyak 32 siswa. Hasil penelitian menunjukkan bahwa berdasarkan hasil pengamatan diketahui bahwa rata-rata aktivitas siswa mengalami peningkatan. Disamping itu metode experiential learning telah terjadi peningkatan jumlah siswa yang mengalami ketuntasan belajar dari siklus I hingga siklus II. Peningkatan kemampuan mengapresiasi puisi tersebut diindikasikan angka ketuntasan mengalami peningkatan, yakni yakni jika refleksi awal sebanyak 50% siswa maka pada siklus I mengalami kenaikan menjadi 72% siswa dan pada siklus II mengalami peningkatan menjadi 91% siswa. Dengan hasil ini maka dapat dikatakan bahwa metode experiential learning efektif dalam meningkatkan prestasi belajar siswa