Narrow Results

It is shown how a noncommutative spacetime leads to an area, mass and entropy quantization condition which allows to derive the Schwarzschild black hole entropy $\frac{A}{4G}$, the logarithmic corrections, and further corrections, from the discrete mass transitions taking place among different mass states in $D=4$. The higher-dimensional generalization of the results in $D=4$ follows. The discretization of the entropy-mass relation $S=S(M)$ leads to an entropy quantization of the form $S=S(M_n)=n$, such that one may always assign $n$ “bits” to the discrete entropy, and in doing so, make contact with quantum information. The physical applications of mass quantization, like the counting of states contributing to the black hole entropy, black hole evaporation, and the direct connection to the black holes-string correspondence [G. Horowitz and J. Polchinski, A correspondence principle for black holes and strings, Phys. Rev. D55 (1997) 6189.] via the asymptotic behavior of the number of partitions of integers, follows. To conclude, it is shown how the recent large $N$ Matrix model (fuzzy sphere) of C.-S. Chu [A matrix model proposal for QG and the QM of black holes, preprint, arXiv:2406.01466] leads to very similar results for the black hole entropy as the physical model described in this work which is based on the discrete mass transitions originating from the noncommutativity of the spacetime coordinates.

The symmetric difference of the $q$-binomial coefficients $F_{n,k}(q)=\left[\genfrac{}{}{0ex}{}{n+k}{k}\right]-q^n\left[\genfrac{}{}{0ex}{}{n+k-2}{k-2}\right]$ was introduced by Reiner and Stanton. They proved that $F_{n,k}(q)$ is symmetric and unimodal for $k\ge 2$ and $n$ even by using the representation theory for Lie algebras. Based on Sylvester’s proof of the unimodality of the Gaussian coefficients, as conjectured by Cayley, we find an interpretation of the unimodality of $F_{n,k}(q)$ in terms of semi-invariants. In the spirit of the strict unimodality of the Gaussian coefficients due to Pak and Panova, we prove the strict unimodality of the symmetric difference $G_{n,k,r}(q)=\left[\genfrac{}{}{0ex}{}{n+k}{k}\right]-q^{nr/2}\left[\genfrac{}{}{0ex}{}{n+k-r}{k-r}\right]$, except for the two terms at both ends, where $n,r\ge 8$, $k\ge r$ and at least one of $n$ and $r$ is even.

Bell polynomials appear in several combinatorial constructions throughout mathematics. Perhaps most naturally in the combinatorics of set partitions, but also when studying compositions of diffeomorphisms on vector spaces and manifolds, and in the study of cumulants and moments in probability theory. We construct commutative and noncommutative Bell polynomials and explain how they give rise to Faà di Bruno Hopf algebras. We use the language of incidence Hopf algebras, and along the way provide a new description of antipodes in noncommutative incidence Hopf algebras, involving quasideterminants. We also discuss Möbius inversion in certain Hopf algebras built from Bell polynomials.

Common meadows are commutative and associative algebraic structures with two operations (addition and multiplication) with additive and multiplicative identities and for which inverses are total. The inverse of zero is an error term $a$ which is absorbent for addition. We study the problem of enumerating all finite common meadows of order n (that is, common meadows with n elements). This problem turns out to be deeply connected with both the number of finite rings of order n and with the number of a certain kind of partition of positive integers.

Hesitant fuzzy sets represent a useful tool in many areas such as decision making or image processing. Finite interval-valued hesitant fuzzy sets are a particular kind of hesitant fuzzy sets that generalize fuzzy sets, interval-valued fuzzy sets or Atanassov’s intuitionistic fuzzy sets, among others. Partitioning is a long-standing open problem due to its remarkable importance in many areas such as clustering. Thus, many different partitioning approaches have been developed for crisp and fuzzy sets. This work presents a partitioning method for the so-called finite interval-valued hesitant fuzzy sets. The definition of this partitioning method involves a definition of an ordering relation for finite interval-valued fuzzy sets membership degrees, i.e, finitely generated sets, as well as the definitions of t-norm and t-conorm for these kinds of sets.

The bm-central limit theorem for positive-definite real symmetric matrices and for homogeneous rooted trees are studied. The combinatorial nature of it is described. In addition, the construction of bm-product of graphs is given, which generalizes comb and boolean products. General properties of bm-extension operators are shown.

We study the properties of the (noncommutative) bm-independence of algebras, indexed by partially ordered sets. The index sets are given by positive cones, in particular the symmetric cones, which include the positive-definite Hermitian matrices with complex or quaternionic entries. We formulate and prove the general versions of the bm-Central Limit Theorems for bm-independent random variables, indexed by lattices in such positive cones. The limit measures we obtain are symmetric and compactly supported on the real line. Their (even) moment sequences (g_n)_n≥0 satisfy the generalized recurrence for the Catalan numbers: , where the coefficients γ(r) are computed by the Euler's beta-function of the first kind, related to the given positive cone. Example of a nonsymmetric cone, the Vinberg's cone, is also studied in this context.

Assume $ZF+AD$. If $𝜖$ is an ordinal and X is a set of ordinals, then ${[X]}_{\ast }^{𝜖}$ is the collection of order-preserving functions $f:𝜖\to X$ which have uniform cofinality $\omega$ and discontinuous everywhere. The weak partition properties on ${\omega }_1$ and ${\omega }_2$ yield partition measures on ${[{\omega }_1]}_{\ast }^{𝜖}$ when $𝜖<{\omega }_1$ and ${[{\omega }_2]}_{\ast }^{𝜖}$ when $𝜖<{\omega }_2$. The following almost everywhere continuity properties for functions on partition spaces with respect to these partition measures will be shown. For every $𝜖<{\omega }_1$ and function $\mathrm{\Phi }:{[{\omega }_1]}^{𝜖}\to {\omega }_1$, there is a club $C\subseteq {\omega }_1$ and a $\zeta <𝜖$ so that for all $f,g\in {[C]}_{\ast }^{𝜖}$, if $f\upharpoonright \zeta =g\upharpoonright \zeta$ and $sup(f)=sup(g)$, then $\mathrm{\Phi }(f)=\mathrm{\Phi }(g)$. For every $𝜖<{\omega }_2$ and function $\mathrm{\Phi }:{[{\omega }_2]}^{𝜖}\to {\omega }_2$, there is an $\omega$-club $C\subseteq {\omega }_2$ and a $\zeta <𝜖$ so that for all $f,g\in {[C]}_{\ast }^{𝜖}$, if $f\upharpoonright \zeta =g\upharpoonright \zeta$ and $sup(f)=sup(g)$, then $\mathrm{\Phi }(f)=\mathrm{\Phi }(g)$. The previous two continuity results will be used to distinguish the cardinalities of some important subsets of $\mathcal{𝒫}({\omega }_2):|{[{\omega }_1]}^{\omega }|<|{[{\omega }_1]}^{<{\omega }_1}|$. $|{[{\omega }_2]}^{\omega }|<|{[{\omega }_2]}^{<{\omega }_1}|<|{[{\omega }_2]}^{{\omega }_1}|<|{[{\omega }_2]}^{<{\omega }_2}|$. $\neg (|{[{\omega }_1]}^{<{\omega }_1}|\le |{[{\omega }_2]}^{\omega }|)$. $\neg (|{[{\omega }_1]}^{{\omega }_1}|\le |{[{\omega }_2]}^{<{\omega }_1}|)$. It will also be shown that ${[{\omega }_1]}^{\omega }$ has the Jónsson property: For every $\mathrm{\Phi }:{}^{<\omega }({[{\omega }_1]}^{\omega })\to {[{\omega }_1]}^{\omega }$, there is an $X\subseteq {[{\omega }_1]}^{\omega }$ with $|X|=|{[{\omega }_1]}^{\omega }|$ so that $\mathrm{\Phi }[{}^{<\omega }X]\ne {[{\omega }_1]}^{\omega }$.

We use vertex operator algebras and intertwining operators to study certain substructures of standard -modules, allowing us to conceptually obtain the classical Rogers–Ramanujan recursion. As a consequence we recover Feigin–Stoyanovsky's character formulas for the principal subspaces of the level 1 standard -modules.

Cross-connection is a construction of regular semigroups using certain categories called normal categories which are abstractions of the partially ordered sets of principal left (right) ideals of a semigroup. We describe the cross-connections in the semigroup $\mathrm{\text{Sing}}(X)$ of all non-invertible transformations on a set $X$. The categories involved are characterized as the powerset category $\mathcal{𝒫}(X)$ and the category of partitions $\mathrm{\Pi }(X)$. We describe these categories and show how a permutation on $X$ gives rise to a cross-connection. Further, we prove that every cross-connection between them is induced by a permutation and construct the regular semigroups that arise from the cross-connections. We show that each of the cross-connection semigroups arising this way is isomorphic to $\mathrm{\text{Sing}}(X)$. We also describe the right reductive subsemigroups of $\mathrm{\text{Sing}}(X)$ with the category of principal left ideals isomorphic to $\mathcal{𝒫}(X)$. This study sheds light into the more general theory of cross-connections and also provides an alternate way of studying the structure of $\mathrm{\text{Sing}}(X)$.

Consider the following social choice problem. A group of individuals seek to classify the elements of X as belonging in one of two sets. The individuals may disagree as to how the elements of X should be classified, and so an aggregation rule is applied to determine a compromise outcome. We require that the social classification should not be imposed, nor should it be manipulable. We prove that the only aggregation rules satisfying these properties are dictatorships.

A q-series identity in four parameters is established and interpreted as a statement about 7-colored overpartitions. As corollaries some overpartition theorems of the Rogers–Ramanujan type and some weighted overpartition theorems are exhibited. Among these are overpartition analogues of classical partition theorems of Schur and Göllnitz.

In this paper we study the general exponential sum related to multiplicative functions f(n) with |f(n)| ≤ 1, namely we study the sum

We give a simple, direct proof of a theorem involving partitions into distinct parts, where multiples of 7 come in two colours.

For a finite set A of positive integers, we study the partition function p_A(n). This function enumerates the partitions of the positive integer n into parts in A. We give simple proofs of some known and unknown identities and congruences for p_A(n). For n in a special residue class, p_A(n) is a polynomial in n. We examine these polynomials for linear factors, and the results are applied to a restricted m-ary partition function. We extend the domain of p_A and prove a reciprocity formula with supplement. In closing we consider an asymptotic formula for p_A(n) and its refinement.

Let S and T be sets of positive integers and let a be a fixed positive integer. An a-shifted partition identity has the form

Modular transformation of certain shifted identities leads to shiftless partition identities. Again let a be a fixed positive integer, and S, T be distinct sets of positive integers. A shiftless partition identity has the form

In this paper, we show, except in one case, how all known 1-shifted and shiftless identities follow from a four-parameter theta-function identity due to Jacobi. New shifted and shiftless partition identities are proved.

We shall develop further N. J. Fine's theory of three parameter non-homogeneous first order q-difference equations. The object of our work is to bring the Rogers–Ramanujan identities within the purview of such a theory. In addition, we provide a number of new identities.

Let b_ℓ(n) denote the number of ℓ-regular partitions of n, where ℓ is prime and 3 ≤ ℓ ≤ 23. In this paper we prove results on the distribution of b_ℓ(n) modulo m for any odd integer m > 1 with 3 ∤ m if ℓ ≠ 3.

The rank of an overpartition pair is a generalization of Dyson's rank of a partition. The purpose of this paper is to investigate the role that this statistic plays in the congruence properties of , the number of overpartition pairs of n. Some generating functions and identities involving this rank are also presented.

Let r_k(n) and t_k(n) denote the number of representations of an integer n as a sum of k squares, and as a sum of k triangular numbers, respectively. We prove that