Pruned Inside-Out Polytopes, Combinatorial Reciprocity Theorems and Generalized Permutahedra

On the basis of a general theory of the operation of passive linear transducers developed by the authors, a proof of the reciprocity theorem for electroacoustic transducers is presented together with the conditions necessary for its validity. The reciprocity theorem states that the ratio of the microphone response to the speaker response for an electroacoustic transducer is a quantity which, under certain general conditions, is independent of the specific nature and construction of the transducer. This quantity is 2d/ρf where ρ is the density of the medium, f is the frequency, and d is the distance from the center of the transducer to the point where the pressure is measured in obtaining the speaker response. The center of the transducer is defined as the point where the incident pressure is measured, in the absence of the transducer, when obtaining the microphone response. The general theory referred to above assumes a pair of the most general equations (in this case, integral equations), relating the pressure and normal velocity at each point on the surface of the transducer and the voltage and current at its terminals. These, together with the appropriate solutions of the wave equation derived by the use of Green's functions and relating the pressure at any point in the medium to the normal velocity distribution on the transducer surface, completely characterize the behavior of the transducer and allow explicit calculation of the responses. The validity of the reciprocity theorem then requires certain symmetry conditions on the parameters entering into the equations. The relation between these symmetry conditions and the physical characteristics of the transducer allow the limits of validity of the reciprocity theorem to be delineated.

The general proof of the Ebers-Moll reciprocity theorem is extended to include high-level injection conditions in bipolar base regions. The theorem, originally derived for the low-level case, is shown to be valid in the high-level limit, as long as the emitter injection efficiency is sufficiently high in both reciprocal configurations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

In this paper the authors propose a new approach to the study of weight systems. Instead of considering graphs whose vertices correspond to the generators of a Lie algebra (as for Cartan subalgebras in the semisimple case), the authors consider the whole weight system. The purpose is to extract information about the weight system from the geometry of the weights. The considerations are restricted to the case where a torus of derivations induces a decomposition of a nilpotent Lie algebra g into one-dimensional weight spaces, none of which is associated with the zero weight. The paper is structured as follows: In Section 2 the most important facts of weight systems of nilpotent Lie algebras and the root system associated to solvable Lie algebras are recalled. In Section 3 the authors formulate their conditions on the weight systems and analyze the consequences of these conditions on the structure of the weight system. They also define associated weight graphs and deduce their elementary geometrical properties. This provides a characterization of the three-dimensional Heisenberg Lie algebra in terms of trees. Section 4 is devoted to the study of certain subgraphs of a weight graph which can be used to reconstruct the weight system from the weight graph. If r is a semidirect product of g and a torus T these subgraphs determine bounds for the solvability class of r . In Section 5 these results are applied to obtain a geometrical proof of the nonexistence of two-step solvable rigid Lie algebras.

While in the last chapter we have focused on measurement aspects and related analytic issues, we will return to more qualitative relations of objects in a Cayley-Klein geometry. In this and the next chapter we will mainly study the projective nature of Cayley-Klein geometries. This chapter focuses on aspects concerning transformations, their (projective) invariants, and the behavior of measurements under these transformations. The next chapter treats geometric objects and their elementary geometric properties, including some elementary geometric theorems. Although Cayley-Klein geometries unify several rather different types of geometric playgrounds in a common projective framework, each type of geometry (in the sense of Section 20.6) has its very special properties. This is due to the fact that the degeneracies of the fundamental conic of the geometry lead to degeneracies in the relevant geometric constructions. For this reason we will sometimes have to perform a special case analysis for the different types of Cayley-Klein geometries. Sometimes one general definition covers a certain effect (for instance reflection) in all types of geometries. Still it may be instructive to consider its particular specialization to certain geometries.

Characterization of embedded special manifolds

We generalize a well-known result proved by Filaseta and Trifonov in 2002 that the Bessel polynomials of all degrees are irreducible over the field of rational numbers. The proof given here of our generalization appears to be simpler than the known proofs of Filaseta-Trifonov Theorem. We use some recent results by Lehmer, Luca, Najman and Shorey regarding the largest prime divisor of a product of consecutive integers, which play a significant role in making the proof shorter. The backbone of our proof is a theorem based on an extension introduced by Ore of the concept of Newton polygons. We illustrate the utility of our generalization by showing that it leads to new classes of monic irreducible polynomials with integer coefficients for which the known irreducibility criteria for polynomials do not seem to be applicable.

1. The problem. In [1], the following problem is posed: Let 0 ,j=O, 1, , n}I Here, we suppose, in addition, without loss of generality, that ao>0, an=. Define (P+ = 6Po+= U 1 6i+. Further, let (P1+ (respectively, 6i2+; 63+) denote the class of polynomials, all of whose roots are real and negative (respectively, are equal and negative; have negative real part). What is the smallest integer Nj = Nj(E) such that (X2 x +e) Qx (X) EZ(P+ where e> 1/4 and QNJ(x) is a polynomial of degree Nj which belongs to (Pt+? Here, j takes any of the values 0, 1, 2, 3. If j = 0, the answer is contained in the statement and proof of Theorem 1 of [2 ]. For j = 1, the smallest integer n (namely N,) for which (x2-x+cE)Hf=l (x+bi)Ez6P+ is exactly the integer N mentioned earlier, as the correspondence y->2Ax, e->(4A2)-l, ai-->Abi shows. Clearly, bi > O, all i and when n = N,, bi > O, i = 1, 2, * * * , N, For j =2, the above question is intimately related to the second part of the research problem while for j = 3, the connection is more remote. All of these questions are rendered easier by an inversion, that is, the fixing of n and the quest for the smallest E = EC such that

The object of the present paper is to give a new set of proofs of some fundamental existence theorems of conformal mapping and potential theory by using the notion of a kernel function introduced by Bergman [2, 3, 4]. The proofs which we present here are a more or less natural outcome of the recent researches done in this theory [5, 6, 7, 11, 12]. Our new attack upon the existence problem has on the one hand the advantage of generality, since it is applicable in the case of partial differential equations of elliptic type and since for functions of several complex variables the theory of the kernel function is well developed, while on the other hand the approach is simple and elementary in comparison with, say, the method using the Dirichlet principle. For potential theory and conformal mapping in multiply-connected plane domains, our proof is probably as simple as any known, and even for simply-connected domains the method has the advantage that it is based on a general theory rather than upon special artifices. A further point which we gain is that the kernel functions associated with the various norms of function theory and elliptic partial differential equations are well adapted to numerical computation, in view of their relation to orthogonal functions, so that we obtain at one and the same time an existence proof and a computational algorithm. The most serious drawback in our method is, perhaps, that we must make assumptions upon the smoothness of the boundary of the domains we consider, so that the general case is reached only after a topological approximation argument is given. From the broader point of view, our treatment is important in that it yields a unified attack upon various existence problems. An extensive class of existence theorems of function theory of one or more variables and of partial differential equations of elliptic type can be developed using one basic procedure. This procedure can be outlined as follows. We set up the reproducing kernel function in a given function space by solving a suitable extremal problem, or alternatively by means of orthogonal functions, and we investigate the scalar product of the kernel function with what may be described as the fundamental singularity associated with this space. A local argument is used to study the behavior of this scalar product as the infinity of the fundamental singularity crosses the boundary of the domain of definition of the functions in our class, and the desired existence theorems follow from the geometric properties thus obtained. The local investigations depend in a general way upon a knowledge of our problem in simple domains, such as, for example, the unit circle.

For lattice polytopes $P_1,\ldots, P_k \subseteq \mathbb{R}^d$, Bihan (2016) introduced the discrete mixed volume $DMV(P_1,\dots,P_k)$ in analogy to the classical mixed volume. In this note we study the associated mixed Ehrhart polynomial $ME_{P_1, \dots,P_k}(n) = DMV(nP_1, \dots, nP_k)$. We provide a characterization of all mixed Ehrhart coefficients in terms of the classical multivariate Ehrhart polynomial. Bihan (2016) showed that the discrete mixed volume is always non-negative. Our investigations yield simpler proofs for certain special cases.We also introduce and study the associated mixed $h^*$-vector. We show that for large enough dilates $r P_1, \ldots, rP_k$ the corresponding mixed $h^*$-polynomial has only real roots and as a consequence the mixed $h^*$-vector becomes non-negative.

This paper concerns a class ${\mathcal {P}_n}$ (defined below) of polynomials of degree less than or equal to $n$ having the properties: each polynomial which is univalent in the unit disk and of degree $n$ or less is in ${\mathcal {P}_n}$ and if $\{ {P_{{n_k}}}\} _{k = 1}^\infty$ is a sequence of polynomials such that ${P_{{n_k}}} \in {\mathcal {P}_{{n_k}}}$ and ${\lim _{k \to \infty }}{P_{{n_k}}} = f$ (uniformly on compact subsets of the unit disk) then $f$ is univalent. The approach is to study the extreme points in ${\mathcal {P}_n}$ ($P \in {\mathcal {P}_n}$ is extreme if $P$ is not a proper convex combination of two distinct elements of ${\mathcal {P}_n}$). Theorem 3 shows that if $P \in {\mathcal {P}_n}$ is extreme then $((n + 1)/n)P(z) - (1/n)zP’(z)$ is univalent and Theorem 6 gives a geometric condition on the image of the boundary of the disk under this mapping in order that $P$ be extreme. Theorem 10 states that the collection of polynomials univalent in the unit disk and having the property $P(z) = z + {a_2}{z^2} + \cdots + {a_n}{z^n},{a_n} = 1/n$, are dense in the class $S$ of normalized univalent functions. These polynomials have the very striking geometric property that the tangent line to the curve $P({e^{i\theta }})$, $0 \leqq \theta \leqq 2\pi$, turns at a constant rate (between cusps) as $\theta$ varies.

SummaryIn 1693, Gottfried Wilhelm Leibniz published in the Acta Eruditorum a geometrical proof of the fundamental theorem of the calculus. It is shown that this proof closely resembles Isaac Barrow's proof in Proposition 11, Lecture 10, of his Lectiones Geometricae, published in 1670. This comparison provides evidence that Leibniz gained substantial help from Barrow's book in formulating and presenting his geometrical formulation of this theorem. The analysis herein also supports the work of J. M. Child, who in 1920 studied the early manuscripts of Leibniz and concluded that he had frequently copied his diagrams from Barrow's book, but without acknowledgement. It is also shown that the diagram of Leibniz associated with his 1693 proof has often been reproduced with errors that make some aspects of his text difficult to comprehend.

A Geometric Proof of the Gap Theorem

In the Fermi Lectures on the obstacle problem in 1998, Caffarelli gave a proof of the mean value theorem which extends to general divergence form uniformly elliptic operators. In the general setting, the result shows that for any such operator L and at any point x0 in the domain, there exists a nested family of sets {Dr(x0)} where the average over any of those sets is related to the value of the function at x0. Although it is known that the {Dr(x0)} are nested and are comparable to balls in the sense that there exists c,C depending only on L such that Bcr(x0) ⊂ Dr(x0) ⊂ BCr(x0) for all r > 0 and x0 in the domain, otherwise their geometric and topological properties are largely unknown. In this paper we begin the study of these topics and we prove a few results about the geometry of these sets and give a couple of applications of the theorems.

Polyhedral Manifolds With Few Vertices

Geometric proofs of some theorems of Schur-Horn type