Mathematics
http://www.mdpi.com/journal/mathematics
Latest open access articles published in Mathematics at http://www.mdpi.com/journal/mathematics<![CDATA[Mathematics, Vol. 2, Pages 232-239: A Conjecture of Han on 3-Cores and Modular Forms]]>
http://www.mdpi.com/2227-7390/2/4/232
In his study of Nekrasov–Okounkov type formulas on “partition theoretic” expressions for families of infinite products, Han discovered seemingly unrelated q-series that are supported on precisely the same terms as these infinite products. In collaboration with Ono, Han proved one instance of this occurrence that exhibited a relation between the numbers a(n) that are given in terms of hook lengths of partitions, with the numbers b(n) that equal the number of 3-core partitions of n. Recently Han revisited the q-series with coefficients a(n) and b(n), and numerically found a third q-series whose coefficients appear to be supported on the same terms. Here we prove Han’s conjecture about this third series by proving a general theorem about this phenomenon.Mathematics2014-12-1924Article10.3390/math20402322322392227-73902014-12-19doi: 10.3390/math2040232Amanda Clemm<![CDATA[Mathematics, Vol. 2, Pages 218-231: Characteristic Variety of the Gauss–Manin Differential Equations of a Generic Parallelly Translated Arrangement]]>
http://www.mdpi.com/2227-7390/2/4/218
We consider a weighted family of \(n\) generic parallelly translated hyperplanes in \(\mathbb{C}^k\) and describe the characteristic variety of the Gauss–Manin differential equations for associated hypergeometric integrals. The characteristic variety is given as the zero set of Laurent polynomials, whose coefficients are determined by weights and the Plücker coordinates of the associated point in the Grassmannian Gr\((k,n)\). The Laurent polynomials are in involution.Mathematics2014-10-1624Article10.3390/math20402182182312227-73902014-10-16doi: 10.3390/math2040218Alexander Varchenko<![CDATA[Mathematics, Vol. 2, Pages 196-217: The Second-Order Shape Derivative of Kohn–Vogelius-Type Cost Functional Using the Boundary Differentiation Approach]]>
http://www.mdpi.com/2227-7390/2/4/196
A shape optimization method is used to study the exterior Bernoulli free boundaryproblem. We minimize the Kohn–Vogelius-type cost functional over a class of admissibledomains subject to two boundary value problems. The first-order shape derivative of the costfunctional is recalled and its second-order shape derivative for general domains is computedvia the boundary differentiation scheme. Additionally, the second-order shape derivative ofJ at the solution of the Bernoulli problem is computed using Tiihonen’s approach.Mathematics2014-09-2624Article10.3390/math20401961962172227-73902014-09-26doi: 10.3390/math2040196Jerico BacaniGunther Peichl<![CDATA[Mathematics, Vol. 2, Pages 172-195: Dynamics of a Parametrically Excited System with Two Forcing Terms]]>
http://www.mdpi.com/2227-7390/2/3/172
Motivated by the dynamics of a trimaran, an investigation of the dynamic behaviour of a double forcing parametrically excited system is carried out. Initially, we provide an outline of the stability regions, both numerically and analytically, for the undamped linear, extended version of the Mathieu equation. This paper then examines the anticipated form of response of our proposed nonlinear damped double forcing system, where periodic and quasiperiodic routes to chaos are graphically demonstrated and compared with the case of the single vertically-driven pendulum.Mathematics2014-09-2223Article10.3390/math20301721721952227-73902014-09-22doi: 10.3390/math2030172Anastasia SofroniouSteven Bishop<![CDATA[Mathematics, Vol. 2, Pages 136-171: Modeling the Influence of Environment and Intervention onCholera in Haiti]]>
http://www.mdpi.com/2227-7390/2/3/136
We propose a simple model with two infective classes in order to model the cholera epidemic in Haiti. We include the impact of environmental events (rainfall, temperature and tidal range) on the epidemic in the Artibonite and Ouest regions by introducing terms in the transmission rate that vary with environmental conditions. We fit the model on weekly data from the beginning of the epidemic until December 2013, including the vaccination programs that were recently undertaken in the Ouest and Artibonite regions. We then modified these projections excluding vaccination to assess the programs’ effectiveness. Using real-time daily rainfall, we found lag times between precipitation events and new cases that range from 3:4 to 8:4 weeks in Artibonite and 5:1 to 7:4 in Ouest. In addition, it appears that, in the Ouest region, tidal influences play a significant role in the dynamics of the disease. Intervention efforts of all types have reduced case numbers in both regions; however, persistent outbreaks continue. In Ouest, where the population at risk seems particularly besieged and the overall population is larger, vaccination efforts seem to be taking hold more slowly than in Artibonite, where a smaller core population was vaccinated. The models including the vaccination programs predicted that a year and six months later, the mean number of cases in Artibonite would be reduced by about two thousand cases, and in Ouest by twenty four hundred cases below that predicted by the models without vaccination. We also found that vaccination is best when done in the early spring, and as early as possible in the epidemic. Comparing vaccination between the first spring and the second, there is a drop of about 40% in the case reduction due to the vaccine and about 10% per year after that.Mathematics2014-09-0523Article10.3390/math20301361361712227-73902014-09-05doi: 10.3390/math2030136Stephen TennenbaumCaroline FreitagSvetlana Roudenko<![CDATA[Mathematics, Vol. 2, Pages 119-135: A Graphical Approach to a Model of a Neuronal Tree with a Variable Diameter]]>
http://www.mdpi.com/2227-7390/2/3/119
Tree-like structures are ubiquitous in nature. In particular, neuronal axons and dendrites have tree-like geometries that mediate electrical signaling within and between cells. Electrical activity in neuronal trees is typically modeled using coupled cable equations on multi-compartment representations, where each compartment represents a small segment of the neuronal membrane. The geometry of each compartment is usually defined as a cylinder or, at best, a surface of revolution based on a linear approximation of the radial change in the neurite. The resulting geometry of the model neuron is coarse, with non-smooth or even discontinuous jumps at the boundaries between compartments. We propose a hyperbolic approximation to model the geometry of neurite compartments, a branched, multi-compartment extension, and a simple graphical approach to calculate steady-state solutions of an associated system of coupled cable equations. A simple case of transient solutions is also briefly discussed.Mathematics2014-07-0923Article10.3390/math20301191191352227-73902014-07-09doi: 10.3390/math2030119Marco Herrera-ValdezSergei SuslovJosé Vega-Guzmán<![CDATA[Mathematics, Vol. 2, Pages 96-118: The Riccati System and a Diffusion-Type Equation]]>
http://www.mdpi.com/2227-7390/2/2/96
We discuss a method of constructing solutions of the initial value problem for diffusion-type equations in terms of solutions of certain Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation is also considered. Examples include, but are not limited to the Fokker-Planck equation in physics, the Black-Scholes equation and the Hull-White model in finance.Mathematics2014-05-1522Article10.3390/math2020096961182227-73902014-05-15doi: 10.3390/math2020096Erwin SuazoSergei SuslovJosé Vega-Guzmán<![CDATA[Mathematics, Vol. 2, Pages 83-95: Traveling Wave Solutions of Reaction-Diffusion Equations Arising in Atherosclerosis Models]]>
http://www.mdpi.com/2227-7390/2/2/83
In this short review article, two atherosclerosis models are presented, one as a scalar equation and the other one as a system of two equations. They are given in terms of reaction-diffusion equations in an infinite strip with nonlinear boundary conditions. The existence of traveling wave solutions is studied for these models. The monostable and bistable cases are introduced and analyzed.Mathematics2014-05-0822Article10.3390/math202008383952227-73902014-05-08doi: 10.3390/math2020083Narcisa Apreutesei<![CDATA[Mathematics, Vol. 2, Pages 68-82: Numerical Construction of Viable Sets for Autonomous Conflict Control Systems]]>
http://www.mdpi.com/2227-7390/2/2/68
A conflict control system with state constraints is under consideration. A method for finding viability kernels (the largest subsets of state constraints where the system can be confined) is proposed. The method is related to differential games theory essentially developed by N. N. Krasovskii and A. I. Subbotin. The viability kernel is constructed as the limit of sets generated by a Pontryagin-like backward procedure. This method is implemented in the framework of a level set technique based on the computation of limiting viscosity solutions of an appropriate Hamilton–Jacobi equation. To fulfill this, the authors adapt their numerical methods formerly developed for solving time-dependent Hamilton–Jacobi equations arising from problems with state constraints. Examples of computing viability sets are given.Mathematics2014-04-1122Article10.3390/math202006868822227-73902014-04-11doi: 10.3390/math2020068Nikolai BotkinVarvara Turova<![CDATA[Mathematics, Vol. 2, Pages 53-67: Convergence of the Quadrature-Differences Method for Singular Integro-Differential Equations on the Interval]]>
http://www.mdpi.com/2227-7390/2/1/53
In this paper, we propose and justify the quadrature-differences method for the full linear singular integro-differential equations with the Cauchy kernel on the interval (–1,1). We consider equations of zero, positive and negative indices. It is shown that the method converges to an exact solution, and the error estimation depends on the sharpness of derivative approximations and on the smoothness of the coefficients and the right-hand side of the equation.Mathematics2014-03-0421Article10.3390/math201005353672227-73902014-03-04doi: 10.3390/math2010053Alexander Fedotov<![CDATA[Mathematics, Vol. 2, Pages 37-52: Bounded Gaps between Products of Special Primes]]>
http://www.mdpi.com/2227-7390/2/1/37
In their breakthrough paper in 2006, Goldston, Graham, Pintz and Yıldırım proved several results about bounded gaps between products of two distinct primes. Frank Thorne expanded on this result, proving bounded gaps in the set of square-free numbers with r prime factors for any r ≥ 2, all of which are in a given set of primes. His results yield applications to the divisibility of class numbers and the triviality of ranks of elliptic curves. In this paper, we relax the condition on the number of prime factors and prove an analogous result using a modified approach. We then revisit Thorne’s applications and give a better bound in each case.Mathematics2014-03-0321Article10.3390/math201003737522227-73902014-03-03doi: 10.3390/math2010037Ping ChungShiyu Li<![CDATA[Mathematics, Vol. 2, Pages 29-36: Some New Integral Identities for Solenoidal Fields and Applications]]>
http://www.mdpi.com/2227-7390/2/1/29
In spaces Rn, n ≥ 2, it has been proved that a solenoidal vector field and its rotor satisfy the series of new integral identities which have covariant form. The interest in them is explained by hydrodynamics problems for an ideal fluid.Mathematics2014-03-0321Article10.3390/math201002929362227-73902014-03-03doi: 10.3390/math2010029Vladimir Semenov<![CDATA[Mathematics, Vol. 2, Pages 12-28: On the Folded Normal Distribution]]>
http://www.mdpi.com/2227-7390/2/1/12
The characteristic function of the folded normal distribution and its moment function are derived. The entropy of the folded normal distribution and the Kullback–Leibler from the normal and half normal distributions are approximated using Taylor series. The accuracy of the results are also assessed using different criteria. The maximum likelihood estimates and confidence intervals for the parameters are obtained using the asymptotic theory and bootstrap method. The coverage of the confidence intervals is also examined.Mathematics2014-02-1421Article10.3390/math201001212282227-73902014-02-14doi: 10.3390/math2010012Michail TsagrisChristina BenekiHossein Hassani<![CDATA[Mathematics, Vol. 2, Pages 1-11: One-Dimensional Nonlinear Stefan Problems in Storm’s Materials]]>
http://www.mdpi.com/2227-7390/2/1/1
We consider two one-phase nonlinear one-dimensional Stefan problems for a semi-infinite material x &gt; 0; with phase change temperature Tf : We assume that the heat capacity and the thermal conductivity satisfy a Storm’s condition. In the first case, we assume a heat flux boundary condition of the type q(t) = q 0 t , and in the second case, we assume a temperature boundary condition T = Ts &lt; Tf at the fixed face. Solutions of similarity type are obtained in both cases, and the equivalence of the two problems is demonstrated. We also give procedures in order to compute the explicit solution.Mathematics2013-12-2721Article10.3390/math20100011112227-73902013-12-27doi: 10.3390/math2010001Adriana BriozzoMaría Natale<![CDATA[Mathematics, Vol. 1, Pages 111-118: Sign-Periodicity of Traces of Singular Moduli]]>
http://www.mdpi.com/2227-7390/1/4/111
Zagier proved that the generating functions of traces of singular values of Jm(z) are weight 3 2 weakly holomorphic modular forms. In this paper we prove that there is the sign-periodicity of traces of singular values of Jm(z).Mathematics2013-10-1514Article10.3390/math10401111111182227-73902013-10-15doi: 10.3390/math1040111Dohoon ChoiByungchan KimSubong Lim<![CDATA[Mathematics, Vol. 1, Pages 100-110: Effective Congruences for Mock Theta Functions]]>
http://www.mdpi.com/2227-7390/1/3/100
Let M(q) =∑ c(n)q n be one of Ramanujan’s mock theta functions. We establish the existence of infinitely many linear congruences of the form: c(An + B) ≡ 0 (mod l j ) where A is a multiple of l and an auxiliary prime, p. Moreover, we give an effectively computable upper bound on the smallest such p for which these congruences hold. The effective nature of our results is based on the prior works of Lichtenstein [1] and Treneer [2].Mathematics2013-09-0413Article10.3390/math10301001001102227-73902013-09-04doi: 10.3390/math1030100Nickolas AndersenHolley FriedlanderJeremy FullerHeidi Goodson<![CDATA[Mathematics, Vol. 1, Pages 89-99: Scattering of Electromagnetic Waves by Many Nano-Wires]]>
http://www.mdpi.com/2227-7390/1/3/89
Electromagnetic wave scattering by many parallel to the z−axis, thin, impedance, parallel, infinite cylinders is studied asymptotically as a → 0. Let Dm be the cross-section of the m−th cylinder, a be its radius and x ^ m = (x m1 , x m2 ) be its center, 1 ≤ m ≤ M , M = M (a). It is assumed that the points, x ^ m , are distributed, so that N(Δ)= 1 2πa ∫ Δ N ( x ^ )d x ^ [1+o(1)] where N (∆) is the number of points, x ^ m , in an arbitrary open subset, ∆, of the plane, xoy. The function, N( x ^ ) ≥0 , is a continuous function, which an experimentalist can choose. An equation for the self-consistent (effective) field is derived as a → 0. A formula is derived for the refraction coefficient in the medium in which many thin impedance cylinders are distributed. These cylinders may model nano-wires embedded in the medium. One can produce a desired refraction coefficient of the new medium by choosing a suitable boundary impedance of the thin cylinders and their distribution law.Mathematics2013-07-1813Article10.3390/math103008989992227-73902013-07-18doi: 10.3390/math1030089Alexander Ramm<![CDATA[Mathematics, Vol. 1, Pages 76-88: On the Distribution of the spt-Crank]]>
http://www.mdpi.com/2227-7390/1/3/76
Andrews, Garvan and Liang introduced the spt-crank for vector partitions. We conjecture that for any n the sequence { N S (m , n) } m is unimodal, where N S (m , n) is the number of S-partitions of size n with crank m weight by the spt-crank. We relate this conjecture to a distributional result concerning the usual rank and crank of unrestricted partitions. This leads to a heuristic that suggests the conjecture is true and allows us to asymptotically establish the conjecture. Additionally, we give an asymptotic study for the distribution of the spt-crank statistic. Finally, we give some speculations about a definition for the spt-crank in terms of “marked” partitions. A “marked” partition is an unrestricted integer partition where each part is marked with a multiplicity number. It remains an interesting and apparently challenging problem to interpret the spt-crank in terms of ordinary integer partitions.Mathematics2013-06-2813Article10.3390/math103007676882227-73902013-06-28doi: 10.3390/math1030076George AndrewsFreeman DysonRobert Rhoades<![CDATA[Mathematics, Vol. 1, Pages 65-75: On the Class of Dominant and Subordinate Products]]>
http://www.mdpi.com/2227-7390/1/2/65
In this paper we provide proofs of two new theorems that provide a broad class of partition inequalities and that illustrate a na¨ıve version of Andrews’ anti-telescoping technique quite well. These new theorems also put to rest any notion that including parts of size 1 is somehow necessary in order to have a valid irreducible partition inequality. In addition, we prove (as a lemma to one of the theorems) a rather nontrivial class of rational functions of three variables has entirely nonnegative power series coefficients.Mathematics2013-05-1512Article10.3390/math102006565752227-73902013-05-15doi: 10.3390/math1020065Alexander BerkovichKeith Grizzell<![CDATA[Mathematics, Vol. 1, Pages 46-64: Stability of Solutions to Evolution Problems]]>
http://www.mdpi.com/2227-7390/1/2/46
Large time behavior of solutions to abstract differential equations is studied. The results give sufficient condition for the global existence of a solution to an abstract dynamical system (evolution problem), for this solution to be bounded, and for this solution to have a finite limit as t → ∞ , in particular, sufficient conditions for this limit to be zero. The evolution problem is: u ˙ = A(t)u + F(t, u) + b(t), t ≥ 0; u(0) = u 0 . (*) Here u ˙ := du dt , u = u(t) ∈ H, H is a Hilbert space, t ∈ R + := [0,∞), A(t) is a linear dissipative operator: Re(A(t)u,u) ≤−γ(t)(u, u) where F(t, u) is a nonlinear operator, ‖ F(t, u) ‖ ≤ c 0 ‖ u ‖ p , p &gt; 1, c 0 and p are positive constants, ‖ b(t) ‖ ≤ β(t) , and β(t)≥0 is a continuous function. The basic technical tool in this work are nonlinear differential inequalities. The non-classical case γ(t) ≤ 0 is also treated.Mathematics2013-05-1312Article10.3390/math102004646642227-73902013-05-13doi: 10.3390/math1020046Alexander Ramm<![CDATA[Mathematics, Vol. 1, Pages 31-45: A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements]]>
http://www.mdpi.com/2227-7390/1/1/31
Let C1 and C2 be algebraic plane curves in ℂ 2 such that the curves intersect in d1 · d2 points where d1, d2 are the degrees of the curves respectively. Oka and Sakamoto proved that π1( ℂ 2 \ C1 U C2)) ≅ π1 ( ℂ 2 \ C1) × π1 ( ℂ 2 \ C2) [1]. In this paper we prove the converse of Oka and Sakamoto’s result for line arrangements. Let A1 and A2 be non-empty arrangements of lines in ℂ 2 such that π1 (M(A1 U A2)) ≅ π1 (M(A1)) × π1 (M(A2)) Then, the intersection of A1 and A2 consists of /A1/ · /A2/ points of multiplicity two.Mathematics2013-03-1411Article10.3390/math101003131452227-73902013-03-14doi: 10.3390/math1010031Kristopher Williams<![CDATA[Mathematics, Vol. 1, Pages 9-30: ρ — Adic Analogues of Ramanujan Type Formulas for 1/π]]>
http://www.mdpi.com/2227-7390/1/1/9
Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form ∑ k = 0 ∞ ( 1 2 ) k ( 1 d ) k ( d - 1 d ) k k ! 3 ( a k + 1 ) ( λ d ) k = δ π for d=2,3,4,6, where łd are singular values that correspond to elliptic curves with complex multiplication, and a,δ are explicit algebraic numbers. In this paper we prove a p-adic version of this formula in terms of the so-called Ramanujan type congruence. In addition, we obtain a new supercongruence result for elliptic curves with complex multiplication.Mathematics2013-03-1311Article10.3390/math10100099302227-73902013-03-13doi: 10.3390/math1010009Sarah ChisholmAlyson DeinesLing LongGabriele NebeHolly Swisher<![CDATA[Mathematics, Vol. 1, Pages 3-8: On Matrices Arising in the Finite Field Analogue of Euler’s Integral Transform]]>
http://www.mdpi.com/2227-7390/1/1/3
In his 1984 Ph.D. thesis, J. Greene defined an analogue of the Euler integral transform for finite field hypergeometric series. Here we consider a special family of matrices which arise naturally in the study of this transform and prove a conjecture of Ono about the decomposition of certain finite field hypergeometric functions into functions of lower dimension.Mathematics2013-02-0511Article10.3390/math1010003382227-73902013-02-05doi: 10.3390/math1010003Michael GriffinLarry Rolen<![CDATA[Mathematics, Vol. 1, Pages 1-2: Mathematics—An Open Access Journal]]>
http://www.mdpi.com/2227-7390/1/1/1
As is widely known, mathematics plays a unique role in all natural sciences as a refined scientific language and powerful research tool. Indeed, most of the fundamental laws of nature are written in mathematical terms and we study their consequences by numerous mathematical methods (and vice versa, any essential progress in a natural science has been accompanied by fruitful developments in mathematics). In addition, the mathematical modeling in various interdisciplinary problems and logical development of mathematics on its own should be taken into account. [...]Mathematics2012-12-2811Editorial10.3390/math1010001122227-73902012-12-28doi: 10.3390/math1010001Sergei Suslov