For a smooth curve C it is known that a very ample line bundle L">L${<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">L</span>}$ on C is normally generated if Cliff(L">L${<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">L</span>}$ ) < Cliff(C) and there exist extremal line bundles L">L${<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">L</span>}$ (:non-normally generated very ample line bundle with Cliff(L">L${<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">L</span>}$ ) = Cliff(C)) with h1(L)&#x2264;1">h1(L)≤1${\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">h</span>}}^{\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">1</span>}}<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">(</span>{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">L</span>}<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">)</span><span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">\le </span>\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">1</span>}$ . However it has been unknown whether there exists an extremal line bundle L">L${<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">L</span>}$ with h1(L)&#x2265;2">h1(L)≥2${\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">h</span>}}^{\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">1</span>}}<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">(</span>{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">L</span>}<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">)</span><span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">\ge </span>\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">2</span>}$ . In this paper, we prove that for any positive integers (g, c) with g =？ 2c +？ 5 and c&#x2261;0">c≡0$\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">c</span>}<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">\equiv </span>\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">0</span>}$ (mod 2) there exists a smooth curve of genus g and Clifford index c carrying an extremal line bundle L">L${<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">L</span>}$ with h1(L)=2">h1(L)=2${\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">h</span>}}^{\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">1</span>}}<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">(</span>{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">L</span>}<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">)</span><span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">=</span>\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">2</span>}$ . In fact, a smooth quadric hypersurface section C of a general projective K3 surface always has an extremal line bundle L">L${<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">L</span>}$ with h1(L)=2">h1(L)=2${\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">h</span>}}^{\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">1</span>}}<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">(</span>{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">L</span>}<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">)</span><span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">=</span>\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">2</span>}$ . More generally, if C has a line bundle M">M${<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">M</span>}$ computing the Clifford index c of C with (3c/2)+3<degM&#x2264;g&#x2212;1">(3c/2)+3<degM≤g？1$<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">(</span>\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">3</span>}\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">c</span>}<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">/</span>\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">2</span>}<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">)</span><span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">+</span>\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">3</span>}<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;"><</span>\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">deg</span>}{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">M</span>}<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">\le </span>\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">g</span>}<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">？</span>\mathrm{<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">1</span>}$ , then C has such an extremal line bundle L">L${<span\; style="font-family:\; MS\; PGothic;\; font-size:\; 12pt;">L</span>}$ .

An optimal control problem for the forced Fisher equation is considered. The control is an artificially introduced genotype and the objective is to match, as well as possible, a specified gene frequency. The existence of a solution of the optimal control problem is proved and an optimality system is derived through the Lagrange multiplier technique. Numerical approximations of the optimality system are defined using finite element methods to effect spatial discretization and a backward Euler method for the time discretiza- tion. Convergence of semi-discrete in time approximations of the state system is proved and a gradient method for solving the nonlinear discrete systems is developed. The results of some preliminary computational experiments are provided.

In this paper, we discuss discrete versions of the heat equations and the wave equations, which are called the ω-diffusion equations and the ω-elastic equations on graphs. After deriving some basic properties, we solve the ω-diffusion equations under (i) the condition that there is no boundary, (ii) the initial condition and (iii) the Dirichlet boundary condition. We also give some additional interesting properties on the ω-diffusion equations, such as the minimum and maximum principles, Huygens property and uniqueness via energy methods. Analogues of the ω-elastic equations on graphs are also discussed.

Keywords: Discrete Laplacian, diffusion kernel, elastic kernel

We propose a new algorithm for two-dimensional magnetotelluric (MT) inversion. Our algorithm is an MT inversion based on the steepest descent method, borrowed from the backpropagation technique of seismic inversion or reverse time migration, introduced in the middle 1980s by Lailly and Tarantola. The steepest descent direction can be calculated efficiently by using the symmetry of numerical Green’s function derived from a mixed finite element method proposed by Nédélec for Maxwell’s equation, without calculating the Jacobian matrix explicitly. We construct three different objective functions by taking the logarithm of the complex apparent resistivity as introduced in the recent waveform inversion algorithm by Shin and Min. These objective functions can be naturally separated into amplitude inversion, phase inversion and simultaneous inversion. We demonstrate our algorithm by showing three inversion results for synthetic data.