Game Edukasi Pengenalan dan Pembelajaran Berhitung untuk Siswa Kelas 1 Sekolah Dasar

Mathematics, especially arithmetic theory, is a difficult subject for most students. Apart from the theory that is difficult to understand, students also have shortcomings in the interest in learning materials and limitations in learning media to teach arithmetic theory in mathematics. The purpose of this research is to produce an educational game as an alternative to studying arithmetic material in mathematics. The target users of this educational game are grade 1 elementary school students. This game has several features, such as displaying material in the form of images and videos in the form of learning to count from numbers 1 to 10, and counting games with drag and drop. The method used to develop this application starts from design and planning, then continues with the material collection, implementation, testing and evaluation, and application maintenance. Based on black box testing, the results show that the educational game has been made as expected, while based on the User Acceptance Test, the results of user perceptions of the game are 94.25% with an indicator of the "Very Good" category which indicates that this educational game can be used as alternative in learning arithmetic theory in mathematics.

Uniform, Integral, and Feasible Proofs for the Determinant Identities

Aiming to provide weak as possible axiomatic assumptions in which one can develop basic linear algebra, we give a uniform and integral version of the short propositional proofs for the determinant identities demonstrated over GF (2) in Hrubeš-Tzameret [15]. Specifically, we show that the multiplicativity of the determinant function and the Cayley-Hamilton theorem over the integers are provable in the bounded arithmetic theory VNC 2 ; the latter is a first-order theory corresponding to the complexity class NC 2 consisting of problems solvable by uniform families of polynomial-size circuits and O (log 2 n )-depth. This also establishes the existence of uniform polynomial-size propositional proofs operating with NC 2 -circuits of the basic determinant identities over the integers (previous propositional proofs hold only over the two-element field).

U-mentalism Patent: The Beginning of Cinematic Supercomputation

This paper discloses in synthesis a super-computation computer architecture (CA) model, presently a provisional Patent Application at INPI (nº 116408). The outline is focused on a method to perform computation at or near the speed of light, resorting to an inversion of the Princeton CA. It expands from isomorphic binary/RGB (typical) digital “images”, in a network of (UTM)s over Turing-machines (M)s. From the binary/RGB code, an arithmetic theory of (typical) digital images permits fully synchronous/orthogonal calculus in parallelism, wherefrom an exponential surplus is achieved. One such architecture depends on any “cell”-like exponential-prone basis such as the “pixel”, or rather the RGB “octet-byte”, limited as it may be, once it is congruent with any wave-particle duality principle in observable objects under the electromagnetic spectrum and reprogrammable designed. Well-ordered instructions in binary/RGB modules are, further, programming composed to alter the structure of the Internet, in virtual/virtuous eternal recursion/recurrence, under man-machine/machine-machine communication ontology.

SUMS OF FOUR SQUARES WITH A CERTAIN RESTRICTION

Z.-W. Sun [‘Refining Lagrange’s four-square theorem’, J. Number Theory175 (2017), 169–190] conjectured that every positive integer n can be written as $ x^2+y^2+z^2+w^2\ (x,y,z,w\in \mathbb {N}=\{0,1,\ldots \})$ with $x+3y$ a square and also as $n=x^2+y^2+z^2+w^2\ (x,y,z,w \in \mathbb {Z})$ with $x+3y\in \{4^k:k\in \mathbb {N}\}$ . In this paper, we confirm these conjectures via the arithmetic theory of ternary quadratic forms.

U-MentalismUtility Patent: an Overview

This paper discloses in synthesis a super-computation computer architecture (CA) model, presently a provisional Patent Application at INPI (nº 116408). The outline is focused on a method to perform computation at or near the speed of light, resorting to an inversion of the Princeton CA. It expands from isomorphic binary/RGB (typical) digital “images”, in a network of (UTM)s over Turing-machines (M)s. From the binary/RGB code, an arithmetic theory of (typical) digital images permits fully synchronous/orthogonal calculus in parallelism, wherefrom an exponential surplus is achieved. One such architecture depends on any “cell”- like exponential-prone basis such as the “pixel”, or rather the RGB “octet-byte”, limited as it may be, once it is congruent with any wave-particle duality principle in observable objects under the electromagnetic spectrum and reprogrammable designed. Well-ordered instructions in binary/RGB modules are, further, programming composed to alter the structure of the Internet, in virtual/virtuous eternal recursion/recurrence, under man-machine/machine-machine communication ontology.

HOW MUCH PROPOSITIONAL LOGIC SUFFICES FOR ROSSER’S ESSENTIAL UNDECIDABILITY THEOREM?

In this paper we explore the following question: how weak can a logic be for Rosser’s essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson’s Q is essentially undecidable in intuitionistic logic, and P. Hájek proved it in the fuzzy logic BL for Grzegorczyk’s variant of Q which interprets the arithmetic operations as nontotal nonfunctional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a much weaker arithmetic theory, a version of Robinson’s R (with arithmetic operations also interpreted as mere relations). Our result is based on a structural version of the undecidability argument introduced by Kleene and we show that it goes well beyond the scope of the Boolean, intuitionistic, or fuzzy logic.

CONSISTENCY OF CIRCUIT EVALUATION, EXTENDED RESOLUTION AND TOTAL NP SEARCH PROBLEMS

We consider sets ${\it\Gamma}(n,s,k)$ of narrow clauses expressing that no definition of a size $s$ circuit with $n$ inputs is refutable in resolution R in $k$ steps. We show that every CNF with a short refutation in extended R, ER, can be easily reduced to an instance of ${\it\Gamma}(0,s,k)$ (with $s,k$ depending on the size of the ER-refutation) and, in particular, that ${\it\Gamma}(0,s,k)$ when interpreted as a relativized NP search problem is complete among all such problems provably total in bounded arithmetic theory $V_{1}^{1}$. We use the ideas of implicit proofs from Krajíček [J. Symbolic Logic, 69 (2) (2004), 387–397; J. Symbolic Logic, 70 (2) (2005), 619–630] to define from ${\it\Gamma}(0,s,k)$ a nonrelativized NP search problem $i{\it\Gamma}$ and we show that it is complete among all such problems provably total in bounded arithmetic theory $V_{2}^{1}$. The reductions are definable in theory $S_{2}^{1}$. We indicate how similar results can be proved for some other propositional proof systems and bounded arithmetic theories and how the construction can be used to define specific random unsatisfiable formulas, and we formulate two open problems about them.

Arithmetic of positive characteristic -series values in Tate algebras

The second author has recently introduced a new class of$L$-series in the arithmetic theory of function fields over finite fields. We show that the values at one of these$L$-series encode arithmetic information of a generalization of Drinfeld modules defined over Tate algebras that we introduce (the coefficients can be chosen in a Tate algebra). This enables us to generalize Anderson’s log-algebraicity theorem and an analogue of the Herbrand–Ribet theorem recently obtained by Taelman.