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Let $\mathcal{C}\subseteq \mathbb{N}^p$ be an integer cone. A $\mathcal{C}$-semigroup $S\subseteq \mathcal{C}$ is an affine semigroup such that the set $\mathcal{C}\setminus S$ is finite. Such $\mathcal{C}$-semigroups are central to our study. We develop new algorithms for computing $\mathcal{C}$-semigroups with specified invariants, including genus, Frobenius element, and their combinations, among other invariants. To achieve this, we introduce a new class of $\mathcal{C}$-semigroups, termed $\mathcal{B}$-semigroups. By fixing the degree lexicographic order, we also research the embedding dimension for both ordinary and mult-embedded $\mathbb{N}^2$-semigroups. These results are applied to test some generalizations of Wilf's conjecture.

The main aim of this work is to introduce and justify the study of semi-covarities. A {\it semi-covariety} is a non-empty family $\mathcal{F}$ of numerical semigroups such that it is closed under finite intersections, has a minimum, $\min(\mathcal{F}),$ and if $S\in \mathcal{F}$ being $S\neq \min(\mathcal{F})$, then there is $x\in S$ such that $S\backslash \{x\}\in \mathcal{F}$. As examples, we will study the semi-covariety formed by all the numerical semigroups containing a fixed numerical semigroup, and the semi-covariety composed by all the numerical semigroups of coated odd elements and fixed Frobenius number., Comment: 12 pages. arXiv admin note: text overlap with arXiv:2302.09121

A numerical semigroup $S$ is coated with odd elements (Coe-semigroup), if $\left\{x-1, x+1\right\}\subseteq S$ for all odd element $x$ in $S$. In this note, we will study this kind of numerical semigroups. In particular, we are interested in the study of the Frobenius number, gender and embedding dimension of a numerical semigroup of this type.

In a recent work, assuming a Beer-Lambert optical absorption and a Gaussian laser time profile, the exact solutions for a 1D-photoacoustic(PA)-boundary value problem predict a null pressure for optically strong absorbent materials. To overcome this, a heuristic correction was introduced by assuming that heat flux travels a characteristic length during the duration of the laser pulse\cite{Ruiz-Veloz2021} $\tau_p$. In this work, we obtained exact solutions in the frequency domain for a 1D-boundary-value-problem for the Dual-Phase-Lag (DPL) heat equation coupled with a 1D PA-boundary-value-problem via the wave-equation. Temperature and pressure solutions were studied by assuming that the sample and its surroundings have a similar characteristic thermal lag response time $\tau_{_T}$, which was assumed to be a free parameter that can be adjusted to reproduce experimental results. Solutions for temperature and pressure were obtained for a three-layer 1D system. It was found that for $\tau_{_T}< \tau_{p}$, the DPL temperature has a similar thermal profile of the Fourier heat equation, however, when $\tau_{_T}\ge \tau_{p}$ this profile is very different from the Fourier case. Additionally, via a numerical Fourier transform the wave-like behavior of DPL temperature is explored, and it was found that as $\tau_{_T}$ increases, thermal wave amplitude is increasingly attenuated. Exact solutions for pressure were compared with experimental signals, showing a close resemblance between both data sets, particularly in the time domain, for an appropriated value of $\tau_{_T}$; the transference function was also calculated, which allowed us to find the maximum response in frequency for the considered experimental setup., Comment: 32 pages, 11 figures

In this work, we will explore the effects of non-commutativity in fractional classical and quantum schemes using the anisotropicc Bianchi Type I cosmological model coupled to a scalar field in the K-essence formalism. We introduce non-commutative variables considering that all minisuperspace variables $q^i_{nc}$ do not commute, so the symplectic structure was modified, resulting in some changes with respect to the traditional formalism. In the quantum regime, the probability density presents a new structure in the scalar field corresponding to the value of the non-commutative parameter., Comment: 28 pages, 23 figures

In this work we shall explore the effects of non commutativity in fractional classical and quantum schemes using the flat Friedmmann-Robertson-Walker (FRW) cosmological model coupled to a scalar field in the K-essence formalism. In previous work we have obtained the commutative solutions in both regimes into the fractional framework. Here we introduce noncommutative variables, considering that all minisuperspace variables $\rm q^i_{nc}$ do not commute, so the symplectic structure was modified. In the quantum regime, the probability density presents new structure in the scalar field corresponding to the value of the non-commutative parameter, in the sense that this probability density undergoes a shift back to the direction of the scale factor, causing classical evolution to arise earlier than in the commutative world., Comment: Latex, 22 pages, 24 figures

In the particular configuration of the scalar field K-essence in the Wheeler-DeWitt quantum equation, for some age in the Bianchi type I anisotropic cosmological model, a fractional differential equation for the scalar field arises naturally. The order of the fractional differential equation is $\beta=\frac{2\alpha}{2\alpha - 1}$. This fractional equation belongs to different intervals, depending on the value of the barotropic parameter; when $\omega_{X} \in [0,1]$, the order belongs to the interval $1\leq \beta \leq 2$, and when $\omega_{X}\in[-1,0)$, the order belongs to the interval $0< \beta \leq 1$. In the quantum scheme, we introduce the factor ordering problem in the variables $(\Omega,\phi)$ and its corresponding momenta $(\Pi_\Omega, \Pi_\phi)$, obtaining a linear fractional differential equation with variable coefficients in the scalar field equation, then the solution is found using a fractional power series expansion. The corresponding quantum solutions are also given. We found the classical solution in the usual gauge N obtained in the Hamiltonian formalism and without a gauge. In the last case, the general solution is presented in a transformed time $T(\tau)$, however in the dust era we found a closed solution in the gauge time $\tau$. Keywords: Fractional derivative, Fractional Quantum Cosmology; K-essence formalism; Classical and Quantum exact solutions., Comment: 26 pages, 8 figures, version to be published in fractal fractionary journal

Let $S$ be a numerical semigroup. We will say that $h\in {\mathbb{N}} \backslash S$ is an {\it isolated gap }of $S$ if $\{h-1,h+1\}\subseteq S.$ A numerical semigroup without isolated gaps is called perfect numerical semigroup. Denote by ${\mathrm m}(S)$ the multiplicity of a numerical semigroup $S$. A covariety is a nonempty family ${\mathscr{C}}$ of numerical semigroups that fulfills the following conditions: there is the minimum of ${\mathscr{C}},$ the intersection of two elements of ${\mathscr{C}}$ is again an element of ${\mathscr{C}}$ and $S\backslash \{{\mathrm m}(S)\}\in {\mathscr{C}}$ for all $S\in {\mathscr{C}}$ such that $S\neq \min({\mathscr{C}}).$ In this work we prove that the set ${\mathscr{P}}(F)=\{S\mid S \mbox{ is a perfect numerical}\ \mbox{semigroup with Frobenius number }F\}$ is a covariety. Also, we describe three algorithms which compute: the set ${\mathscr{P}}(F),$ the maximal elements of ${\mathscr{P}}(F)$ and the elements of ${\mathscr{P}}(F)$ with a given genus. A ${\mathrm{Parf}}$-semigroup (respectively, ${\mathrm{Psat}}$-semigroup) is a perfect numerical semigroup that in addition is an Arf numerical semigroup (respectively, saturated numerical semigroup). We will prove that the sets: ${\mathrm{Parf}}(F)=\{S\mid S \mbox{ is a ${\mathrm{Parf}}$-numerical semigroup with Frobenius number} F\}$ and ${\mathrm{Psat}}(F)=\{S\mid S \mbox{ is a ${\mathrm{Psat}}$-numerical semigroup with Frobenius number } F\}$ are covarieties. As a consequence we present some algorithms to compute ${\mathrm{Parf}}(F)$ and ${\mathrm{Psat}}(F)$., Comment: arXiv admin note: text overlap with arXiv:2302.09121, arXiv:2303.12470, arXiv:2305.02070, arXiv:2305.13881

In this work we will show that if $F$ is a positive integer, then ${\mathrm{Sat}}(F)=\{S\mid S \mbox{ is a saturated numerical semigroup with Frobenius number } F\}$ is a covariety. As a consequence, we present two algorithms: one that computes ${\mathrm{Sat}}(F),$ and the other which computes all the elements of ${\mathrm{Sat}}(F)$ with a fixed genus. If $X\subseteq S\backslash \Delta(F)$ for some $S\in {\mathrm{Sat}}(F),$ then we will see that there is the least element of ${\mathrm{Sat}}(F)$ containing a $X$. This element will denote by ${\mathrm{Sat}}(F)[X].$ If $S\in{\mathrm{Sat}}(F),$ then we define the ${\mathrm{Sat}}(F)$-rank of $S$ as the minimum of $\{\mbox{cardinality}(X)\mid S={\mathrm{Sat}}(F)[X]\}.$ In this paper, also we present an algorithm to compute all the element of ${\mathrm{Sat}}(F)$ with a given ${\mathrm{Sat}}(F)$-rank., Comment: arXiv admin note: text overlap with arXiv:2303.12470, arXiv:2305.02070

In this work we will introduce the concept of ratio-covariety, as a nonempty family $\mathscr{R}$ of numerical semigroups verifying certain properties. This concept will allow us to: \begin{enumerate} \item Describe an algorithmic process to compute $\mathscr{R}.$ \item Prove the existence of the smallest element of $\sR$ that contains a set of positive integers. \item Talk about the smallest ratio-covariety that contains a finite set of numerical semigroups. \end{enumerate} In addition, in this paper we will apply the previous results to the study of the ratio-covariety $\mathscr{R}(F,m)=\{S\mid S \mbox{ is a numerical semigroup with Fro-} \mbox {benius number }F \mbox{ and multiplicity }m\}.$, Comment: arXiv admin note: text overlap with arXiv:2303.12470, arXiv:2302.09121

In this work we will show that if $F$ is a positive integer, then the set ${\mathrm{Arf}}(F)=\{S\mid S \mbox{ is an Arf numerical semigroup with Frobenius number } F\}$ verifies the following conditions: 1) $\Delta(F)=\{0,F+1,\rightarrow\}$ is the minimum of ${\mathrm{Arf}}(F),$ 2) if $\{S, T\} \subseteq {\mathrm{Arf}}(F)$, then $S \cap T \in {\mathrm{Arf}}(F),$ 3) if $S \in {\mathrm{Arf}}(F),$ $S\neq \Delta(F)$ and ${\mathrm m}(S)=\min (S \backslash \{0\})$, then $S\backslash \{{\mathrm m}(S)\} \in {\mathrm{Arf}}(F)$. The previous results will be used to give an algorithm which calculates the set ${\mathrm{Arf}}(F).$ Also we will see that if $X\subseteq S\backslash \Delta(F)$ for some $S\in {\mathrm{Arf}}(F),$ then there is the smallest element of ${\mathrm{Arf}}(F)$ containing $X.$, Comment: 15 pages

Context: A detailed study of the close interacting binary V4142\,Sgr based on photometric and spectroscopic analysis is presented.This system belongs to the enigmatic class of Algol-like variables showing a long photometric cycle of unknown nature. Aims: Performing photometric data-mining and spectroscopic observations covering the orbital cycle, we obtain the orbital parameters and the stellar properties of the binary system, along with the physical properties of the accretion disk located around the hot star. Insights on the evolutive path of the system are obtained. Methods: The light curve was modeled through an inverse modeling method using a theoretical light curve of the binary system, considering the light curve contribution of both stars and the accretion disk of the hot star to obtain the fundamental parameters. To constrain the main stellar parameters the mass ratio was fixed, as well as the donor temperature using the obtained values from our spectroscopic analysis including deblending methods to isolate the spectral lines of the stellar components. The system parameters were compared with a grid of binary star evolutive models in order to get insights on the evolutionary history of the system. Results: The orbital period and the long cycle were re-calculated and found to be of $30.633 \pm 0.002 ~\mathrm{days}$ and $1201 \pm 14 ~\mathrm{days}$. The spectral analysis reveals H$\alpha$ double emission with a persistent $V \leq R$ asymmetry which is considered evidence of a possible wind emergin from the hotspot region...

Denote by $\mathrm m(S)$ the multiplicity of a numerical semigroup $S$. A covariety is a nonempty family $\mathscr{C}$ of numerical semigroups that fulfills the following conditions: there is the minimum of $\mathscr{C},$ the intersection of two elements of $\mathscr{C}$ is again an element of $\mathscr{C}$ and $S\backslash \{\mathrm m(S)\}\in \mathscr{C}$ for all $S\in \mathscr{C}$ such that $S\neq \min(\mathscr{C}).$ In this work we describe an algorithmic procedure to compute all the elements of $\mathscr{C}.$ We prove that there exists the smallest element of $\mathscr{C}$ containing a set of positive integers. We show that $\mathscr{A}(F)=\{S\mid S \mbox{ is a numerical semigroup with Frobenius number }F\}$ is a covariety, and we particularize the previous results in this covariety. Finally, we will see that there is the smallest covariety containing a finite set of numerical semigroups., Comment: 13 pages

Using a particular form of the quantum K-essence scalar field, we show that in the quantum formalism, a fractional differential equation in the scalar field variable, for some epochs in the Friedmann-Lema\^itre-Robertson-Walker (FLRW) model (radiation and inflation-like epochs, for example), appears naturally. In the classical analysis, the kinetic energy of scalar fields can falsify the standard matter in the sense that we obtain the time behavior for the scale factor in all scenarios of our Universe by using the Hamiltonian formalism, where the results are analogous to those obtained by an algebraic procedure in the Einstein field equations with standard matter. In the case of the quantum Wheeler-DeWitt (WDW) equation for the scalar field $\phi$, a fractional differential equation of order $\beta=\frac{2\alpha}{2\alpha-1}$ is obtained. This fractional equation belongs to different intervals, depending on the value of the barotropic parameter; that is to say, when $ \omega_X \in [0,1]$, the order belongs to the interval $1\leq \beta \leq 2$, and when $ \omega_X \in [-1,0)$, the order belongs to the interval $0< \beta \leq 1$. The corresponding quantum solutions are also given., Comment: 26 pages, 21 figures, We have add references and substantial changes, version published in Universe Journal

Let $S$ and $\Delta$ be numerical semigroups. A numerical semigroup $S$ is an $\mathbf{I}(\Delta)$-{\it semigroup} if $S\backslash \{0\}$ is an ideal of $\Delta$. We will denote by $\mathcal{J}(\Delta)=\{S \mid S \text{ is an $\mathbf{I}(\Delta)$-semigroup} \}.$ We will say that $\Delta$ is {\it an ideal extension of } $S$ if $S\in \mathcal{J}(\Delta).$ In this work, we present an algorithm that allows to build all the ideal extensions of a numerical semigroup. We can recursively denote by $\mathcal{J}^0(\mathbb{N})=\mathbb{N},$ $\mathcal{J}^1(\mathbb{N})=\mathcal{J}(\mathbb{N})$ and $\mathcal{J}^{k+1}(\mathbb{N})=\mathcal{J}(\mathcal{J}^{k}(\mathbb{N}))$ for all $k\in \mathbb{N}.$ The complexity of a numerical semigroup $S$ is the minimun of the set $\{k\in \mathbb{N}\mid S \in \mathcal{J}^k(\mathbb{N})\}.$ In addition, we will give an algorithm that allows us to compute all the numerical semigroups with fixed multiplicity and complexity.

We present a detailed photometric and spectroscopic analysis of DD CMa, based on published survey photometry and new spectroscopic data. We find an improved orbital period of $P_\mathrm{o}= 2.0084530 \pm 0.0000006 ~\mathrm{d}$. Our spectra reveal H$\beta$ and H$\alpha$ absorptions with weak emission shoulders and we also find color excess in the WISE multiband photometry, interpreted as signatures of circumstellar matter. We model the $V$-band orbital light curve derived from the ASAS and ASAS-SN surveys, assuming a semidetached configuration and using the mass ratio and temperature of the hotter star derived from our spectroscopic analysis. Our model indicates that the system consists of a B 2.5 dwarf and a B 9 giant of radii 3.2 and 3.7 $\mathrm{R_{\odot}}$, respectively, orbiting in a circular orbit of radius 6.75 $\mathrm{R_{\odot}}$. We also found $M_{\mathrm{c}} = 1.7 \pm 0.1 ~\mathrm{M_{\odot}}$, $T_{\mathrm{c}} = 11350 \pm 100 ~\mathrm{K}$ and $M_{\mathrm{h}} = 6.4 \pm 0.1 ~\mathrm{M_{\odot}}$, $T_{\mathrm{h}} = 20000 \pm 500 ~\mathrm{K}$, for the cooler and hotter star, respectively. We find broad single emission peaks in H$\alpha$ and H$\beta$ after subtracting the synthetic stellar spectra. Our results are consistent with mass exchange between the stars, and suggest the existence of a stream of gas being accreted onto the early B-type star., Comment: 14 pages with 13 figures

We present a photometric and spectroscopic study of HD 50526, an ellipsoidal binary member of the group Double Periodic Variable stars. Performing data-mining in photometric surveys and conducting new spectroscopic observations with several spectrographs during 2008 to 2015, we obtained orbital and stellar parameters of the system. The radial velocities were analyzed with the genetic PIKAIA algorithm, whereas Doppler tomography maps for the H$\alpha$ and H$\beta$ lines were constructed with the Total Variation Minimization code. An optimized simplex-algorithm was used to solve the inverse-problem adjusting the light curve with the best stellar parameters for the system. We find an orbital period of $6.701 \pm 0.001 ~\mathrm{d}$ and a long photometric cycle of $191 \pm 2 ~\mathrm{d}$. We detected the spectral features of the coldest star, and modeled it with a $\log{g} = 2.79 \pm 0.02 ~\mathrm{dex}$ giant of mass $1.13 \pm 0.02 ~\mathrm{M_{\odot}}$ and effective temperature $10500 \pm 125 ~\mathrm{K}$. In addition, we determine a mass ratio $q= 0.206 \pm 0.033$ and that the hot star is a B-type dwarf of mass $5.48 \pm 0.02 ~\mathrm{M_{\odot}}$. The $V$-band orbital light curve can be modeled including the presence of an accretion disk around the hotter star. This fills the Roche lobe of the hotter star, and has a radius $14.74 \pm 0.02 ~\mathrm{R_{\odot}}$ and temperature at the outer edge $9400 ~\mathrm{K}$. Two bright spots located in the disk account for the global morphology of the light curve. The Doppler tomography maps of H$\alpha$ and H$\beta$, reveal complex structures of mass fluxes in the system., Comment: 16 pages with 14 figures

We report the discovery of 3 new Double Periodic Variables based on the analysis of ASAS-SN light curves: GSD J11630570-510306, V593 Sco and TYC 6939-678-1. These systems have orbital periods between 10 and 20 days and long cycles between 300 and 600 days.

We study the structure of the family of numerical semigroups with fixed multiplicity and Frobenius number. We give an algorithmic method to compute all the semigroups in this family. As an application we compute the set of all numerical semigroups with given multiplicity and genus.

In this paper we study those submonoids of $\mathbb{N}^d$ which a non-trivial pseudo-Frobenius set. In the affine case, we prove that they are the affine semigroups whose associated algebra over a field has maximal projective dimension possible. We prove that these semigroups are a natural generalization of numerical semigroups and, consequently, most of their invariants can be generalized. In the last section we introduce a new family of submonoids of $\mathbb{N}^d$ and using its pseudo-Frobenius elements we prove that the elements in the family are direct limits of affine semigroups.

DQ Velorum is a galactic double periodic variable (DPV), this system is a semi-detached binary comprised of a B-type gainer and an A-type donor star plus an extended accretion disc around the gainer. The system also presents an orbital period of $6.08337$~days and a long period of $189$~days whose origin is still under debate. Here we studied the possibility that this period may be driven by a magnetic dynamo investigating the entire evolution of the system. The model matches in a very good way the current state of the system and it can potentially be used to describe the evolution of DQ Velorum. It also predicts an increase of the dynamo number of the donor during epochs of high mass transfer in this system, and a theoretical long/orbital period ratio very close to the observed one at the present system age., Comment: 3 pages, 1 figure, conference proceeding

We present a simple model for the Double Periodic Variable (DPV) V495 Cen, which evolves as a binary system of intermediate mass, where the gainer cannot accrete at high rate, limited by the Eddington accretion rate, leading to the formation of an accretion disc. The theoretical model begins at the zero age main sequence considering the rotation for both stars. For this purpose we used the stellar evolution code MESA, developed to calculate the evolution of stars in a wide range of parameters. We started the model adjusting fundamental parameters published for this system through a chi-square optimization algorithm, and adopting an initial orbital period of 3.9 days and initial masses for the primary component $M_{i,d} = 3.40$ $M_{\odot}$ and $M_{i,g}= 3.18$ $M_{\odot}$ for the gainer, with a metallicity associated to this type of DPV of $Z = 0.02$. The method converged successfully for eight free degrees and 5% of confidence with a chi-square of $\Delta \chi^{2}_{0.95,8}= 0.212$. We describe each evolutionary stage of both components until that the donor reaches 20% core helium depletion as stop criterion. We offer a complementary analysis for understanding the mechanism of the magnetic dynamos inside the donor star using the Tayler-Spruit formalism. Currently, the theoretical model is consistent with the fundamental parameters published for V495 Cen and we discuss how our predictions can help to develop efficients theoretical models for DPV stars.

We have performed a new search for DPVs of short period in the ASAS catalog (Pojmanski, G., 1997), focusing on those stars with orbital periods between 2 to 3 days which also show variations in their brightness. From a total of 244 objects, we have found another candidate to DPV, one whose mean brightness is gradually decreasing. By fitting a 3rd order polynomial to the mean magnitude and then moving it to zero for a second analysis, a gradual decrease over 2500 days was revealed. During the last 1000 days of this decrease, a 42% increase in the variation between the minimum and maximum values of the magnitude was observed. We determined the orbital period by using the PDM IRAF software (Stellingwerf 1978) and estimated the errors for the orbital period and long cycle by visual inspection of the light curves phased with trial periods near the minimum of the periodogram given by PDM., Comment: 3 pages, 4 figures

We give two algorithmic procedures to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number and type, and the whole set of almost symmetric numerical semigroups with fixed Frobenius number. Our algorithms allow to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number with similar or even higher efficiency that the known ones. They have been implemented in the GAP (http://www.gap-system.org) package NumericalSgps (http://www.gap-system.org/Packages/numericalsgps.html)., Comment: 12 pages, 1 figure. References updated. A conjecture added (Remark 20). Accepted for publication in Journal of Algebra and its Applications

Double Periodic Variables (DPV) are among the new enigmas of semi-detached eclipsing binaries. These are intermediate-mass binaries characterized by a long photometric period lasting on average 33 times the orbital period. We present a spectroscopic and photometric study of the DPV V495 Cen based on new high-resolution spectra and the ASAS V-band light curve. We have determined an improved orbital period of $33.492 \pm 0.002$ d and a long period of 1283 d. We find a cool evolved star of $M_{2}=0.91\pm 0.2 M_{\odot}$, $T_{2}= 6000\pm 250 K$ and $R_{2}=19.3 \pm 0.5 R_{\odot}$ and a hot companion of $M_{1}= 5.76\pm 0.3 M_{\odot}$, $T_{1}=16960\pm 400 K$ and $R=4.5\pm0.2 R_{\odot}$. The mid-type B dwarf is surrounded by a concave and geometrically thick disc, of radial extension $R_{d}= 40.2\pm 1.3 R_{\odot}$ contributing $\sim$ 11 percent to the total luminosity of the system at the V band. The system is seen under inclination $84.\!\!^{\circ}8$ $\pm$ $0.\!\!^{\circ}6$ and it is at a distance $d= 2092 \pm 104.6$ pc. The light curve analysis suggests that the mass transfer stream impacts the external edge of the disc forming a hot region 11 % hotter than the surrounding disc material. The persistent $V

Fixed two positive integers m and e, some algorithms for computing the minimal Frobenius number and minimal genus of the set of numerical semigroups with multiplicity m and embedding dimension e are provided. Besides, the semigroups where these minimal values are achieved are computed too.

Given $m\in \mathbb{N},$ a numerical semigroup with multiplicity $m$ is called packed numerical semigroup if its minimal generating set is included in $\{m,m+1,\ldots, 2m-1\}.$ In this work, packed numerical semigroups are used to built the set of numerical semigroups with fixed multiplicity and embedding dimension, and to create a partition in this set. Moreover, Wilf's conjecture is checked in the tree associated to some packed numerical semigroups.

Behavior of the photoacoustic signal produced by nanoparticles as a function of their concentration was studied in detail. As the concentration of nanoparticles is increased in a sample, the peak-to-peak photoacoustic amplitude increases linearly up to a certain value, after which an asymptotic saturated behavior is observed. To elucidate the mechanisms responsible for these observations, we evaluate the effects of nanoparticles concentration, the optical attenuation and the effects of heat propagation from nano-sources to their surroundings. We found that the saturation effect of the photoacoustic signal as a function the concentration of nanoparticles is explained by a combination of two different mechanisms. As has been suggested previously, but not modeled correctly, the most important mechanism is attributed to optical attenuation. The second mechanism is due to an interference destructive process attributed to the superimposition of the photoacoustic amplitudes generated for each nanoparticle, this explanation is reinforced through our experimental and simulations results; based on this, it is found that the linear behavior of the photoacoustic amplitude could be restricted to optical densities $\le0.5$., Comment: 5 pages, 5 figures, 1 supplemental data