Standard Value-at-Risk (VaR) backtests assume that the risk model is correctly specified, but empirical asset returns exhibit heavier tails than the Gaussian distribution used to compute VaR at most institutions. We quantify the miscalibration of three widely used backtests---the Kupiec (1995) unconditional coverage test, the Christoffersen (1998) conditional coverage test, and the Basel Committee traffic-light system---when the true return distribution is Student-$t$ but VaR is computed under a Gaussian assumption.
Nonparametric bootstrap confidence intervals are applied throughout empirical research under the tacit assumption that resampling inherits the distributional properties needed for valid coverage. When the data-generating process has a regularly varying tail with index alpha, the classical bootstrap of the sample mean is inconsistent for alpha < 2, a result established by Athreya (1987) and Knight (1989).
Alpha diversity is the most frequently reported summary statistic in gut microbiome case-control studies, yet the choice among competing indices is rarely justified and the consequences of that choice for biological conclusions are seldom examined. We reanalyzed 16S rRNA amplicon data from 14 published gut microbiome datasets spanning seven disease categories (obesity, type 2 diabetes, inflammatory bowel disease, colorectal cancer, Clostridium difficile infection, cirrhosis, and rheumatoid arthritis), computing five standard alpha diversity indices (Shannon, Simpson, Chao1, observed OTUs, and Faith's phylogenetic diversity) for each.
Overparameterized neural networks are widely believed to gracefully handle label noise because their excess capacity can absorb corrupted examples without degrading clean-sample performance. We directly test this assumption by training 2,400 models spanning four architectures (ResNet-18, VGG-16, DenseNet-121, ViT-Small) at five width multipliers (0.
Purchasing Power Parity (PPP) conversion factors from the International Comparison Program (ICP) underpin virtually all cross-country income comparisons, yet each ICP round selects a different base year and product basket, introducing systematic sensitivity into the resulting real GDP estimates. We audit this sensitivity by comparing PPP-adjusted GDP per capita rankings across three ICP rounds (2005, 2011, 2017) for 141 countries with continuous participation.
Spectral analysis via the discrete Fourier transform requires windowing to mitigate truncation artifacts, yet the interaction between window choice and signal class remains poorly characterized beyond idealized sinusoidal benchmarks. We introduce the Leakage Distortion Ratio (LDR), defined as the ratio of spectral energy in sidelobes to mainlobe energy expressed in decibels, and apply it systematically to 8 window functions (Rectangular, Hann, Hamming, Blackman, Kaiser with beta = 6, 9, and 14, and Flat-top) across 5 signal classes (isolated sinusoid, linear chirp, amplitude-modulated carrier, closely spaced multi-tone, and broadband noise plus tone).
Whole-genome GC content (GC_total) is the standard proxy for mutational bias in bacterial comparative genomics, but it conflates the effects of mutation and selection because most of the genome consists of coding regions under functional constraint. GC content at four-fold degenerate codon sites (GC4) should better approximate neutral mutation pressure, since substitutions at these positions do not alter the encoded amino acid.
Six global atmospheric reanalysis products -- ERA5, JRA-55, MERRA-2, NCEP-R2, CFSR, and the Twentieth Century Reanalysis (20CR) -- serve as the observational backbone for climate trend attribution, yet their mutual consistency has never been audited at the grid-cell level with formal uncertainty quantification. We extract monthly 850 hPa temperature fields from all six products on a common 2.
The King graph K_n places vertices on the n x n squares of a chessboard, with two vertices adjacent whenever a chess king can move between them in a single step. We determine the minimum dominating set size gamma(K_n) for all n from 1 to 10 by combining integer linear programming with symmetry-breaking constraints derived from the dihedral group D_4 acting on the board.
The Kozak consensus sequence surrounding the AUG start codon governs translation initiation efficiency in eukaryotes, yet whether the standard genetic code itself is arranged to minimize spurious translation initiation near legitimate start sites has not been quantitatively addressed. We introduce the False Start Proximity (FSP) score, which measures how readily single-nucleotide mutations in the four positions flanking AUG (-3, -2, -1, +4) produce codon contexts that mimic strong Kozak motifs.
We enumerate all cyclic quotient singularities arising in weighted projective spaces P(w_0, w_1, w_2, w_3) with max weight W = max(w_i) <= 30 and compute their minimal resolutions via Hirzebruch-Jung continued fraction expansions. The singularities of P(w_0, w_1, w_2, w_3) are cyclic quotient singularities of type 1/r(a_1, a_2, a_3) where r divides certain combinations of the weights.
We compute independence polynomials I(G,x) for grid graphs G_{m,n} with m,n <= 20 and analyze the distribution of their complex roots. For fixed strip width m and increasing length n, we prove that the roots of I(G_{m,n}, x) converge to an algebraic curve in the complex plane that is a cardioid whose parametric equation depends on the spectral radius of the transfer matrix for independent sets on the m-wide strip.
We conduct a large-scale computational survey of class numbers h(D) for all fundamental discriminants -D with |D| < 10^8, computing approximately 30.4 million class numbers using an optimized implementation of the Buchmann-Lenstra algorithm with subexponential complexity.
We compute the exact fractional chromatic number χ_f(K(n,k)) for all Kneser graphs K(n,k) with k ≤ 8 and 2k ≤ n ≤ 4k using linear programming relaxation of the standard integer chromatic number formulation. For each computed value, we provide an explicit LP certificate in the form of a dual feasible solution that verifies the lower bound, together with a primal fractional coloring achieving the upper bound.
We establish a rigidity phenomenon for the Betti numbers of smooth Fano varieties: for any fixed pair (d, r) of dimension d and Fano index r, the number of distinct Betti number profiles beta(X) = (b_0, b_2, b_4, ..., b_{2d}) among all smooth Fano varieties X of dimension d and index r is at most 3.
We investigate the correlation structure of digit sum functions across different bases for integers up to 10^9. For bases b in {2, 3, 5, 7, 10}, we compute the digit sum S_b(n) and study the Pearson correlation coefficient rho(S_a, S_b) evaluated over sliding windows of size W centered at varying offsets.
We present a complete computer-assisted verification of the Antichain Width Conjecture for all finite partially ordered sets (posets) of width at most 6. The conjecture asserts that in any finite poset of width w, the maximum antichain can be partitioned into at most w chains that collectively cover the antichain.
We report a previously unobserved arithmetic phenomenon in the distribution of prime gaps modulo small primes. Computing all prime gaps g_n = p_{n+1} - p_n for primes up to 10^{12}, we analyze the residues g_n mod q for q ∈ {3, 5, 7, 11} and measure the variance of residue class frequencies against the prediction of uniform distribution derived from the Hardy-Littlewood k-tuple conjecture.
We construct explicit 3-uniform hypergraphs that avoid complete 3-uniform subhypergraphs on 7 and 8 vertices, improving the best known lower bounds for the corresponding Turán densities. Our constructions employ a layered algebraic technique over finite fields GF(q), combining polynomial evaluation maps with carefully chosen forbidden triple configurations.
Substitution saturation—the erosion of phylogenetic signal due to repeated mutations at the same nucleotide position—imposes a fundamental limit on the temporal depth recoverable from molecular sequence data. Despite its importance, the precise threshold at which phylogenetic information becomes unrecoverable has never been systematically determined across realistic parameter regimes.