From the uniform convergence of Vn towards V it follows by Theorem A.5 1. that U νnm (s) = Z log |t − s|−1 dνnm (t) converges uniformly (on C) towards U ν (s) := R log |t − s|−1 dν (t).
Particle Systems with Repulsion Exponent \beta\ and Random Matrices
Furthermore, deﬁne for µ ∈ M1 (Σ) the energy functional IQ (µ) := Z Q(t)dµ(t) + Z Z log |s − t|−1 dµ(s)dµ(t).
Particle Systems with Repulsion Exponent \beta\ and Random Matrices
For α, β ∈ C and δ ∈ Z/2Z, the function (sgn f )δ |f |α(log |f |)β is smooth on the complement of S .
Distributions and Analytic Continuation of Dirichlet Series
Let Ssis (R) denote the linear span of all products (sgn x)η (log |x|)k |x|β f (x), with β ∈ C, η ∈ Z/2Z, k ≥ 0, and f ∈ S (R).
Automorphic Distributions, L-functions, and Voronoi Summation for GL(3)
In particular, one can check very explicitly for the Gaussian model that these subleading contributions in Sef f (s∗ ) cancel the ˆN log ˆN and the ˆN 2 log t pieces from the determinant. A remaining ˆN -dependent (but ν - and t-independent) constant could have been absorbed in the overall normalisation of Z .
Supersymmetric Gauge Theories from String Theory
Connection critical points of s are the same as ordinary critical points of log |s(Z )|h .
Counting String Vacua
Using classical tools, one can show that p (C, t) ⇒ ∃b > 0 s.t. Z eθp (ax) dµ(x) < +∞. ∃C, t > 0 s.t. µ satisﬁes LS Iθ∗ Consequently, a Log-concave measure µ satisﬁes the inequality LS Iθ∗ p (C, t) if and only if there is some b > 0 such that R eθp (ax) dµ(x) < +∞.
Characterization of Talagrand's Like Transportation-Cost Inequalities on the Real Line
For each i, in Lemma 3.5 we replace most of the X ’s with Xi , U with U/√2i , Z with Z/√2i ; the X ’s we don’t replace are the cardinality of the family (which we divide by in the end) and the log X which occurs when we evaluate the test function bg at log p/ log X .
A Symplectic Test of the L-Functions Ratios Conjecture
It is worth noting that the calibration of the S-L relation based only on the oETGs data is: log(Re ) = −0.28MR (z = 0) − 5.6 (dashed (red) line in Fig. 8), a relation with the same slop of the one at z ∼ 0 but with an offset of ∼ 0.5 in the zero point corresponding to a factor ∼ 3 in Re .
The population of early-type galaxies at 1
T /V )· log Z , where mud is the mass of the light u,d quarks, N s is the spatial extension, Nτ euclidean time extension, and V the system volume.
QCD Phase Diagram: Phase Transition, Critical Point and Fluctuations
Consider noisy ABC, where sobs = S (yobs ) + hx, where x is drawn from K (·), then the expected noisy-ABC log-likelihood, E (cid:8)log[p(θ|Sobs )](cid:9) = Z Z log[p(θ|S(y) + hx)]π(y|θ0)K (x)dydx, has its maximum at θ = θ0 .
Constructing Summary Statistics for Approximate Bayesian Computation: Semi-automatic ABC
We see that residual NP effects in Ok = log (W11) are expected to be tiny; the resulting subtraction in fact produces a shift of only 0.0001 in αs(M 2 Z ) .
Status of the Lattice and Tau Decay Determinations of alpha_s
Figure 8 displays (on a log scale) estimates of the relevant knotting probabilities (along with those for L− and the unsigned S and Z ) versus polygon length n1 .
Knotting probabilities after a local strand passage in unknotted self-avoiding polygons
We shall indicate the dependence of various objects on m by the subscript m , e.g., Pmt denotes the FCS of Qm and emt(s) its Rényi entropic functional. emt(1 − s) = log Z e−tsφdPmt (φ) is the cumulant generating function of Pmt and e−tφdPmt (φ) = dµωmt |ωm (−tφ).
Quantum Hypothesis Testing and Non-Equilibrium Statistical Mechanics
This ﬂuctuation relation has the following equivalent ref ormulations: et (s) = log Z e−tsφdPt (φ). dPt (−φ) = e−tφdPt (φ).
Quantum Hypothesis Testing and Non-Equilibrium Statistical Mechanics
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