Professor of Mathematics, LaGuardia Community College, CUNY · PhD, Columbia University · author of Physics with MAPLE (Wiley-VCH) · associate editor of Numeracy · recipient of the CUNY Chancellor's Award for Undergraduate Mathematics Instruction. Research in differential geometry, dynamical systems, and the mathematics of physics pedagogy. A long-time friend of V5AI Studio, and one of the quiet contributors whose care for computer algebra and quantitative reasoning makes the ground that systems like ours can stand on.
Papers
2026 · April · NEW
Comment on "Specific heat of an ideal Bose gas above the Bose condensation temperature"
submitted · American Journal of Physics
A close reading of Einstein's 1925 derivation of the specific heat of an unsaturated ideal Bose gas — guiding the reader through the Taylor expansion Einstein sketched (with Jakob Grommer's help), correcting several numerical errors that propagated into later teaching literature, and comparing Einstein's T → ∞ expansion with a T ≈ Tc expansion from Wang's own 2004 paper. Closes with a historical arc from Kapitza and London (1938) through Flexner's "Usefulness of Useless Knowledge" (1939) to Stone's Einstein and the Quantum (2013).
pedagogic essay · V5AI Studio · interactive edition
A narrative walk through the acoustic oscillations imprinted on the cosmic microwave background — how primordial plasma rang like a bell, why the first peak's angular scale pinned the geometry of the universe to flatness, and what the higher peaks still tell us. Built for students who have met calculus but not cosmology.
Open links to the primary sources cited in the Einstein 1925 Comment. Archival scans and open-access papers where available.
F. Wang, "Specific heat of an ideal Bose gas above the Bose condensation temperature,"Am. J. Phys. 72(9), 1193–1194 (2004)
A. Einstein, The Essential Einstein, edited by D. K. Buchwald and T. Sauer (Princeton University Press, 2025)
A. Einstein, "Quantentheorie des einatomigen idealen Gases. Zweite Abhandlung,"Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin, Physikalisch-Mathematische Klasse, 3–14 (1925) · open archival PDF (Münster)
S. N. Bose (translated by O. Theimer and B. Ram), "Planck's law and the light quantum hypothesis,"Am. J. Phys. 44(11), 1056–1057 (1976)
F. London, "The λ-phenomenon of liquid helium and the Bose-Einstein degeneracy,"Nature 141, 643–644 (1938)
A. Flexner, "The Usefulness of Useless Knowledge,"Harper's Magazine 179, 544–555 (1939) · open PDF (IAS)
F. London, "On the Bose-Einstein Condensation,"Phys. Rev. 54, 947–954 (1938)
F. London, Superfluids (Wiley, 1954)
A. D. Stone, Einstein and the Quantum (Princeton University Press, 2013)