Lifting inequalities for polytopes

We present a method of lifting linear inequalities for the flag f-vector of polytopes to higher dimensions. Known inequalities that can be lifted using this technique are the non-negativity of the toric g-vector and that the simplex minimizes the cd-index. We obtain new inequalities for six-dimensional polytopes. In the last section we present the currently best known inequalities for dimensions 5 through 8.     

Inequalities for <Emphasis Type="Bold">cd</Emphasis>-Indices of Joins and Products of Polytopes

The cd-index is a polynomial which encodes the flag f-vector of a convex polytope. For polytopes U and V, we determine explicit recurrences for computing the cd-index of the free join and the cd-index of the Cartesian product U x V. As an application of these recurrences, we prove the inequality involving the cd-indices of three polytopes.     

Quadratic Gröbner bases for smooth 3×3 transportation polytopes

The toric ideals of 3×3 transportation polytopes T_rc\mathsf{T}_{\mathbf{rc}} are quadratically generated. The only exception is the Birkhoff polytope B ₃. If T_rc\mathsf{T}_{\mathbf{rc}} is not a multiple of B ₃, these ideals even have square-free quadratic initial ideals. This class contains all smooth 3×3 transportation polytopes.     

The Partial-Fractions Method for Counting Solutions to Integral Linear Systems

We present a new tool to compute the number $\phi_{\bf A} (b)$ of integer solutions to the linear system $$ x \geq 0, A x = b, $$ where the coefficients of $A$ and $b$ are integral. $\phi_{\bf A} (b)$ is often described as a vector partition function. Our methods use partial fraction expansions of Eulers generating function for $\phi_{\bf A} (\b)$. A special class of vector partition functions are Ehrhart (quasi-)polynomials counting integer points in dilated polytopes.     

Asymptotically Maximal Families of Hypersurfaces in Toric Varieties

A real algebraic variety is maximal (with respect to the Smith-Thom inequality) if the sum of the Betti numbers (with coefficients) of the real part of the variety is equal to the sum of Betti numbers of its complex part. We prove that there exist polytopes that are not Newton polytopes of any maximal hypersurface in the corresponding toric variety. On the other hand we show that for any polytope Δ there are families of hypersurfaces with the Newton polytopes that are asymptotically maximal when λ tends to infinity. We also show that these results generalize to complete intersections.     

Invariant Theory for Non-Associative Real Two-Dimensional Algebras and Its Applications

The set ${\mathcal A}$ of all non-associative algebra structures on a fixed 2-dimensional real vector space $A$ is naturally a ${\mbox{\rm GL}}(2,{\mbox{\bf R}})$-module. We compute the ring of ${\mbox{\rm SL}}(2,{\mbox{\bf R}})$-invariants in the ring of polynomial functions, ${\mathcal P}$, on ${\mathcal A}$. We use invariant theory to compute the exact number of nonzero idempotents of an arbitrary 2-dimensional real division algebra. We show that the absolute invariants (i.e.,the ${\mbox{\rm GL}}(2, {\mbox{\bf R}})$-invariants in the field of fractions of ${\mathcal P}$) distinguish the isomorphism classes of 2-dimensional non-associative real division algebras. We show that the (open) set $\Omega^+\subset{\mathcal A}$ of all division algebra structures on $A$ has four connected components. A similar result is proved for another class of regular 2-dimensional real algebras (the principal isotopes of the algebra ${\mbox{\bf R}}\oplus{\mbox{\bf R}}$).     

eO-代数簇公理

扩张Ockham代数簇$e{\bf O}$是由所有$(L;\wedge,\vee, f, k,0,1)$所组成的代数类,其中$(L;\wedge,\vee,0,1)$是有界分配格, $f$是$L$上的偶同态, $k$是$L$ 是$L$上的同态且满足条件: $fk=kf$. 在本文中,我们把Urquhart定理推广到$e{\bf O}$-代数类,并特别考虑$e{\bf O}$-代数的子代数类 $e_2{\bf M}$.在子代数类$e_2{\bf M}$中, $f$和$k$满足条件: $f^{2}=id_L$及$k^{2}=id_L$. 我们证明: 在子代数类$e_2{\bf M}$中,有19个非等价公理.同时我们给出其蕴含关系的表达图式.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-Functional Inequalities and Weighted Porous Media Equations

Using measure-capacity inequalities we study new functional inequalities, namely L ^q -Poincaré inequalities

A Continuous <Emphasis Type="Italic">d</Emphasis>-Step Conjecture for Polytopes

The curvature of a polytope, defined as the largest possible total curvature of the associated central path, can be regarded as a continuous analogue of its diameter. We prove an analogue of the result of Klee and Walkup. Namely, we show that if the order of the curvature is less than the dimension d for all polytopes defined by 2d inequalities and for all d, then the order of the curvature is less that the number of inequalities for all polytopes.     

Flag enumerations of matroid base polytopes

In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the cd-index of a polytope can be expressed when a polytope is split by a hyperplane, and apply these to the cd-index of a matroid base polytope of a rank 2 matroid.     

全矩阵代数的抛物子代数上具有可导性的非线性映射

Let F be a field of characteristic 0, Mn（F） the full matrix algebra over F, t the subalgebra of Mn（F） consisting of all upper triangular matrices. Any subalgebra of Mn（F） containing t is called a parabolic subalgebra of Mn（F）. Let P be a parabolic subalgebra of Mn（F）. A map φ on P is said to satisfy derivability if φ（x·y） = φ（x）·y＋x·φ（y） for all x,y ∈ P, where φ is not necessarily linear. Note that a map satisfying derivability on P is not necessarily a derivation on P. In this paper, we prove that a map φ on P satisfies derivability if and only if φ is a sum of an inner derivation and an additive quasi-derivation on P. In particular, any derivation of parabolic subalgebras of Mn（F） is an inner derivation.     

A graph associated with the p\pi-character degrees of a group

Let G be a group and $\pi$ be a set of primes. We consider the set ${\rm cd}^{\pi}(G)$ of character degrees of G that are divisible only by primes in $\pi$. In particular, we define $\Gamma^{\pi}(G)$ to be the graph whose vertex set is the set of primes dividing degrees in ${\rm cd}^{\pi}(G)$. There is an edge between p and q if pq divides a degree $a \in {\rm cd}^{\pi}(G)$. We show that if G is $\pi$-solvable, then $\Gamma^{\pi}(G)$ has at most two connected components.     

Inverse problems in magneto-electroscanning (in encephalography, for magnetic microscopes, etc.)

Contrary to the prevailing opinion about the incorrectness of the inverse MEEG-problem, we prove its unique solvability in the framework of the system of Maxwell''s equations [3]. The solution of this problem is the distribution of ${\bf y} \mapsto {\bf q}({\bf y})$ current dipoles of brain neurons that occupies the region $Y \subset \mathbb{R}^3 $. It is uniquely determined by the non-invasive measurements of the electric and magnetic fields induced by the current dipoles of neurons on the patient"s head. The solution can be represented in the form ${\bf q}={\bf q}_0+{\bf p}_0\delta\Big|_{\partial Y}$, where ${\bf q}_0$ is the usual function defined in $Y,$ and ${\bf p}_0\delta\Big|_{\partial Y} $ is a $\delta$-function on the boundary of the domain $Y$ with a certain density ${\bf p}_0$. It is essential that ${\bf p}_0$ and ${\bf q}_0$ are interrelated. This ensures the correctness of the inverse MEEG-problem. However, the components of the required 3-dimensional distribution $ {\bf q} $ must turn out to be linearly dependent if only the magnetic field ${\bf B}$ is taken into account. This question is considered in detail in a flat model of the situation.     

Isometry-Invariant Valuations on Hyperbolic Space

Hyperbolic area is characterized as the unique continuous isometry-invariant simple valuation on convex polygons in We then show that continuous isometry-invariant simple valuations on polytopes in for are determined uniquely by their values at ideal simplices. The proofs exploit a connection between valuation theory in hyperbolic space and an analogous theory on the Euclidean sphere. These results lead to characterizations of continuous isometry-invariant valuations on convex polytopes and convex bodies in the hyperbolic plane a partial characterization in and a mechanism for deriving many fundamental theorems of hyperbolic integral geometry, including kinematic formulas, containment theorems, and isoperimetric and Bonnesen-type inequalities.     

Schur–Weyl Theory for C*‐algebras

To each irreducible infinite dimensional representation $(\pi ,\mathcal {H})$ of a C*‐algebra $\mathcal {A}$, we associate a collection of irreducible norm‐continuous unitary representations $\pi _{\lambda }^\mathcal {A}$ of its unitary group ${\rm U}(\mathcal {A})$, whose equivalence classes are parameterized by highest weights in the same way as the irreducible bounded unitary representations of the group ${\rm U}_\infty (\mathcal {H}) = {\rm U}(\mathcal {H}) \cap (\mathbf {1} + K(\mathcal {H}))$ are. These are precisely the representations arising in the decomposition of the tensor products $\mathcal {H}^{\otimes n} \otimes (\mathcal {H}^*)^{\otimes m}$ under ${\rm U}(\mathcal {A})$. We show that these representations can be realized by sections of holomorphic line bundles over homogeneous Kähler manifolds on which ${\rm U}(\mathcal {A})$ acts transitively and that the corresponding norm‐closed momentum sets $I_{\pi _\lambda ^\mathcal {A}}^{\bf n} \subseteq {\mathfrak u}(\mathcal {A})^{\prime }$ distinguish inequivalent representations of this type.     

A classification of smooth convex 3-polytopes with at most 16 lattice points

We provide a complete classification up to isomorphism of all smooth convex lattice 3-polytopes with at most 16 lattice points. There exist in total 103 different polytopes meeting these criteria. Of these, 99 are strict Cayley polytopes and the remaining four are obtained as inverse stellar subdivisions of such polytopes. We derive a classification, up to isomorphism, of all smooth embeddings of toric threefolds in ? ^N where N≤15. Again we have in total 103 such embeddings. Of these, 99 are projective bundles embedded in ? ^N and the remaining four are blow-ups of such toric threefolds.     

On convex optimization without convex representation

We consider the convex optimization problem P:min_x {f(x) : x ? K}{{\rm {\bf P}}:{\rm min}_{\rm {\bf x}} \{f({\rm {\bf x}})\,:\,{\rm {\bf x}}\in{\rm {\bf K}}\}} where f is convex continuously differentiable, and K ì \mathbb Rⁿ{{\rm {\bf K}}\subset{\mathbb R}^n} is a compact convex set with representation {x ? \mathbb Rⁿ : g_j(x) 3 0, j = 1,?,m}{\{{\rm {\bf x}}\in{\mathbb R}^n\,:\,g_j({\rm {\bf x}})\geq0, j = 1,\ldots,m\}} for some continuously differentiable functions (g _j ). We discuss the case where the g _j ’s are not all concave (in contrast with convex programming where they all are). In particular, even if the g _j are not concave, we consider the log-barrier function f_m{\phi_\mu} with parameter μ, associated with P, usually defined for concave functions (g _j ). We then show that any limit point of any sequence (x_m) ì K{({\rm {\bf x}}_\mu)\subset{\rm {\bf K}}} of stationary points of f_m, m? 0{\phi_\mu, \mu \to 0} , is a Karush–Kuhn–Tucker point of problem P and a global minimizer of f on K.     

The decidability of some classes of Stone algebras

Iterating the triple construction applied consecutively to n Boolean algebras, we introduce two finitely axiomatizable subclasses SAⁱ_n{{\bf SA}^{\rm i}_n} and SA^s_n{{\bf SA}^{\rm s}_n} of the class SA _n of all Stone algebras of degree n with all the structure homomorphisms in their P-product representation injective or surjective, respectively. Then the class of all Post algebras of degree n is definitionally equivalent to the intersection SAⁱ_n ?SA^s_n{{\bf SA}^{\rm i}_{n} \cap {\bf SA}^{\rm s}_{n}}. We show that for each n ≥ 2 the class SAⁱ_n{{\bf SA}^{\rm i}_n} is hereditarily undecidable while SA^s_n{{\bf SA}^{\rm s}_{n}} is decidable. As a consequence we obtain several (un)decidability results for various axiomatic classes of Stone algebras: among them the decidability of the class of all Stone algebras of degree n which are dually pseudocomplemented and form a dual Stone algebra under the operation of dual pseudocomplement, and undecidability of the class of all Stone algebras with Boolean dense set. On the other hand, the class of all finite members in SA _n is decidable.     

Graph homology of the moduli space of pointed real curves of genus zero

The moduli space parameterizes the isomorphism classes of S-pointed stable real curves of genus zero which are invariant under relabeling by the involution σ. This moduli space is stratified according to the degeneration types of σ-invariant curves. The degeneration types of σ-invariant curves are encoded by their dual trees with additional decorations. We construct a combinatorial graph complex generated by the fundamental classes of strata of . We show that the homology of is isomorphic to the homology of our graph complex. We also give a presentation of the fundamental group of .