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Register a new userA modified Euler-Maclaurin formula in 1D and 2D with applications in statistical physics
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The Euler-Maclaurin summation formula is generalized to a modified form by expanding the periodic Bernoulli polynomials as its Fourier series and taking cuts, which includes both the Euler-Maclaurin summation formula and the Poission summation formula as special cases. By making use of the modified formula, a numerical summation method is obtained and the error can be controlled. The modified formula is also generalized from one dimention to two dimentions. Examples of its applications in statistical physics are also discussed.
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We give a local Euler-Maclaurin formula for rational convex polytopes in a rational euclidean space . For every affine rational polyhedral cone C in a rational euclidean space W, we construct a differential operator of infinite order D(C) on W with constant rational coefficients, which is unchanged when C is translated by an integral vector. Then for every convex rational polytope P in a rational euclidean space V and every polynomial function f (x) on V, the sum of the values of f(x) at the integral points of P is equal to the sum, for all faces F of P, of the integral over F of the function D(N(F)).f, where we denote by N(F) the normal cone to P along F.
We discuss certain ternary algebraic structures appearing more or less naturally in various domains of theoretical and mathematical physics. Far from being exhaustive, this article is intended above all to draw attention to these algebras, which may find more interesting applications in the years to come.
Symmetry lies at the heart of todays theoretical study of particle physics. Our manuscript is a tutorial introducing foundational mathematics for understanding physical symmetries. We start from basic group theory and representation theory. We then introduce Lie Groups and Lie Algebra and their properties. We next discuss with detail two important Lie Groups in physics Special Unitary and Lorentz Group, with an emphasis on their applications to particle physics. Finally, we introduce field theory and its version of the Noether Theorem. We believe that the materials cover here will prepare undergraduates for future studies in mathematical physics.
In this work we study the tau-function $Z^{1D}$ of the KP hierarchy specified by the topological 1D gravity. As an application, we present two types of algorithms to compute the orbifold Euler characteristics of $overline{mathcal M}_{g,n}$. The first is to use (fat or thin) topological recursion formulas emerging from the Virasoro constraints for $Z^{1D}$; and the second is to use a formula for the connected $n$-point functions of a KP tau-function in terms of its affine coordinates on the Sato Grassmannian. This is a sequel to an earlier work.
We get a generalization of Kreins formula -which relates the resolvents of different selfadjoint extensions of a differential operator with regular coefficients- to the non-regular case $A=-partial_x^2+( u^2-1/4)/x^2+V(x)$, where $0< u<1$ and $V(x)$ is an analytic function of $xinmathbb{R}^+$ bounded from below. We show that the trace of the heat-kernel $e^{-tA}$ admits a non-standard small-t asymptotic expansion which contains, in general, integer powers of $t^ u$. In particular, these powers are present for those selfadjoint extensions of $A$ which are characterized by boundary conditions that break the local formal scale invariance at the singularity.
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