scholarly journals Concentration for Random Product Formulas

PRX Quantum ◽  
2021 ◽  
Vol 2 (4) ◽  
Author(s):  
Chi-Fang Chen ◽  
Hsin-Yuan Huang ◽  
Richard Kueng ◽  
Joel A. Tropp
Keyword(s):  
Random Product ◽  
Product Formulas

scholarly journals Tails of Stopped Random Products: The Factoid and Some Relatives

2008 ◽  
Vol 45 (04) ◽  
pp. 1161-1180
Author(s):  
Anthony G. Pakes
Keyword(s):  
Tail Behaviour ◽  
Random Product ◽  
Random Products ◽  
Heavy Tailed ◽  
First Time ◽  
Independent And Identically Distributed

The upper tail behaviour is explored for a stopped random product ∏j=1NXj, where the factors are positive and independent and identically distributed, andNis the first time one of the factors occupies a subset of the positive reals. This structure is motivated by a heavy-tailed analogue of the factorialn!, called the factoid ofn. Properties of the factoid suggested by computer explorations are shown to be valid. Two topics about the determination of the Zipf exponent in the rank-size law for city sizes are discussed.

scholarly journals Random product of quasi-periodic cocycles

2020 ◽  
pp. 1
Author(s):  
Jamerson Bezerra ◽  
Mauricio Poletti
Keyword(s):  
Random Product

scholarly journals Augmented Rook Boards and General Product Formulas

10.37236/809 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Brian K. Miceli ◽  
Jeffrey Remmel
Keyword(s):  
Product Formula ◽  
Combinatorial Proof ◽  
Rook Theory ◽  
Special Cases ◽  
Ferrers Board ◽  
General Product ◽  
Product Formulas

There are a number of so-called factorization theorems for rook polynomials that have appeared in the literature. For example, Goldman, Joichi and White showed that for any Ferrers board $B = F(b_1, b_2, \ldots, b_n)$, $$\prod_{i=1}^n (x+b_i-(i-1)) = \sum_{k=0}^n r_k(B) (x)\downarrow_{n-k}$$ where $r_k(B)$ is the $k$-th rook number of $B$ and $(x)\downarrow_k = x(x-1) \cdots (x-(k-1))$ is the usual falling factorial polynomial. Similar formulas where $r_k(B)$ is replaced by some appropriate generalization of the $k$-th rook number and $(x)\downarrow_k$ is replaced by polynomials like $(x)\uparrow_{k,j} = x(x+j) \cdots (x+j(k-1))$ or $(x)\downarrow_{k,j} = x(x-j) \cdots (x-j(k-1))$ can be found in the work of Goldman and Haglund, Remmel and Wachs, Haglund and Remmel, and Briggs and Remmel. We shall refer to such formulas as product formulas. The main goal of this paper is to develop a new rook theory setting in which we can give a uniform combinatorial proof of a general product formula that includes, as special cases, essentially all the product formulas referred to above. We shall also prove $q$-analogues and $(p,q)$-analogues of our general product formula.

scholarly journals Lozenge Tiling Function Ratios for Hexagons with Dents on Two Sides

10.37236/9363 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Daniel Condon
Keyword(s):  
Triangular Lattice ◽  
Lozenge Tilings ◽  
Simple Product ◽  
Two Sides ◽  
Product Formulas

We give a formula for the number of lozenge tilings of a hexagon on the triangular lattice with unit triangles removed from arbitrary positions along two non-adjacent, non-opposite sides. Our formula implies that for certain families of such regions, the ratios of their numbers of tilings are given by simple product formulas.

scholarly journals Evaluating the Numbers of some Skew Standard Young Tableaux of Truncated Shapes

10.37236/3890 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Ping Sun
Keyword(s):  
Young Tableaux ◽  
Multiple Integrals ◽  
Standard Young Tableaux ◽  
Product Formulas

In this paper the number of standard Young tableaux (SYT) is evaluated by the methods of multiple integrals and combinatorial summations. We obtain the product formulas of the numbers of skew SYT of certain truncated shapes, including the skew SYT $((n+k)^{r+1},n^{m-1}) / (n-1)^r $ truncated by a rectangle or nearly a rectangle, the skew SYT of truncated shape $((n+1)^3,n^{m-2}) / (n-2) \backslash \; (2^2)$, and the SYT of truncated shape $((n+1)^2,n^{m-2}) \backslash \; (2)$.

scholarly journals Burgess-like subconvexity for

2019 ◽  
Vol 155 (8) ◽  
pp. 1457-1499 ◽  
Author(s):  
Han Wu
Keyword(s):  
Eisenstein Series ◽  
Number Field ◽  
Main Tool ◽  
Previous Method ◽  
Triple Product ◽  
Extended Theory ◽  
Product Formulas ◽  
Cuspidal Representations

We generalize our previous method on the subconvexity problem for $\text{GL}_{2}\times \text{GL}_{1}$ with cuspidal representations to Eisenstein series, and deduce a Burgess-like subconvex bound for Hecke characters, that is, the bound $|L(1/2,\unicode[STIX]{x1D712})|\ll _{\mathbf{F},\unicode[STIX]{x1D716}}\mathbf{C}(\unicode[STIX]{x1D712})^{1/4-(1-2\unicode[STIX]{x1D703})/16+\unicode[STIX]{x1D716}}$ for varying Hecke characters $\unicode[STIX]{x1D712}$ over a number field $\mathbf{F}$ with analytic conductor $\mathbf{C}(\unicode[STIX]{x1D712})$ . As a main tool, we apply the extended theory of regularized integrals due to Zagier developed in a previous paper to obtain the relevant triple product formulas of Eisenstein series.

Thermal Sensitivity of Neosartorya fischeri Ascospores in Regular and Reduced-Sugar Grape Jelly

1997 ◽  
Vol 60 (12) ◽  
pp. 1577-1579 ◽  
Author(s):  
LARRY R. BEUCHAT ◽  
GERALD D. KUHN
Keyword(s):  
Thermal Processing ◽  
Thermal Inactivation ◽  
Thermal Sensitivity ◽  
Neosartorya Fischeri ◽  
Concord Grape ◽  
Grape Jelly ◽  
Made In ◽  
Product Formulas

A study was conducted to determine the thermal sensitivity of Neosartorya fischeri ascospores in regular (standard) and reduced-sugar formulas of Concord grape jelly. Ascospores were inactivated more rapidly by heating in the reduced-sugar formula than in the regular formula. Protection against inactivation in the regular formula may be due to its having a higher pH (3.27) than the pH (3.14) of the reduced-sugar formula. The lower aw (0.78) of the regular jelly formula compared to that of the reduced-sugar formula (aw, 0.82) would also protect against thermal inactivation. These results demonstrate the need to consider the efficacy of thermal processing schemes in achieving desired levels of sterility when changes are made in product formulas.

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