Multiplicative Congruences with Variables from Short Intervals

Recently, several bounds have been obtained on the number of solutions to congruences of the type $$(x_1+s)...(x_{\nu}+s)\equiv (y_1+s)...(y_{\nu}+s)\not\equiv0 \pmod p$$ modulo a prime $p$ with variables from some short intervals. Here, for almost all $p$ and all $s$ and also for a fixed $p$ and almost all $s$, we derive stronger bounds. We also use similar ideas to show that for almost all primes, one can always find an element of a large order in any rather short interval

Stravinsky Fest, September 21, 1988

This is the concert program of the Stravinsky Fest performance on Wednesday, September 21, 1988 at 8:30 p.m., at the Boston University Concert Hall, 855 Commonwealth Avenue, Boston, Massachusetts. Works performed were eighth note equals 160 from Three Pieces for Clarinet Solo, Duet for Two Bassoons, Epitaphium für das Grabmal des Prinzen Max Egon zu Fürstenberg, and Pastorale by Igor Stravinsky; Canon for 3 in Memoriam of Igor Stravinsky by Elliott Carter; and Octet for Wind Instruments and The Soldier's Tale by Igor Stravinsky. Digitization for Boston University Concert Programs was supported by the Boston University Humanities Library Endowed Fund

Products in Residue Classes

We consider a problem of P. Erdos, A. M. Odlyzko and A. Sarkozy about the representation of residue classes modulo m by products of two not too large primes. While it seems that even the Extended Riemann Hypothesis is not powerful enough to achieve the expected results, here we obtain some unconditional results ``on average'' over moduli m and residue classes modulo m and somewhat stronger results when the average is restricted to prime moduli m = p. We also consider the analogous question wherein the primes are replaced by easier sequences so, quite naturally, we obtain much stronger results.Comment: 18 page

On Congruences with Products of Variables from Short Intervals and Applications

We obtain upper bounds on the number of solutions to congruences of the type $$(x_1+s)...(x_{\nu}+s)\equiv (y_1+s)...(y_{\nu}+s)\not\equiv0 \pmod p$$ modulo a prime $p$ with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M. Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. B. Friedlander and H. Iwaniec and some results of M.-C. Chang and A. A. Karatsuba on character sums twisted with the divisor function

Institutional Underpinnings of the Minimum Wage Fixing Machinery: The Role of Social Dialogue

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Opera Scenes, February 25, 1998

This is the concert program of the Opera Scenes performance on Wednesday, February 25, 1998 at 4:00 p.m., at Room 171, 855 Commonwealth Avenue. Scenes performed were from The Rake's Progress by Igor Stravinsky and The Secret Garden by Lucy Simon. Digitization for Boston University Concert Programs was supported by the Boston University Humanities Library Endowed Fund

Boston University Symphony Orchestra, March 4, 2004

This is the concert program of the Boston University Symphony Ochestra performance on Thursday, March 4, 2004 at 8:00 p.m., at the Tsai Performance Center, 685 Commonwealth Avenue, Boston, Massachusetts. Works performed were Ma Mère l'oye by Maurice Ravel and Pétrouchka by Igor Stravinsky. Digitization for Boston University Concert Programs was supported by the Boston University Center for the Humanities Library Endowed Fund