Related papers: Representations of almost regular m-gonal forms I

It is known that any $m$-gonal form of rank $n \ge 5$ is almost regular. On the other words, any $m$-gonal form of rank $n \ge 5$ represents every sufficiently large integer which is locally represented. In this article, we study the…

In this article, we consider the representation of $m$-gonal forms over $\mathbb N_0$. We show that any $m$-gonal forms over $\mathbb N_0$ of rank $\ge 5$ is almost regular and ponder the sufficiently large integers which are indeed…

In this paper, we show that for a fixed rank $n$, there are only finitely many $m$ for which there is a regular $m$-gonal form of rank $n$ and determine every type of the (generalized) regular $m$-gonal form for every sufficiently large…

The maximal $n$ which admits a non-almost regular $m$-gonal form of rank $n$ is $4$. In this article, we consider the $m$-gonal forms of rank $4$ which is not almost regular.

In this article, we consider the rank of universal $m$-gonal forms for all sufficiently large $m$. Especially, we determine the minimal rank of universal $m$-gonal form and the maximal rank of kinds of proper universal $m$-gonal form.

In this paper, we consider representations of integers as sums of at most four distinct $m$-gonal numbers (allowing a fixed number of repeats of each polygonal number occurring in the sum). We show that the number of such representations…

In this paper, we classify the $(a_1,a_2,a_3,a_4,a_5)$ for which the universality of an $m$-gonal form $F_m(\mathbf x)$ having its first five coefficients as $(a_1,a_2,a_3,a_4,a_5)$ is characterized as the representability of positive…

In this paper, we study the set of positive integers that characterize the universality of $m$-gonal form.

In this paper, we show that for any $m$-gonal form $F_m(\mathbf x)$ with $m \ge 12$ which represents every positive integer up to $m-4$, by putting together only unary $m$-gonal form, we may complete an universal form.

A (positive definite and integral) quadratic form $f$ is called regular if it represents all integers that are locally represented. It is known that there are only finitely many regular ternary quadratic forms up to isometry. However, there…

For a natural number $m$, generalized $m$-gonal numbers are those numbers of the form $p_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in \mathbb Z$. In this paper we establish conditions on $m$ for which the ternary sum $p_m(x)+p_m(y)+p_m(z)$ is…

Among all restricted integer compositions with at most $m$ parts, each of which has size at most $l$, choose one uniformly at random. Which integer does this composition represent? In the current note, we show that underlying distribution…

In this paper, for n a positve integer, we compute the number of n degree representations for a dihedral group G of order 2m, m \geq 3 and the dimensions of the corresponding spaces of G invariant bilinear forms over a complex field C. We…

In this paper, we consider sums of three generalized $m$-gonal numbers whose parameters are restricted to integers with a bounded number of prime divisors. With some restrictions on $m$ modulo $30$, we show that a density one set of…

We give explicit structure of the graded ring of modular forms with respect to Gamma(N) (N=1,2,3,4,5,6,7,8,9,10,12,16,18) and for some other congruence groups. We also study the modular forms of half-integer weight for certain groups.

A positive-definite integral quadratic form is called regular if it represents every positive integer which is locally represented. In this article, we classify all regular diagonal quadratic forms of rank greater than 3.

In this article, we consider weighted sums of generalized polygonal numbers with coefficients $1$ or $2$. We show that for any $m\ge10$, those weighted sums of generalized $m$-gonal numbers represent every non-negative integers if they only…

All quasi-affine connected Generalized Dynkin Diagrams with rank $> 5$ are found. All quasi-affine Nichols (Lie braided) algebras with rank $> 5$ are also found.

We investigate pairwise quasi-orthogonal subalgebras in $M_{p^{kn}}$ which are isomorphic to $M_{p^{k}}$ for $k \ge 1$, $n \ge 2$ and a prime number $p$ with $p \ge 3$. We prove there exist $p^{2kn}-1/p^{2k}-1$ such subalgebras and they…

This is the second part of our study of epimorphisms with source a thick generalized $m$-gon and target a thin generalized $m$-gon. We classify the case $m = 8$ when the polygons are finite (in the first part [15] we handled the cases $m =…