Related papers: The Delta Conjecture at $q=1$
We generalize previous definitions of Tesler matrices to allow negative matrix entries and negative hook sums. Our main result is an algebraic interpretation of a certain weighted sum over these matrices, which we call the Tesler function.…
In his paper \cite{MR1}, Markus Reineke proposed a conjecture that there exists a stable weight system $\Theta$ for every indecomposable representation of Dynkin type quiver. In this paper, we showed this conjecture is true for quivers of…
In this paper, we establish a $q$-integral formula by using the orthogonality relation, and also provide a new proof of the $q$-orthogonality relation for the continuous $q$-ultraspherical polynomials. A new $q$-beta integral with five…
We consider the elliptic system $$-\Delta u_i = u_i^3+\sum\limits_{j=1\atop j\not=i}^{q+1}{ \beta_{ij}}u_i u_j^2\ \hbox{in}\ \mathbb R^4, \ i=1,\dots,q+1.$$ when $\alpha:=\beta_{ij}$ and $\beta:=\beta_{i(q+1)}=\beta_{(q+1)j}$ for any…
This paper was motivated by a conjecture of Br\"{a}nd\'{e}n (European J. Combin. \textbf{29} (2008), no.~2, 514--531) about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the…
In two previous papers the author introduced a multiplication of distributions in one dimension and he proved that two one-dimensional Dirac delta functions and their derivatives can be multiplied, at least under certain conditions. Here,…
For a family of weight functions, $h_\kappa$, invariant under a finite reflection group on $\RR^d$, analysis related to the Dunkl transform is carried out for the weighted $L^p$ spaces. Making use of the generalized translation operator and…
We present a recent analysis of epsilon'/epsilon in the 1/N_c expansion. We show that the 1/N_c corrections to the matrix element of Q_6 are large and positive, indicating a Delta I = 1/2 enhancement similar to the one of Q_1 and Q_2 which…
We study the action of the derived Hecke algebra on the space of weight one forms. By analogy with the topological case, we formulate a conjecture relating this to a certain Stark unit. We verify the truth of the conjecture numerically, for…
We prove a conjecture that arose in the context of a subspace enumeration problem over finite fields. We prove, more generally, a bibasic, double-sum identity, which extends a $q$-analogue of the (terminating) binomial theorem.
We improve unconditional estimates on $\Delta_k(x)$, the remainder term of the generalised divisor function, for large $k$. In particular, we show that $\Delta_k(x) \ll x^{1 - 1.889k^{-2/3}}$ for all sufficiently large fixed $k$.
In this paper, we introduce a simple Bessel $\delta$-method to the theory of exponential sums for $\rm GL_2$. Some results of Jutila on exponential sums are generalized in a less technical manner to holomorphic newforms of arbitrary level…
The well-known Dixmier conjecture asks if every algebra endomorphism of the first Weyl algebra over a characteristic zero field is an automorphism. We bring a hopefully easier to solve conjecture, called the $\gamma,\delta$ conjecture, and…
The Stanley-Stembridge conjecture asserts that the chromatic symmetric function of a $(3+1)$-free graph is $e$-positive. Recently, Hikita proved this conjecture by giving an explicit $e$-expansion of the Shareshian-Wachs $q$-chromatic…
The Jacobian Conjecture has been reduced to the symmetric homogeneous case. In this paper we give an inversion formula for the symmetric case and relate it to a combinatoric structure called the Grossman-Larson Algebra. We use these tools…
We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence \{\lambda_k\}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we…
We study a certain class of arithmetic functions that appeared in Klurman's classification of $\pm 1$ multiplicative functions with bounded partial sums, c.f., Comp. Math. 153 (8), 2017, pp. 1622-1657. These functions are periodic and…
A triangle group is denoted by $\Delta(p,q,r)$ and has finite presentation $$ \Delta(p,q,r)=\langle x,y | x^p=y^q=(xy)^r=1 \rangle .$$ We examine a method for composition of permutation representations of a triangle group $\Delta(p,q,r)$…
We use the theory of symmetric functions to enumerate various classes of alternating permutations w of {1,2,...,n}. These classes include the following: (1) both w and w^{-1} are alternating, (2) w has certain special shapes, such as…
We prove the consistency of the existence of a $Q$-set whose square is not a $\Delta$-set and that if there is a $\Delta$-set, then there exists a $\Delta$-set whose all finite powers are $\Delta$-sets.