Coefficients of Sylvester's Denumerant

For a given sequence $\mathbf{\alpha} = [\alpha_1,\alpha_2,\dots,\alpha_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(\mathbf{\alpha})(t)$ that counts the nonnegative integer solutions of the equation $\alpha_1x_1+\alpha_2 x_2+\cdots+\alpha_{N} x_{N}+\alpha_{N+1}x_{N+1}=t$, where the right-hand side $t$ is a varying nonnegative integer. It is well-known that $E(\mathbf{\alpha})(t)$ is a quasi-polynomial function in the variable $t$ of degree $N$. In combinatorial number theory this function is known as Sylvester's denumerant. Our main result is a new algorithm that, for every fixed number $k$, computes in polynomial time the highest $k+1$ coefficients of the quasi-polynomial $E(\mathbf{\alpha})(t)$ as step polynomials of $t$ (a simpler and more explicit representation). Our algorithm is a consequence of a nice poset structure on the poles of the associated rational generating function for $E(\mathbf{\alpha})(t)$ and the geometric reinterpretation of some rational generating functions in terms of lattice points in polyhedral cones. Our algorithm also uses Barvinok's fundamental fast decomposition of a polyhedral cone into unimodular cones. This paper also presents a simple algorithm to predict the first non-constant coefficient and concludes with a report of several computational experiments using an implementation of our algorithm in LattE integrale. We compare it with various Maple programs for partial or full computation of the denumerant.Comment: minor revision, 28 page

Top Coefficients of the Denumerant

International audienceFor a given sequence $\alpha = [\alpha_1,\alpha_2,\ldots , \alpha_N, \alpha_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(\alpha)(t)$ that counts the nonnegative integer solutions of the equation $\alpha_1x_1+\alpha_2 x_2+ \ldots+ \alpha_Nx_N+ \alpha_{N+1}x_{N+1}=t$, where the right-hand side $t$ is a varying nonnegative integer. It is well-known that $E(\alpha)(t)$ is a quasipolynomial function of $t$ of degree $N$. In combinatorial number theory this function is known as the $\textit{denumerant}$. Our main result is a new algorithm that, for every fixed number $k$, computes in polynomial time the highest $k+1$ coefficients of the quasi-polynomial $E(\alpha)(t)$ as step polynomials of $t$. Our algorithm is a consequence of a nice poset structure on the poles of the associated rational generating function for $E(\alpha)(t)$ and the geometric reinterpretation of some rational generating functions in terms of lattice points in polyhedral cones. Experiments using a $\texttt{MAPLE}$ implementation will be posted separately.Considérons une liste $\alpha = [\alpha_1,\alpha_2,\ldots , \alpha_N, \alpha_{N+1}]$ de $N+1$ entiers positifs. Le dénumérant $E(\alpha)(t)$ est lafonction qui compte le nombre de solutions en entiers positifs ou nuls de l’équation $\sum^{N+1}_{i=1}x_i\alpha_i=t$, où $t$ varie dans les entiers positifs ou nuls. Il est bien connu que cette fonction est une fonction quasi-polynomiale de $t$, de degré $N$. Nous donnons un nouvel algorithme qui calcule, pour chaque entier fixé $k$ (mais $N$ n’est pas fixé, les $k+1$ plus hauts coefficients du quasi-polynôme $E(\alpha)(t)$ en termes de fonctions en dents de scie. Notre algorithme utilise la structure d’ensemble partiellement ordonné des pôles de la fonction génératrice de $E(\alpha)(t)$. Les $k+1$ plus hauts coefficients se calculent à l’aide de fonctions génératrices de points entiers dans des cônes polyèdraux de dimension inférieure ou égale à $k$

Caracterização físico-química e desenvolvimento pós-colheita de jabuticabas (Plinia peruviana e P. cauliflora)

Plinia peruviana e P. cauliflora pertencem à família Myrtaceae, apresentam potencial para exploração como frutífera e/ou para arborização urbana. Contudo, pouco se conhece sobre suas características, sendo o objetivo deste trabalho avaliar morfometria (dimensão, massa fresca de frutos, polpa, casca e sementes, rendimento de polpa, número de sementes por fruto e cor dos frutos), constituição química (sólidos solúveis SS, acidez titulável AT, Ratio e teor vitamina C) e desenvolvimento pós-colheita (constituição química e perda de massa fresca) dos frutos destas espécies. As amostras estudadas são arredondadas, com alto teor de água (83 %), alto rendimento de polpa (entre 67 e 76 %), massa fresca entre 5,30 e 6,82 g e mais de uma semente por fruto. Apresentaram entre 11,4 e 12,7 SS, baixa AT e 19 mg 100 g de polpa-1 de vitamina C. Plinia peruviana e P. cauliflora apresentam características físico-químicas similares, podendo ser armazenadas por 28 dias sob-refrigeração (≈2ºC).Characterization and post-harvest behavior of jabuticabas Plinia peruviana and P. cauliflora. Plinia peruviana and P. cauliflora are Myrtaceae with potential for agricultural exploitation or urban afforestation. However, there is not much information about its characteristics; consequently, making the objective of this work to evaluate morphometry (size, fresh fruit mass, pulp, bark and seeds, pulp yield, number of seeds per fruit and fruit color), chemical composition (soluble solids, acid titratable AT, Ratio and vitamin C content) and post-harvest behavior (chemical composition and loss of fresh mass) of its fruits. The samples studied were rounded, with high water content (83%), high yield of pulp (between 67 and 76%), fresh mass between 5.30 and 6.82 g and more than one seed per fruit. They presented between 11.4 and 12.7 SS, low AT and 19 mg 100 g of pulp-1 of vitamin C. Plinia peruviana and P. cauliflora have shown similar physicochemical characteristics and can be stored for 28 days under refrigeration (≈2ºC)

Urinary endogenous peptides as biomarkers for prostate cancer

Prostate cancer (PCa) is one of the most prevalent types of cancer in men worldwide; however, the main diagnostic tests available for PCa have limitations and a biopsy is required for histopathological confirmation of the disease. Prostate specific antigen (PSA) is the main biomarker used for the early detection of PCa, but an elevated serum concentration is not cancer specific. Therefore, there is a need for the discovery of new non invasive biomarkers that can accurately diagnose PCa. The present study used trichloroacetic acid induced protein precipitation and liquid chromatography mass spectrometry to profile endogenous peptides in urine samples from patients with PCa (n=33), benign prostatic hyperplasia (n=25) and healthy individuals (n=28). Receiver operating characteristic curve analysis was performed to evaluate the diagnostic performance of urinary peptides. In addition, Proteasix tool was used for in silico prediction of protease cleavage sites. Five urinary peptides derived from uromodulin were revealed to be significantly altered between the study groups, all of which were less abundant in the PCa group. This peptide panel showed a high potential to discriminate between the study groups, resulting in area under the curve (AUC) values between 0.788 and 0.951. In addition, urinary peptides outperformed PSA in discriminating between malignant and benign prostate conditions (AUC=0.847), showing high sensitivity (81.82%) and specificity (88%). From in silico analyses, the proteases HTRA2, KLK3, KLK4, KLK14 and MMP25 were identified as potentially involved in the degradation of uromodulin peptides in the urine of patients with PCa. In conclusion, the present study allowed the identification of urinary peptides with potential for use as non invasive biomarkers in PCa diagnosis